Quantum geometry and quiver gauge theories

TCD-MATH-13-17, HMI-13-02 QUANTUM GEOMETRY AND QUIVER GAUGE THEORIES

arXiv:1312.6689v1 [hep-th] 23 Dec 2013

NIKITA NEKRASOV, VASILY PESTUN, AND SAMSON SHATASHVILI Abstract. We study macroscopically two dimensional N = (2, 2) supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product Td × R2ǫ of a d-dimensional torus and a two dimensional cigar with Ω-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories, quantization of the moduli spaces of instantons on R2−d × T2+d and singular monopoles on R2−d × T1+d , for d = 0, 1, 2, and the Yangian Yǫ (gΓ ), quantum affine ell algebra Uaff q (gΓ ), or the quantum elliptic algebra Uq,p (gΓ ) associated to Kac-Moody algebra gΓ for quiver Γ.

Contents 1. Introduction 1.1. Gauge theories 1.1.1. Prepotentials and special geometry 1.1.2. Departure from the Seiberg-Witten theory. 1.1.3. Effective twisted superpotential. 1.1.4. Bethe equations and vacua 1.1.5. Focus of the paper 1.1.6. Chiral ring generating functions 1.2. Quantum groups 1.3. Main result 1.4. Conventions 1.4.1. Quantum parameter 1.4.2. Quantum algebras 1.5. Acknowledgments 2. The partition Z-function of quiver gauge theories 2.1. The Lagrangian data 2.1.1. Six dimensions and anomalies 2.2. The Z-function 2.3. Index computation 2.4. Four dimensions 2.4.1. The Y-operators in four dimensions 2.5. Five dimensions 2.5.1. The Y-operators in five dimensions 2.5.2. Asymptotics of the Y-operators Date: December 23, 2013. 1

3 3 3 4 4 5 5 6 6 7 7 7 8 8 9 9 12 13 15 16 16 16 17 17

2

NIKITA NEKRASOV, VASILY PESTUN, AND SAMSON SHATASHVILI

2.6. Partition function is a sum over partitions 2.6.1. Plethystic exponents and the asymptotics ǫ2 → 0 2.6.2. The bi-fundamental contribution 2.6.3. The fundamental and the anti-fundamental hypermultiplet contributions 2.6.4. The vector multiplet contribution 2.6.5. The topological contributions 2.6.6. Six dimensional case 2.6.7. The gauge and modular anomalies b0 theory in six dimensions 2.6.8. Example: the A 2.6.9. Example: the A1 theory in six dimensions 3. The twisted superpotential W of quiver theories 3.1. The limit shape 3.2. Entropy estimates 3.3. Analysis of the limit shape equations 3.3.1. On-shell versus off-shell 4. Quantum geometry 4.1. The slope estimates 4.2. The convergence of q-characters 4.3. The observables 4.4. Superpotential 4.5. Dual periods 5. Quantum groups 5.1. Introduction 5.1.1. Five dimensional version 5.1.2. Four dimensional version 5.1.3. Six dimensional version 5.1.4. Dimension uniform description 5.1.5. The twist mass for the A-type affine quivers 5.1.6. Quantum toroidal algebras 5.1.7. Other topics 5.2. Quantum group interpretation of the gauge theory results 5.2.1. First interpretation of the gauge theory q-character equations 6. Examples and discussions 6.1. Finite Ar quiver 6.2. A∞ quiver 6.2.1. Details on the A1 quiver 6.2.2. Details on the A3 = D3 and Ar quivers 6.3. From Baxter equation to Y -functions from D-Wronskian b∗ quiver 6.4. A 0 7. Four dimensional limit and the opers. 7.1. The A1 case. 7.2. Other examples 7.2.1. q-characters for the A2 theory b0 case. 7.2.2. Type II* A

17 19 19 20 21 21 22 23 23 24 25 25 27 28 30 32 35 35 35 36 37 37 37 37 38 38 39 39 40 42 42 43 46 46 46 48 49 51 52 55 55 56 56 58

QUANTUM GEOMETRY AND QUIVER GAUGE THEORIES

7.2.3. D and E quivers Appendix A. Quantum affine algebras A.1. Quantum affine algebra A.1.1. Conventions A.1.2. Chevalley-Drinfeld-Jimbo construction A.2. Drinfeld current realization A.2.1. Drinfeld currents A.2.2. Yangian and elliptic version A.3. Level zero representations A.3.1. Level zero representations and generalized weights A.3.2. Level zero fundamental highest weight modules A.4. Loop weights A.5. Universal R-matrix A.6. Dual interpretation A.6.1. Twisted q-characters References

3

59 59 59 59 60 61 61 62 63 63 65 65 67 68 69 69

1. Introduction The Bethe/gauge correspondence between the supersymmetric gauge theories and quantum integrable systems is a subject of research spanning over a decade [1–9] and even longer in the context of topological gauge theories [10–13]. It can also be viewed as a correspondence between the supersymmetric gauge theories and representation theory of infinite dimensional algebras typical for the two dimensional conformal field theories or integrable deformations thereof. This relation, dubbed the BPS/CFT correspondence in [14], following the prior work in [15–21], became a subject of intense development after the seminal work [22] where the instanton partition functions of the S-class N = 2 gauge theories in the Ω-background [19] were conjectured to be the Liouville (and, more generally, the ADE Toda) conformal blocks. 1.1. Gauge theories. Recently [23] the Seiberg-Witten geometry, the curves [24, 25] and the integrable systems Q associated to all N = 2 (affine) ADE quiver gauge theories, whose gauge group Gv = SU(vi ) is a product of unitary groups, were systematically found. The direct application of quantum field theory techniques leads to the equivariant integration over the instanton moduli on R4 [1, 19, 21, 26, 27]. The above mentioned geometry emerges in the flat space limit ǫ1 , ǫ2 → 0 of the Ω-deformed gauge theory [19]. 1.1.1. Prepotentials and special geometry. The prepotential F (a, m; q) of the effective low-energy theory relates to the gauge theory partition Z-function [19] in the flat space limit as follows: F (a, m; q) = − lim ǫ1 ǫ2 log Z(a, m; q; ǫ1 , ǫ2 ) (1.1) ǫ1 ,ǫ2 →0

where q here denotes the set of gauge coupling constants of the theory, m the set of masses of the hypermultiplets fields, and a is the set of the flat special coordinates

4


on the moduli space of vacua M on the Coulomb branch of the theory. The latter is identified with the asymptotics of the scalar fields of N = 2 vector multiplets at infinity of the Euclidean space-time. The geometry of M and F (a, m; q) are captured by a complex analogue of a classical integrable system, an algebraic integrable system. The phase space P of that system admits a partial compactification which is a complex symplectic manifold with the holomorphic symplectic form and projection to M with Lagrangian fibers, which are, generically, abelian varieties. The gauge theories studied in [23] and in this paper are labeled by the classes of representations Γ(v, w, w) ˜ of quivers which correspond to the Dynkin diagrams Γ of simply laced finite-dimensional or affine Kac-Moody algebras gΓ . The dimension vectors v = (vi ), w = (wi ), w ˜ = (w ˜ i ) encode the number of colors, the number of fundamental flavors and the number of anti-fundamental flavors charged with respect to the gauge group factor U(vi ), respectively. For affine Kac-Moody gΓ , let gΓ′ be the underlying finite-dimensional simple Lie algebra. The corresponding phase spaces P turn out to be the moduli spaces of solutions to the BPS equations, GΓ -monopoles (for finite quivers) on Rd × T3−d , or GΓ′ -instantons (for affine quivers) on Rd ×T4−d . The gauge group GΓ (GΓ′ ) for these BPS equations is not the original gauge group Gv of the quiver theory. Yet the solution of the quantum Gv -gauge theory is captured by the moduli space of particular classical solutions to the GΓ - (GΓ′ )-gauge theory. 1.1.2. Departure from the Seiberg-Witten theory. In this paper, we extend the results of [23] in two ways. First of all, we study the effective low-energy theory of the two dimensional N = (2, 2) supersymmetric theory which is obtained from the original four dimensional N = 2 theory by Ω-deformation, affecting two out of four space-time dimensions. Secondly, we focus more on the theories in five dimensions, compactified on a circle. The four dimensional case is obtained by sending the radius of the circle to zero. 1.1.3. Effective twisted superpotential. In terms of the general Ω-backgrounds [19], we send to zero only one equivariant parameter, say ǫ2 = 0, while keeping the other parameter ǫ1 = ǫ finite. Thus we work in the limit discussed in [5] and study the universal part of the effective twisted superpotential W(a, m; q, ǫ) = − lim ǫ2 log Z(a, m; q; ǫ1 = ǫ, ǫ2 ) , ǫ2 →0

(1.2)

identified in the Refs. [4], [5], [6], [7]. Recall that the twisted superpotential is a specific F -term in the Lagrangian of a N = (2, 2) supersymmetric theory in two dimensions. A four dimensional gauge theory can be viewed as a two dimensional theory with an infinite number of degrees of freedom. The two dimensional Ω-deformation of the type we study in this paper breaks the four dimensional N = 2 supersymmetry down to two dimensional N = 2 supersymmetry. However, in order to view the resulting theory as the two dimensional theory with the well-defined effective Lagrangian we need to specify the boundary conditions at infinity. One may view the choice of the boundary conditions at infinity as a choice of a three dimensional supersymmetric theory compactified on a circle,


5

coupled to the original four dimensional theory on the product of a cigar-like twodimensional geometry and the two dimensional Minkowski world-sheet of the resulting theory. The three dimensional theory at infinity is compactified on a circle because of the asymptotics of the cigar-like geometry, which looks like R1 × S1 at infinity. The contribution W ∞ (a, m; ǫ) of the three dimensional theory at infinity of the cigar is purely perturbative and is given by a sum of dilogarithm functions linear(a,m) X W ∞ (a, m; ǫ) ∼ ǫ Li2 e ǫ According to the Refs. [8, 9], the twisted superpotential W eff (a, m; q, ǫ) of the effective two-dimensional theory is the sum: W eff (a, m; q, ǫ) = W(a, m; q, ǫ) + W ∞ (a, m; ǫ)

(1.3)

which can be identified with the Yang-Yang function of some quantum integrable system [4], [5], [6], [7]. The Ω-deformation parameter ǫ plays the rôle of a Planck constant. It is easy to see by comparing the Eqs. (1.2) and (1.1) that in the limit ǫ → 0, the superpotential W(a, m; q, ǫ) behaves as 1 F (a, m; q) ǫ and the quantum integrable system approaches the classical one with the phase space P, and the prepotential F (a, m; q) describing the special geometry of its base M. 1.1.4. Bethe equations and vacua. More precisely, the Bethe equations of that quantum integrable system read as follows: exp

∂W eff (a, m; q, ǫ) = 1, ∂ai

i = 1, . . . , dimM

(1.4)

The Eq. (1.4) describes in some specific Darboux coordinates, e.g. [9], the intersection of two Lagrangian subvarieties, cf. [8], one with the generating function W(a) and another, with the generating function −W ∞ (a). 1.1.5. Focus of the paper. In this paper we explore a particular formalism for the systematic computation of the universal part of the superpotential, W(a, m; q, ǫ). We shall not discuss the choices of boundary conditions and representations of the noncommutative deformations of the algebra of functions on P. Another aspect in which this paper develops the Ref. [23] is its focus on the five dimensional version of the theory. The N = 2 gauge theory in four dimensions can be canonically lifted to the N = 1 theory in five dimensions. The five dimensional N = 1 supersymmetric gauge theory compactified on a circle S1ℓ of circumference ℓ can be viewed [28] as a particular deformation of the four dimensional N = 2 theory. One studies the theory with twisted boundary conditions, such that in going around the circle the remaining four dimensional Euclidean space-time is rotated in the two orthogonal two-planes, by the angles ℓǫ1 and ℓǫ2 , respectively. For q1 = eıℓǫ1 ,

q2 = eıℓǫ2 ,

q = q1 q2

(1.5)

6


we are interested in the limit q2 → 1, q = q1 and for convergence issues we shall assume in this paper

(1.6)

|q|< 1

(1.7)

For the quiver which is a Dynkin graph of (affine) ADE Lie algebra gΓ , we find that the ǫ-deformed limit shape partition profile equations of [21] can be mapped to the Bethe equations of a formal trigonometric XXZ gΓ -spin chain. Despite the similarity to the problems (for finite Ar quivers in four dimensions) studied in [29–31] and also in [2, 3, 5, 32–46], there are important conceptual differences which we shall explain below. In fact, the universal part of the gauge theory twisted superpotential does not correspond to any spin chain. It corresponds to a commutative subalgebra of noncommutative associative algebra, which is the quantum affine algebra Uaff q (gΓ ) associated to gΓ . One can then study the spectrum of this subalgebra in any representation of Uaff q (gΓ ), which would lead to the ordinary Bethe equations. 1.1.6. Chiral ring generating functions. In gauge theory, the functions Yi (x), where i ∈ IΓ runs through the set IΓ of nodes of Γ, encode the expectation values of chiral ring observables. For a point u ∈ M on the Coulomb branch moduli space M and a simple gauge group factor U(vi ) we define the generating function, in the four and five dimensional versions given by 4d : Yi (x − 2ǫ ; u) = exp( hlog det(x − φi )iu ) (1.8)

− 21 5d : Y+ ; u) = exp( log det(1 − eıℓ(φi −x) ) u ), ξ = eıℓx i (ξq

respectively, with φi the complex adjoint scalar in the SU(vi ) gauge vector multiplet, hiu denotes the expectation value at the point u ∈ M of the moduli space of vacua, and ℓ is the circumference of the circle S1ℓ . We give the geometric definition below, cf. Eqs. (2.36), (2.43)). 1.2. Quantum groups. In this note we will be dealing with quantum algebras Uǫ gΓ (Cx ) :

Yǫ (gΓ ),

Uaff q (gΓ ),

Uell q,p (gΓ )

(1.9)

associated to the quiver theories on R4 , on R4 × S1ℓ and on R4 × T2ℓ,−ℓ/τp , respectively. Here ℓ is the circumference of the circle S1ℓ on which we compactify the 5d gauge theory, and (ℓ, −ℓ/τp ) are the periods of the torus T2ℓ,−ℓ/τp on which we compactify the six dimensional gauge theory. The parameters ǫ, or q q = eıℓǫ ,

(1.10)

are the quantization parameters (Planck constant) of quantum algebras (1.9). The algebras (1.9) are defined using second Drinfeld realization [47], i.e. quantum gΓ currents (ψi± (x), e± (x))i∈IΓ , see A.2.1. The domain Cx of the additive spectral parameter x ∈ Cx in quantum gΓ -currents, for the four, five and the six dimensional case, is • a plane Cx = R2 ∼ = C for gΓ -Yangian Yǫ (gΓ ), • a cylinder Cx = R2 /( 2π Z) ∼ = C× for gΓ -quantum affinization Uaff = R1 ×S1 ∼ q (gΓ ), ℓ


7

• a torus Cx = T2τp ∼ )(Z + τp Z) ∼ = R2 /( 2π = C× /pZ = Ep for gΓ -quantum elliptic ℓ 2πıτp algebra Uell is the multiplicative elliptic q,p (gΓ ); where the parameter p = e modulus In the classical limit ǫ = 0, quantum algebras (1.9) reduce to the (universal enveloping of) classical current algebra UgΓ (Cx ) on the spectral domain Cx , and the results of this note reduce to the results of [23]. Namely, in [23], to each point on the Coulomb moduli space M of the gauge theory, one associated a current h(x) ∈ UgΓ (Cx ), defined in terms of the functions Yi (x) and gauge theory data. It is shown in [23] that h(x) satisfies the equation χi (h(x)) = Ti (x), i ∈ IΓ (1.11) where χi is a (twisted) character of the i-th fundamental gΓ -module, and Ti (x) is a polynomial of degree vi . 1.3. Main result. In this paper we show, the result (1.11) of [23] has natural quantum generalization 4: for ǫ 6= 0, the algebra UgΓ (Cx ) is replaced by its quantum version Uǫ gΓ (Cx ). We construct a quantum current h(x) ∈ Uǫ gΓ (Cx ), defined in terms of the gauge theory data and the generating functions Yi (x) such that the character equation (1.11) still holds, but χi are replaced by the characters for quantum algebras known as q-characters [48, 49]. 1.4. Conventions. 1.4.1. Quantum parameter. In this paper, the q-numbers are defined by [n]q =

n

n

1

1

q 2 − q− 2 q 2 − q− 2

(1.12)

Our conventions for the quantum parameter q = eıℓǫ of the quantum groups differ from [49] or [50], so that our q is the q 2 of [49] or [50]. Our conventions for q agree with e.g. [51–54] so that the SU(n) three dimensional Chern-Simons theory at the level k relates to the quantum group Uq (sln ) with parameter 2πı q = exp , h∨sln = n (1.13) h∨ + k and the invariant of the unknot in fundamental representation n is hWn ( )i = [n]q in convention (1.12). The three dimensional Chern-Simons theory has been related to N = 4 SYM on a four-manifold with a boundary in [55], and the parameter q in the normalization (1.13) naturally counts instantons in the N = 4 SYM and enjoys Langlands duality, or modular transformation: for a simply-laced gauge group for q = eıℓǫ = e2πıτǫ the Langlands dual τǫL = −1/τǫ . It would be interesting to explore the modular properties of the ǫ-‘modular’ parameter τǫ =

ℓ ǫ 2π

(1.14)

in the context of this paper, dealing with quantum algebras (1.9), in particular in view of the applications to S-duality [56, 57]. We leave this task for the future.

8


1.4.2. Quantum algebras. There is a variety of conventions for naming the quantum algebras Uǫ (Cx ) (1.9) in the literature. In this note we follow the naming scheme in which the quiver Γ = A1 = • that has one node and no edges, associated to the Lie algebra gΓ = sl2 , corresponds to • 4d: the sl2 Yangian Yǫ (sl2 ); XXX sl2 -spinchain for 4d c • 5d: the sl2 quantum affine algebra Uaff q (sl2 ) ≃ Uq (sl2 ); XXZ sl2 -spinchain ell ∼ • 6d: the sl2 quantum elliptic algebra Uq,p (sl2 ) = Eǫ,τp (sl2 ); XYZ sl2 -spinchain

It is known that the Yangian Yǫ (gΓ ) or the quantum affinization Uaff q (gΓ ) can be defined using second Drinfeld current realization [47], see A.2.1, for any generalized symmetrizable affine Kac-Moody algebra gΓ . In particular, if gΓ is an affine Kac-Moody Lie algebra gΓ = b g where g is a finite dimensional simple Lie algebra, then Uaff q (gΓ ) aff b g) Uaff g) = Uq (b q (gΓ ) = Uq (b

(1.15)

b g is often called in the literature g quantum toroidal algebra because of double loop b and Yǫ (gΓ ) = Yǫ (b g) is called affine Yangian. In this note we will be using consistent naming convention • gΓ -Yangian for Yǫ (gΓ ), • gΓ -quantum affine algebra for Uaff q (gΓ ), • gΓ -quantum elliptic algebra for Uell q,p (gΓ ). br , is associated in our For example, the necklace quiver with r + 1 nodes, gΓ = A [ conventions with gl r+1 quantum affine algebra, or, equivalently, with glr+1 quantum toroidal algebra, see [58]. In terms of Schur-Weyl duality, quantum deformations of g-Weyl groups are called g-Hecke algebras. If g is a finite-dimensional simple Lie group, then • Uq (g) is Schur-Weyl dual to g-Hecke algebra Meanwhile, for the Cx = C× versions of quantum algebras (1.9), considered in this note, associated to gΓ = g or gΓ = b g where g is finite dimensional simple Lie algebra • Uaff q (g) is Schur-Weyl dual to g-affine Hecke algebra • Uaff g) is Schur-Weyl dual to g-double affine Hecke algebra (DAHA), see review q (b [59] See section 5 for more details and literature review on representation theory of quantum algebras. 1.5. Acknowledgments. We are grateful to I. Cherednik, E. Frenkel, V. Kazakov, H. Nakajima, A. Okounkov, F. Smirnov, V. Smirnov, V. Tarasov, Z. Tsuboi and A. Zayakin for discussions. Research of NN was supported in part by RFBR grants 12-02-00594, 12-01-00525, by Agence Nationale de Recherche via the grant ANR 12 BS05 003 02, by Simons Foundation, and by Stony Brook Foundation. VP gratefully acknowledges support from Institute for Advanced Study, NSF grant PHY-1314311, PHYS-1066293, Rogen Dashen Membership, MSERF 14.740.11.0081,


9

NSh 3349.2012.2, RFBR 10-02-00499, 12-01-00482, 13-02-00478 А and the hospitality of the Aspen Center for Physics. The work of SSh is supported in part by the Science Foundation Ireland under the RFP program and by the ESF Research Networking Programme “Low-Dimensional Topology and Geometry with Mathematical Physics (ITGP)”. The results of the paper were reported in 2012 at [60–64] and in 2013 at [65, 66]. We thank the organizers of these conferences and the institutions for the invitations and the participants for the questions and the comments. After we reported on the main results of the present paper we received a preprint [67] which partially overlaps with our section 4. 2. The partition Z-function of quiver gauge theories We start with the class of supersymmetric quiver gauge theories studied in [23], that is the N = 2 supersymmetric four-dimensional gauge theories with a Lagrangian microscopic description, with a gauge group Gv = ×i SU(vi ) and matter hypermultiplets in the fundamental, the bifundamental or the adjoint representations of Gv , such that the theory is asymptotically free or conformal in the ultraviolet. 2.1. The Lagrangian data. The theory is encoded in the following equipped quiver Γ(v, w, w), ˜ cf. [23]: • An oriented graph Γ whose vertices label gauge groups while edges label bifundamental multiplets. • The set of vertices of Γ is denoted by IΓ . For our purposes it is sufficient to consider non-multiple edges, so the set of edges comes as EΓ ⊂ IΓ × IΓ with the two natural projection maps s and t on the first and second factor, respectively, with s(e) called the source of the edge e and t(e) called the target of the edge e. ˜ For the vertex i we • There are three Z-valued functions on IΓ , v, w and w. ˜ i = w(i) ˜ denote vi = v(i) ∈ Z>0 , wi = w(i) ∈ Z≥0 and w ∈ Z≥0 . Then vi is the number of colors in the factor SU(ni ) of the gauge group associated to the i-th vertex, wi and w ˜ i is the number of fundamental and anti-fundamental multiplets charged under the gauge group factor SU(vi ), respectively. Note that in the gauge theory on flat space-time there is no distinction between the fundamental and anti-fundamental hypermultiplets. The distinction comes upon the coupling to the Ω-background. • Each edge e ∈ EΓ describes a bifundamental hypermultiplet transforming in (¯ vs(e) , vt(e) ) (or adjoint hypermultiplet if the edge is a loop with s(e) = t(e)). Let Iij be the incidence matrix of the graph γ with Iij = # s−1 (i) ∩ t−1 (j) + # s−1 (j) ∩ t−1 (i) equal to the number of oriented edges connecting the vertices i and j with either orientation. The Cartan matrix of Γ is aij = 2δij − Iij . The integers (vi , wi , w ˜ i)

10


should satisfy the inequality X

aij vj ≥ wi + w ˜i

(2.1)

j∈IΓ

which implies that the four dimensional theory is sensibly defined in the ultraviolet as a quantum field theory. This implies that aij is the finite or affine ADE Cartan matrix and Γ is the finite or affine ADE Dynkin diagram. If the inequality (2.1) is saturated, we say that RΓ(v,w,w) ˜ is in conformal class X aij vj = wi + w ˜ i ↔ RΓ(v,w,w) (2.2) ˜ is in conformal class j∈IΓ

All affine type ADE quivers Γ saturate the inequality (2.1) at wi = w ˜i = 0 and hence belong to conformal class automatically. The assigned dimensions vi for affine ADE are uniquely determined by a single integer N and are given by vi = Nai P where ai are Dynkin marks on the vertices of Γ; so that the imaginary root δ = i ai αi where αi are simple roots. • The set κ = (κi )i∈IΓ , where κi is the level of the five-dimensional Chern-Simons term for the i-th gauge group factor SU(vi ) Z 1 1 3 1 5 κi trvi (2.3) A ∧ dA ∧ dA + A ∧ dA + A 3 4 5 Each level κi is constrained by the inequality + κ− i ≤ κi ≤ κi

where κ− ˜i + i = −vi + w κ+ i = vi − wi −

X

X

j∈t(s−1 (i)) j∈s(t−1 (i))

(2.4) vj (2.5) vj

The range of allowed Chern-Simons couplings (2.4) is non-empty under the conditions (2.1) when the four dimensional theory with the same quiver is in the asymptotically free or conformal class. • qi ∈ C× , 0 < |qi |< 1 is the exponentiated complexified coupling constant of the SU(vi ) gauge factor for the vertex i ∈ IΓ . The coupling constants q encode the Yang-Mills coupling gYM and the θ-parameter for each gauge group factor: qi = exp(2πıτi ),

τ=

4πı θ + 2 gYM 2π

(2.6)

• ai,a, i ∈ IΓ , a = 1, . . . , vi are the eigenvalues of the scalars in the vector multiplet serving as the special coordinates on the classical Coulomb moduli space ˜ fi,f ) are the masses of the fundamental (anti-fundamental) matter multi• mfi,f (m plets for the vertex i with f = 1, . . . , wi (f = 1, . . . w ˜ i) bf • me is the mass of the bifundamental matter multiplet at edge e ∈ EΓ


11

We will also use exponentiated notations for the order parameters of the theory wi,a = eıℓai,a q1 = eıℓǫ1 ,

q2 = eıℓǫ2

(2.7)

bf

µe = eıℓme , f

µfi,f = eıℓmi,f ,

f

˜ i,f µ ˜fi,f = eıℓm

Let RΓ(v,w,w) ˜ that is the representation of ˜ be the representation of quiver Γ(v, w, w), the gauge group Gv = ×i SU(vi ) (2.8) for the matter hypermultiplets in the theory M M M ˜ i ), ¯ (¯ vi ⊗ w (2.9) ( w ⊗ v ) ⊕ (¯ v ⊗ v ) ⊕ RΓ(v,w,w) = i i s(e) t(e) ˜ e∈EΓ

i∈IΓ

i∈IΓ

where slightly abusing notations by vi we denote complex vector space of dimension vi , etc. It may be useful to think about wi fundamental matter multiplets for SU(vi ) as a bifundamental hypermultiplet with an auxiliary frozen gauge group SU(wi ), and the role of masses mfi,f is played by the frozen scalars of the SU(wi ) vector multiplet. Equivalently, M M M ˜ i) Hom(vi , w (2.10) Hom(wi , vi ) ⊕ RΓ(v,w,w) Hom(vs(e) , vt(e) ) ⊕ ˜ = e∈EΓ

i∈IΓ

i∈IΓ

The hypermultiplet space RΓ(v,w,w) ˜ is naturally acted by flavor symmetry group for the fundamental fields Gw,w˜ = ×i U(wi ) × ×i U(w ˜ i) (2.11) and bifundamentals Gedge = ×e∈EΓ U(1) (2.12) The total flavor symmetry group is Gf = Gedge × Gw,w˜

(2.13)

The theories corresponding to two equipped quivers differing by the orientation of some arrows (or the choice of the fundamental/anti-fundamental multiplets) are equivalent after certain adjustments of the couplings κi and qi . The rationale for the equation (2.1) is to require the non-positive beta-function βi ≤ 0 for the four dimensional running of the i-th gauge coupling constant: X d Λuv τi = βi = wi + w ˜i − aij vj . dΛuv j

The actual reason for this requirement is not so much that the gauge theory is welldefined in the ultraviolet, since we are going to study the five dimensional theory whose definition requires some completion at high energies. The class of theories which can be sensibly completed at high energy by embedding in some brane configuration in string theory or by geometrical engineering in M-theory or by compactification of the

12


(0, 2)-theory from six dimensions seems to be much larger. However, there are reasons to suspect that once we go beyond the realm of the theories which are sensible quantum field theories in four dimensions, the microscopic physics of M-theory will not decouple. For example, the instanton partition function defined entirely in terms of the gauge theory degrees of freedom will diverge, or will cease to be gauge invariant. 2.1.1. Six dimensions and anomalies. Formally one can define and compute the partition Z-function even for the six dimensional theory compactified on the two-torus T2ℓ,−ℓ/τp , as the equivariant elliptic genus of the four dimensional Gv instanton moduli spaces [68], [69]. The Coulomb parameters a of the six dimensional theory compactified on T2ℓ,−ℓ/τp parametrize holomorphic Gv C -bundles on T2ℓ,−ℓ/τp . The latter is a quotient of the abelian variety by the action of finite group (the Weyl group of Gv ). One has to check that the partition function Z is invariant with respect to large gauge transformations on T2ℓ,−ℓ/τp . Indeed, we find that if βi = 0 then Z is invariant, provided the coupling constants qi of SU(vi ) transform in a suitable way under such large gauge transformations. This has the following explanation. The six dimensional N = (1, 0) theory for affine ADE quiver Γ with gauge group Gv = ×SU(vi ) where vi = Nai and ai are Dynkin marks can be realized on the stack of N D5 branes in IIB on C2 /Γ, where Γ ⊂ SU(2) is the finite subgroup associated to affine ADE quiver Γ by McKay correspondence. In addition to the vector multiplets of the gauge group Gv = ×SU(vi ) one has r tensor multiplets coming from the reduction of the IIB potentials on the r cycles generating H2 (C2 /Γ). Now, the theory is not anomalous (see p.11 of [70] in the A1 case and p.8 of [71] for the more general case) if the scalar fields of the tensor multiplets are coupled suitably to the gauge fields. Because of this coupling, indeed, we need to compensate the large gauge transformation, acting by the shift of a by the change of qi when we compute the Z-function of the six dimensional theory on T2ℓ,−ℓ/τp . We recall the computation from [72] adopting to our notations. Consider N = 1 theory in six dimensions with the gauge group Gv and the hypermultiplet in the quiver representation RΓ(v,w,w) ˜ . The quartic anomaly cancellation condition from [73] requires 4

4

tradj F − trR F =

r˜ X

˜

(αi j tr Fj )2

(2.14)

˜i=1

where F is Gv curvature, index j labels the simple factors in the gauge group Gv and r˜ is equal to the number of tensor multiplet that we need to add to cancel the anomaly in the vector and hypermultiplet. A useful identity for SU(vi ) gauge groups is tradj Fi4 = 2vi tr Fi4 + 6(tr Fi2 )2

(2.15)

Let xi,α be Chern roots of Fi . Each (i, j) bifundamental hypermultiplet contributes −

X α,β

(xiα − xjβ )4 = −(vj tr Fi4 + 6 tr Fi2 tr Fj2 + vi tr Fj4 )

(2.16)


13

For a quiver Γ with Cartan matrix aij we find that vector multiplet and all bifundamental together contribute X aij (vi tr Fj4 + 3 tr Fi2 tr Fj2 ) (2.17) ij

Adding up the contributions from all the fundamental and the anti-fundamental multiplets we find, for the total anomaly X X aij tr Fi2 tr Fj2 (2.18) (vi aij − wj − w ˜ j ) tr Fj4 + 3 i,j

i,j

4 For RΓ(v,w,w) ˜ in the conformal class (2.2) the dangerous tr F term vanishes. To cancel the remaininganomaly by the r˜ tensor multiplets we want to present it as a sum of r˜ 2 P˜ ˜ ˜ squares ˜ri=1 αi j tr Fj2 with some real coefficients αi j , so that the two-forms B˜i of the tensor multiplets interact with the gauge fields by means of the coupling ˜

αi j B˜i ∧ tr Fj2 shifting the gauge coupling τj → τj +

X

˜ αi j

(2.19)

Z

T2

˜i

B˜i

(2.20)

˜

In other words, we need to find r˜ one-forms αi j , ˜i = 1, . . . , r˜ such that aij =

r˜ X

˜

˜

αi i αi j

(2.21)

˜i=1

The equation exactly coincides with the definition of the Cartan matrix aij of quiver P ˜i Γ in terms of simple roots αj = ˜i α j e˜i expressed in some orthonormal basis e˜i . Therefore, r˜ = r where r denotes the rank of Cartan matrix aij . Recall that the Γ of finite type has r = #IΓ nodes while the Γ of affine type has r = #IΓ − 1 nodes. ˜ ˜ If Γ is of affine type, then αji for ˜i = 1, . . . , r are the coefficients of simple roots αi j in the expansion in the basis (e˜i )˜i=0,...,r , where e˜0 = δ is the imaginary root of the affine system [74], and (e˜i )|˜i=1,...,r is the orthonormal basis for the root system for the underlying quiver Γ′ of finite type obtained by removing the affine node ’0’ from Γ. We conclude that the lift to the six dimensional N = (1, 0) theories of all of the quiver theories RΓ(v,w,w) ˜ of the conformal class (2.2) is not prohibited by the quartic anomaly [73]. 2.2. The Z-function. To the quiver representation RΓ(v,w,w) ˜ , the parameters q, m, a and two additional parameters ǫ = (ǫ1 , ǫ2 ) introduced in [1, 26, 27] we associate the partition function Z as the formal twisted Witten index of the five dimensional gauge theory [19, 21]: ˜ Z(t) = TrH (−1)F eıℓH t, t∈T (2.22) It is the partition function of the five dimensional theory on the five-dimensional man1 4 ×S1 ifold which is the twisted bundle R^ hǫ1 ,ǫ2 ;ℓi mentioned earlier, over the circle Shℓi of

14


circumference ℓ, with the fibers being R4 . The boundary conditions involve the twists by flavor symmetry, the R-symmetry, the asymptotic gauge transformation, and the rotation of R4 . The couplings qi are actually the expectation values of the complexified Wilson loop of the five dimensional vector multiplet which couples to the conserved topological current Ji = ⋆trvi F ∧ F Therefore the couplings qi and the couplings µe , µi,f are of somewhat similar nature. In fact, in some theories (and most likely in all theories studied in our paper) the global symmetry of the theory extends to include the transformations generated by ˜ is the maximal torus of the group Geq × Gtop = the topological charges. The torus T Gv × Gf × GL × Gtop , defined below. The group Gtop is generated by the topological charges. For the theories in this paper the it is U(1)IΓ . The factor Gv is the group of global constant gauge transformations (or changing of the framing at infinity of the framed gauge bundle used to describe instantons on R4 ), the Gf is the flavor symmetry group acting on hypermultiplets, the GL = SO(4) is the R4 rotation group. The fourdimensional instantons are lifted to particles in five dimensions. The partition function Z factorizes as the product of the tree level, the perturbative and the instanton factors: Z = Z tree Z pert Z inst .

(2.23)

The perturbative part is ambiguous, as we mentioned above, it depends on the choice of boundary conditions. The instanton part is unambiguous and can be defined mathematically precisely as the Geq -character of the graded vector space Hkinst X (t) (2.24) Z inst (q; t) = qk strHinst k k

evaluated on the group element t of the maximal torus T of the equivariant group Geq = Gv × Gf × GL . The group element t ∈ T is parametrized by (a, m, ǫ). The Coulomb parameters a = (ai,a ) are coordinates in the Cartan algebra of global constant gauge transformations acting at the framing of the gauge bundle at infinity Y SU(vi ) (2.25) Gv = i∈IΓ

f ˜ fi,f ) are the coordinates on the Cartan subalgebra the mass parameters m = (mbf e , mi,f , m of the flavor symmetry group (2.13)

Gf = ×e∈EΓ U(1) ×i∈IΓ U(wi ) ×i∈IΓ U(w ˜ i)

(2.26)

and the epsilon parameters ǫ = (ǫ1 , ǫ2 ) are the coordinates on the Cartan subalgebra of the four dimensional rotation group GL = SO(4)

(2.27)

acting on R4 by rotations about a particular point 0. The set of integers k ∈ ZI+Γ = {ki ∈ Z≥0 | i ∈ IΓ } encodes the four dimensional instanton charges (the second Chern classes c2 ) for the gauge group factors SU(vi ) for i ∈ IΓ . More precisely, R Hkinst = H • (Minst k ,E )

(2.28)


15

˜ is the sheaf cohomology of the virtual matter bundle EbRΓ(v,w,w) → Minst defined as k inst follows. The base space Mk is the framed Gv -instanton moduli space on R4 . Let AGv be the space of connections on a principal Gv -bundle over R4ǫ1 ,ǫ2 with fixed framing at infinity and fixed second Chern class k. Then

Minst = {A ∈ AGv | FA+ = 0}/G∞ k

(2.29)

where G∞ denotes the group of framed gauge transformations, i.e. trivial at ∞ = S4 \R4ǫ1 ,ǫ2 . The fiber of the virtual matter bundle EAR at the point A ∈ Minst is the k virtual space of the zero modes of Dirac operator in representation RΓ(v,w,w) in the ˜ 4 background of the instanton connection A in the gauge bundle on S , that is /R /R EbAR = ker D A − coker D A

(2.30)

The equivariant partition function Z(q, a, m; ǫ) is analytic in the equivariant parameters (a, m; ǫ) which henceforth are treated as complex variables. 2.3. Index computation. In this section we give only the brief review of the gauge theory partition Z-function, mostly in order to set the notations. We shall skip the arguments of the partition function Z. For more details one can consult [1, 19, 21, 26, 27, 75–79]. The T-equivariant path integral of the supersymmetric Ω-deformed gauge S theory on R4 localizes to the TIΓ equivariant integral over the instanton moduli space k Minst k , where k = (ki )i∈IΓ ∈ Z+ is the second Chern class of the anti-self-dual Gv -connection on R4 and Y Mvi ,ki (2.31) Minst = k i∈IΓ

Then, using the Atiyah-Bott theorem one reduces the integral over Minst to the sum k over the set of fixed points of the locally defined expressions. The fixed point formula is similar to the calculation of an integral of a meromorphic form via residues. To apply Atiyah-Bott localization we need to partly compactify the moduli space of instantons to the Gieseker-Nakajima moduli space Minst in which vector bundles are replaced by k the torsion free sheaves, or, in differential-geometric language, by the noncommutative instantons [80]. Thus, in particular, Mvi ,ki is the moduli space of framed torsion free rank vi sheaves Ei on CP2 , trivialized over a complex line CP1∞ at infinity, with the second Chern characters ch2 (Ei ) = ki . Let E i denote the universal torsion free sheaf over Mv,k × CP2 in the i-th factor, and let π : Mv,k × CP2 → Mv,k be the projection. For a quiver representation RΓ(v,w,w) ˜ the matter sheaf cf. (2.10)) over Mv,k × CP2 is M M M ˜ ˜ i) Hom(Es(e) , Et(e) ) ⊕ Hom(wi , Ei ) ⊕ Hom(Ei , w (2.32) = E RΓ(v,w,w) e∈EΓ

Let

i∈IΓ

i∈IΓ

˜ ˜ = Rπ∗ E RΓ(v,w,w) EbRΓ(v,w,w) be the derived pushforward of the matter sheaf to Mv,k , which means that the vir˜ | tual fiber EbRΓ(v,w,w) m at a point m ∈ Mv,k is the graded space of sheaf cohomology RΓ(v,w,w) • 2 ˜ H (CP , E |m ).

16


2.4. Four dimensions. The instanton part of the four dimensional gauge theory partition function is Z ∞ X k inst ˜ q eT (EbRΓ(v,w,w) ) (2.33) 4d : Z = Mv,k

k=0

R

where equivariant integration Mv,k denotes the pushforward of the T-cohomologies of Mv,k by projection map from Mv,k to a point, eT is the T-equivariant Euler class, and Y k qk = (2.34) qi i i∈IΓ

2.4.1. The Y-operators in four dimensions. The geometric definition of generating functions Yi (x) (1.8) is conveniently given in terms of the spectral Chern class c(x, E). For xI the virtual Chern roots of a vector bundle E, define the rational x-spectral Chern class c(x, E) as Y c(x, E) = (x − xI ) (2.35) I

so that c(1, E) = c(E ∨ ) is the ordinary total Chern class. Then P∞ k R ˜ ) c (x, E˚i )eT (EbRΓ(v,w,w) k=0 q Mv,k T ǫ 4d : Yi (x − 2 ) = P∞ k R ˜ ) e (EbRΓ(v,w,w) k=0 q Mv,k T

(2.36)

and E˚i is the restriction of the sheaf Ei at the T-fixed point 0 ∈ C2 . The restriction is defined for sheaves using the Koszul resolution complex and derived pushforward map (2.37) E˚i = Rπ∗ Ei ⊗ (Λ2 T ∗2 → T ∗2 → O) C

C

2.5. Five dimensions. The five dimensional, K-theoretic, version of the partition function is X ˜ 5d : Z inst = qk chT HT• (Mv,k , I(EbRΓ(v,w,w) )) (2.38) k

where I is the index functor defined on sheaves by ∞ X (−1)l Λl E I:E →

(2.39)

l=0

which satisfies

I [E1 ⊕ E2 ] = I [E1 ] I [E2 ]

(2.40)

and

−1 I [E ∨ ] = I [E]∨ = (−1)rkE ΛrkE E ⊗ I [E] At the level of Chern characters, with xI the virtual Chern roots, " # " # X Y I exI = (1 − exI ) I

(2.41)

(2.42)

I

Remember, that in our theories we have the mass parameters, i.e. the equivariant parameters for the symmetries, which rescale the fibers of various summands in the matter sheaf (2.32). Thus the sum in (2.42) is more refined then it may appear.


17

2.5.1. The Y-operators in five dimensions. The five-dimensional, K-theoretic geometric definition of the Yi (ξ) function is P∞ k R ˜ ) tdT (T Mv,k )b cT (ξ, E˚i )b cT (1, EbRΓ(v,w,w) 1 k=0 q Mv,k −2 (2.43) Yi (ξq ) = P∞ k R bRΓ(v,w,w) ˜ ) q td (T M )b c (1, E T v,k T k=0 Mv,k

where the K-theoretic trigonometric ξ-spectral characteristic class b c(ξ, E) is defined in terms of the T-equivariant Chern roots xI of the sheaf E by Y b cT (ξ, E) = (1 − ξI /ξ) ξ = eıℓx , ξI = eıℓxI (2.44) I

where ℓ is the circumference of the 5d compactification circle S1ℓ . The class b cT (ξ, E) has the following transformation property: b cT (ξ, E ∨) = e−c1 (E) (−ξ)−rkE b cT (ξ −1 , E)

(2.45)

2.5.2. Asymptotics of the Y-operators. Using the definitions above, and the fact that rk Ei = vi , we get: 4d : Yi (x) → xvi , x→∞ ( (2.46) Yi (ξ) → 1, ξ→∞ Qvi 5d : − 21 −vi Yi (ξ) → ξ ), ξ→0 a=1 (−wi,a q In the limit q = 0 the functions Yi (x), Yi (ξ) are (Laurent) polynomials vi Y ǫ (x − ai,a ) 4d : lim Yi (x − 2 ) = q→0

5d :

lim Yi (ξq

a=1

− 21

q→0

)=

vi Y

(2.47)

(1 − wi,a /ξ)

a=1

1

The shifts x 7→ x − 2ǫ , ξ 7→ ξq − 2 are introduced here for a later convenience of relating Yi to the representation theory of quantum groups. 2.6. Partition function is a sum over partitions. The equivariant integration over Mv,k reduces to the sum over the space MTv,k of T-fixed points by Atiyah-Bott localization X X Z inst = Zλ (2.48) k

Note that Zλ in (2.48) contains the factor qk =

λ∈MT v,k

Y

qki i

i∈IΓ

which helps to make the instanton sum a convergent series, at least for |qi |≪ 1. The T-fixed points λ are the sheaves which split as the sum of the T-equivariant rank one torsion free sheaves. The latter are the finite codimension monomial ideals I in the ring of polynomials in two variables C[z1 , z2 ]. The latter are in one-to-one correspondence with partitions of the size codim I . The use of torsion free sheaves in lieu of the honest

18


bundles reduces the rotation symmetry group from SO(4) to U(2), since it requires a choice of complex structure on R4 ≈ C2 . Thus the symmetry group for the Mvi ,ki moduli space is U(vi ) × U(2). The maximal torus T is unchanged by this reduction. The set of fixed points MTv,k is a two indexed set of partitions labeled by the quiver node i ∈ IΓ and by the color a = 1, . . . , vi of the SU(vi )-factor in Gv : MTv,k = {λi,a | i ∈ IΓ , a = 1, . . . , vi },

(2.49)

where each partition λi,a is a non-increasing sequence of non-negative integers λi,a = (λi,a,k )k∈N = (λi,a,1 ≥ λi,a,2 ≥ . . . ≥ 0 = 0 = . . .)

(2.50)

whose size is defined as: |λi,a |=

∞ X

λi,a,k

k=1

Let us denote by λi = and

i (λi,a )va=1

the colored partition corresponding to U(vi ) (cf. [21]), |λi |=

vi X

|λi,a |

a=1

The fixed point λ ∈ Mv,k corresponding to λ = (λi )i∈IΓ (we thus use the same letter λ to denote both the fixed point in the moduli space of sheaves and the quiver-colored partition it corresponds to) is the collection (Ei,λi )i∈IΓ of the sheaves which are the direct sums of the monomial ideals corresponding to λi,a . The topology of the sheaf determines the sizes of the colored partitions: |λi |= ki

(2.51)

Let Ei,λi be the U(vi ) × U(2) equivariant Chern character of Ei,λi restricted at 0 ∈ C2 . Let (ai,a ), ǫ = (ǫ1 , ǫ2 ) be coordinates in the Cartan algebra of SU(vi ) × SO(4). From the ADHM [81] construction one derives (cf. [75]) Ei,λi = Wi − (1 − q1 )(1 − q2 )Vi,λi

(2.52)

where q1 , q2 are defined in (2.7), Wi =

vi X

eıℓai,a

a=1

Vi,λi =

vi X X

(2.53) ıℓci,a,s

e

a=1 s∈λi,a

where s ∈ λi,a labels boxes in the partition λi,a so that s = (s1 , s2 ) is a pair of integers with s1 = 1, 2, . . . , and s2 = 1, . . . , λi,a,s1 , and ci,a,s is the ai,a -shifted content [21] of the box s ci,a,s = ai,a + (s1 − 1)ǫ1 + (s2 − 1)ǫ2

(2.54)


19

The contribution of the fixed point λ to the partition function is a product of factors labeled by the vector and hypermultiplets ! ! Y Y bf af cs top vec f (2.55) Ze,λ Zi,λ Zi,λ Zi,λ Zi,λZi,λ Zλ = e∈EΓ

i∈IΓ

2.6.1. Plethystic exponents and the asymptotics ǫ2 → 0. The contributions to the five dimensional index Zλ are found by applying the plethystic exponent (index functor) I to the corresponding characters χλ . Applying I to χ is equivalent to using AtiyahSinger formula to compute the T-equivariant index of the Dolbeault operator ∂¯ for the complex of (0, •)-forms on (virtual) vector space which is T-module with Chern character χ. First, we introduce the q-analogue of the log Γ function Y ξ γq (ξ) = log I = log (1 − ξq n ) (2.56) 1−q n≥0 To compute Z in the limit ǫ2 → 0 we will need the limit of γq2 (x) function for q2 = eıℓǫ2 → 1. We find ∞ X 1 1 ξ m ǫ2 →0 Li2 (ξ) (2.57) ≈ exp − γq2 (ξ) = m m 1 − q ıℓǫ 2 2 m=1 We now proceed with the definitions of each factor in (2.55).

2.6.2. The bi-fundamental contribution. First, we describe explicitly the bi-fundamental bf factor Ze,λ . Let / e,λ χe,λ = ind D (2.58) be the index of the Dirac operator on R4 in the bifundamental representation e = ¯ s(e) ) evaluated in the background of the connection associated to the fixed (vt(e) ), v point λ ∈ Minst k . From the equivariant Atiyah-Singer theorem we find the character 1

χe,λ =

−µ−1 e

(q1 q2 ) 2 ∨ Et(e),λt(e) Es(e),λ s(e) (1 − q1 )(1 − q2 )

(2.59)

which contains the overall factor µ−1 (2.7) accounting for the bi-fundamental mass. e It is the U(1)e flavor symmetry group element of the bi-fundamental hypermultiplet corresponding to the edge e. By E ∨ we denote the equivariant Chern character of the dual bundle E∨ in which all weights are the negatives of the weights in E. The character Ei,λ of the universal bundle Ei,λ can be represented by the infinite sum Ei,λ =

∞ X ∞ X

(1 − q1 )eıℓxi,a,k

(2.60)

a=1 k=1

where xi,a,k

xi,a,k = ai,a + ǫ1 (k − 1) + ǫ2 λi,a,k

(2.61)

20


denotes the ai,a -shifted content of the box s = (s1 , s2 ) = (k, λi,a,k + 1) (which is right outside the Young diagram of the partition λi,a ). The profile of the partition λi,a can be parametrized by the infinite sequence ξi,a = (ξi,a,1, ξi,a,2, . . .) ξi,a,k = eıℓxi,a,k

(2.62)

Ξi = (ξi,a,k )a=1,...,vi ;k=1,2,...

(2.63)

Let denote the set of all ξi,a,k with fixed i ∈ IΓ and 1 ≤ a ≤ vi , k ∈ N. Then 1/2

χe,λ =

q1

1/2 q2

X

−1/2

− q1 −

−1/2 q2 (ξt ,ξs )∈Ξt(e) ×Ξs(e)

ξt(e) µe ξs(e)

(2.64)

bf The contribution Ze,λ of the bi-fundamental hypermultiplets, computed from the Chern character χe,λ (2.64), asymptotes to   X ξt  1 bf L Ze,λ ∼ exp − (2.65) ıℓǫ2 µe ξ s (ξt ,ξs )∈Ξt(e) ×Ξs(e)

where

1

−1

L(ξ) = Li2 (q12 ξ) − Li2 (q1 2 ξ) (2.66) Actually, both (2.64) and (2.65) are formal expressions at this stage, as the convergence of the series for Ei,λi requires |q1 |< 1. Of course, the original formula (2.59) makes sense, as it operates with finite expressions, so that the correct formula is χe,λ =

1 (q1 q2 ) 2 ∨ + Wt(e),λt(e) Ws(e),λ s(e) (1 − q1 )(1 − q2 ) 1 −1 ∨ − 21 ∨ 2 µe (q1 q2 ) Wt(e),λt(e) Vs(e),λs(e) + (q1 q2 ) Vt(e),λt(e) Ws(e),λs(e) −

−µ−1 e

1

−1

1

−1

∨ 2 2 2 2 − µ−1 e (q1 − q1 )(q2 − q2 )Vt(e),λt(e) Vs(e),λs(e) (2.67)

One can actually make sense out of (2.64) and (2.65) using a prescription where one sums over ξs first, viewing it as a telescopic sum. The telescopic sum is actually finite. The remaining sum over ξt is convergent. One should use a similar prescription for the vec evaluation of Zi,λ below. One can actually avoid using the formal expressions whatsoever, by working with the finite formulae like (2.67). Below we write the formal expressions for the sake of brevity. 2.6.3. The fundamental and the anti-fundamental hypermultiplet contributions. The f contribution of the fundamentals Zi,λ is obtained by applying the I-functor to the simpler version of the character (2.59) χfi

=−

wi X f=1

1

(q1 q2 ) 2 Ei µ−1 (1 − q1 )(1 − q2 ) i,f

(2.68)


Thus, in the ǫ2 → 0 limit,



f Zi,λ ∼ exp −

1 ıℓǫ2

X

(ξt ,ξs )∈Ξi ×{µi,f }

and, similarly, the anti-fundamental contribution is  X 1 af Zi,λ = exp − ıℓǫ2

(ξt ,ξs )∈{˜ µi,f }×Ξi



L(ξt /ξs ) 

L(ξt /ξs )

21

(2.69)

(2.70)

2.6.4. The vector multiplet contribution. To compute the vector multiplet contribution vec Zi,λ we need to find the Geq -character of the tangent space Tλ Minst to the moduli k inst space at the fixed point λ ∈ Mk . From the deformation theory, the character of the tangent space is dual to the index of Dirac operator in the adjoint representation twisted by the square root of the canonical bundle. Thus, q1 q2 χvec = Ei E ∨ (2.71) i (1 − q1 )(1 − q2 ) i Therefore, in the ǫ2 → 0 limit,



vec Zi,λ = exp 

1 ıℓǫ2

X

(ξt ,ξs )∈Ξi ×Ξi

1 2



L(q1 ξt /ξs )

(2.72)

top 2.6.5. The topological contributions. The factor Zi,λ accounts for the topological instanton charge given by the second Chern class ki for the i-th gauge group factor U(vi ) ! ∞ vi 2πıτi X X ξi,a,k top ki Zi,λ = qi = exp log (2.73) ˚ ıℓǫ2 k=1 a=1 ξi,a,k

where

˚ Ξi = {˚ ξi,a,k = wi,aq1k−1 }

(2.74)

corresponds to the empty partition (λi,a,k ) = (0, 0, . . . , ). The second Chern class of Ei,λi is the size |λi | of the colored partition ki =

vi X ∞ X

λi,a,k

(2.75)

a=1 k=1

cs Zi,λ

Finally, the factor accounts for the contribution from the Chern-Simons coupling. It is the descendant of the cubic term in the tree level prepotential Z ıℓκi tr Φ3i (2.76) Lcs,i = −κi d5 xd4 ϑ tr Φ3i ∼ 3! ǫ1 ǫ2

The fixed point contribution of the cubic term 16 tr Φ3i is equal to the third component of the Chern character of the universal bundle, restricted at the fixed point λ × 0 ∈

22


2 Minst k ×C :

1 tr Φ3i 3!

=



X a3i,a  a

λ×{0}

6

X ǫ1 ǫ2 (ǫ1 + ǫ2 ) cs  |λi,a |−ǫ1 ǫ2 2 s∈λ

−

i,a

which gives, in terms of the ξi,a,k variables cs Zi,λ



κi X log(ξ/˚ ξ) = exp 2ǫ2 ξ∈Ξ

log(ξ ˚ ξ) + ǫ1 + ǫ2 ıℓ

i

!

(2.77)

(2.78)

2.6.6. Six dimensional case. For a six dimensional theory compactified on the torus Tp = C/(ℓZ ⊕ −ℓ/τp Z)

(2.79)

with the metric ds2p =

ℓ2 (Imτp dzd¯ z) , |τp |2

z∼z+

ℓ (n1 + n2 τp ) τp

and the nome p = e2πıτp , the elliptic analogue of the index functor (2.42) is " # X Y Ip exI = θ1 (exI ; p) I

where

θ1 (ξ; p) = −ı θ(ξ; p) =

X

X

(2.80)

(2.81)

I

r2

1

1

1

(−1)r− 2 ξ r p 2 = ıp 8 ξ − 2 θ(ξ; p)

r∈Z+ 21 n n

(−1) ξ p

1 n(n−1) 2

(2.82) −1

= (ξ, p)∞ (p, p)∞ (pξ , p)∞

n∈Z

In terms of the additive arguments ξ = eıℓx ,

p = e2πıτp

(2.83) (2.84)

θ1 (ξ; p) = ϑ1 (x; τp ) Under the shifts

2π (n1 + n2 τp ), ℓ the function ϑ1 (x; τp ) transforms as

n1 , n2 ∈ Z

(2.85)

ϑ1 (x; τp ) 7→ e−(πıτp n2 +πı(n1 +n2 )+ıℓn2 x) ϑ1 (x; τp )

(2.86)

x 7→ x˜ = x +

2

For future use, let us rewrite the multiplier for the ϑ1 transformation under the shifts by (n1 , n2 ) = (n, 0) in the form: ıℓ

T1n : e−ıπn = e− 2 (˜x−x)

(2.87)

under the shifts by (n1 , n2 ) = (0, n) in the form: T2n : e−(πıτp n

2 +ıℓnx+ıπn)

2

ıℓ − 4πτ

=e

p

(x˜2 −x2 )− 2τıℓp (˜x−x)

(2.88)


Under the modular transformations of the parameter τp : 1 ıℓ2 x2 1 3 x 1 ϑ1 (x, τp + 1) = e 4 πı ϑ1 (x, τp ) ϑ1 ;− = e− 4 πı τp2 e 4πτp ϑ1 (x; τp ) τp τp

23

(2.89)

2.6.7. The gauge and modular anomalies. Define " # X X eıℓwI := wId I

I

d

The transformation of Ip [χ] under the large gauge transformations, which act on the Coulomb moduli ai,a by the shifts: ai,a 7→ a˜i,a = ai,a +

2π (ni,a + mi,a τp ), ℓ

ni,a , mi,a ∈ Z

(2.90)

is given by Ip [χ] 7→ Ip [χ], ˜ cf. Eqs. (2.87), (2.88) ıℓ ([χ] ˜ 1 − [χ]1 ) 2 ıℓ ıℓ2 ([χ] ˜ 2 − [χ]2 ) − ([χ] ˜ 1 − [χ]1 ) T2n : Ip [χ] ˜ − Ip [χ] = − 4πτp 2τp T1n : Ip [χ] ˜ − Ip [χ] = −

(2.91)

Proposition. The six dimensional theory compactified on a torus is gauge invariant, if the right hand side of (2.91) vanishes for all (ni,a , mi,a ), or if the right hand side of (2.91) can be compensated by some consistent transformation of the coupling constants, i.e. qi ’s. For that to be the case, the right hand side of (2.91) must be proportional to ki and should not depend on the individual colored partitions λi . The transformation of Ip [χ] under τp → −1/τp , i.e. the modular anomaly, is proportional to the exponent of [χ]2 . Proposition. For the theory compactified on the torus to make sense on its own, the partition function must be modular invariant, perhaps at the expense of making the gauge couplings qi transform under the modular group. The latter means that [χ]2 can be linear in ki , but should not depend on any moduli such as ai,a . b0 theory in six dimensions. For the A b0 = (N = 2∗ ) theory the 2.6.8. Example: the A set of vertices consists of one element, so we skip the index i in what follows. We have q1 q2 χvec = EE ∨ (2.92) (1 − q1 )(1 − q2 ) and 1

adj

χ

(q1 q2 ) 2 µ−1 =− EE ∨ (1 − q1 )(1 − q2 )

(2.93)

Then χvec,inst = −W V ∨ − q1 q2 W ∨ V + (1 − q1 )(1 − q2 )V V ∨

(2.94)

24


b0 theory we find, in the k-instanton sector For the A i h X adj (aa − cb,s )2 + (2ǫ+ + cb,s − aa )2 + = − χvec + χ λ λ 2

+

X

a,b,s∈λb

a,b,s∈λb

(aa − cb,s − ǫ+ − m)2 + (ǫ+ + cb,s − aa − m)2 =

= 2kv(m + ǫ+ )(m − ǫ+ ) (2.95)

Since this is a-independent, there is no gauge anomaly. In addition, using (2.95) we see that under τp → −1/τp ıℓ2 v(m2 − ǫ2+ ) Zλ 7→ exp |λ| Zλ (2.96) 2πτp b0 theory in six dimensions can be rendered The instanton partition function of the A modular invariant by requiring the gauge coupling to transform ıℓ2 v(m2 − ǫ2+ ) q 7→ exp − q (2.97) 2πτp under the τp → −1/τp transformation. The factor m2 − ǫ2+ should be related to the anomaly cancellation condition dH = F 2 − R2 ∝ m2 − ǫ2+

(2.98)

where F and R are the equivariant curvatures of the R-symmetry connection and spin-connection, respectively, on the compactification torus Tℓ,−ℓ/τp . 2.6.9. Example: the A1 theory in six dimensions. Here is an example of potentially sick theory. Again, we skip the i index below. The SU(v) theory with v fundamental hypermultiplets and v anti-fundamental hypermultiplets has, in the instanton sector: X vec 2 2 2 (a − c ) + (2ǫ + c − a ) + χλ + χfλ + χaf = 2ǫ ǫ k − a b,s + b,s a 1 2 λ 2 +

a,b,s∈λb

X

(cb,s + ǫ+ − mf )2 + (ǫ+ + m ˜ f − cb,s )2 =

b,f,s∈λb

X

= 2ǫ1 ǫ2 k 2 + 2

cb,s

b,s∈λb

+k

v X a=1

!

v X a=1

(2aa − 2ǫ+ − ma − m ˜ a)

!

−2(aa − ǫ+ )2 + m2a + m ˜ 2a + 2ǫ+ (m ˜ a − ma )

(2.99)

To make the right hand side gauge invariant we need to insist on the following traceless constraint: v X (2aa − 2ǫ+ − ma − m ˜ a) = 0 (2.100) a=1


25

which means that only the SU(v) part of the gauge group is possibly anomaly free, hardly a surprise. In addition, we must make the gauge coupling q transform under the gauge transformations, so that the combination v ıℓ2 X −2(aa − ǫ+ )2 + m2a + m ˜ 2a + 2ǫ+ (m ˜ a − ma ) q exp − 4πτp a=1

(2.101)

is invariant. However the modular invariance cannot be fixed because of the first term ∝ k2 . 3. The twisted superpotential W of quiver theories We now wish to evaluate the Z-partition function in the limit ǫ2 → 0 with ǫ1 = ǫ fixed. On the physical grounds (the extensitivity of the partition function of the instanton gas) we expect the following behavior: W(a, m; q, ǫ) Z(a, m; q, ǫ1 = ǫ, ǫ2 → 0) ∼ exp − (3.1) ǫ2

Our task is to demonstrate the validity of the asymptotics (3.1) and to evaluate W.

3.1. The limit shape. We shall now investigate the limit ǫ2 → 0. Observe that each contribution to the full partition function (2.55), for small ǫ2 , if expressed through the quantities xi,a,k , contains the factor ǫ12 in the exponential. This suggests to look for a dominant contribution to (2.55) among the configurations with finite (xi,a,k ), for all i ∈ IΓ , a = 1, . . . , vi , k = 1, 2, . . .. Recall that for finite sums ! X Sλ lim ǫ2 log exp (3.2) = Sλ∗ ǫ2 →0 ǫ2 λ∈Λ

where λ∗ is the element of the set Λ on which Sλ∗ Re is maximal (3.3) ǫ2 under the assumption that the phases Im Sǫ2λ of the contributions of the configurations λ with Re Sǫ2λ close the maximal one, are also close to each other. A typical example is the evaluation of the series ∞ X (q/ǫ2 )n n! n=0 for Re(q/ǫ2 ) > 0. The dominant contribution comes from n ∼ q/ǫ2 , i.e. from x = ǫ2 n ∼ q. To make the similar evaluation for all values of q/ǫ2 we use the analyticity of the series (assuming it converges, which in the present example is easy to establish), and then evaluate it with an arbitrary accuracy for q/ǫ2 ∈ R+ . In the case at hand we need to show that there is a domain in the space of parameters a, m, ǫ1 , ǫ2 such that the phases of all contributions to the partition function in the vicinity of the dominant one are aligned. We shall make the argument in the four

26


˜ i = 0, leaving the general five dimensional case to a curious dimensional case, with w student. It is convenient, for this purpose, to choose all the parameters in the problem: ai,a , ǫ1,2 , mf , mbf e , qi etc., to be real. We shall also assume that ǫ1 = ǫ ≫ −ǫ2 > 0, and that ai,a − ai,b ≫ ǫ for a 6= b for all i. We shall also assume that all masses are real, and much larger then |ai,a | for all (i, a). The contribution (2.55) Zλ of a set λ = (λi,a ) of quiver colored partitions to the partition function Z is then also real. We only need to make sure it is positive for all λ’s close to the dominant one. Let us now compare the contributions of two close configurations, namely λ and λ′ obtained from λ by adding one square to the λi,a partition. The ratio can be easily evaluated to be (cf. the Eq. (3.23) below) Zλ′ /Zλ = (−1)vi −1+

P

e∈s−1 (i)

vt(e)

×

(ǫ1 + ǫ2 ) × ǫ1 ǫ2 Q Q e∈t−1 (i) Ys(e) (x + me + ǫ1 + ǫ2 ) e∈s−1 (i) Yt(e) (x − me ) qi Pi (x)

Y′i (x)Yi (x + ǫ1 + ǫ2 ) Y ×

e∈s−1 (i)∩t−1 (i)

(me + ǫ1 )(me + ǫ2 ) (3.4) me (me + ǫ1 + ǫ2 )

where x is one of the zeroes of Yi . It determines the position of the box one can add to λi,a , in fact, x is equal to the ai,a -shifted content of that box. By taking |me |≫ ǫ we can neglect the last line in (3.4). Also, by taking the masses of the fundamentals to be much larger then the Coulomb moduli, we can drop the fundamental polynomial Pi (x), replacing it by (−1)wi . Using the fact that the zeroes and the poles of Yi interlace for ǫ1 /ǫ2 < 0, and assuming |ai,a − ai,b|≫ ǫ, the sign of the denominator of (3.4) is the same as the sign of ǫ1 + ǫ2 . Now, the final adjustment is the choice of the chamber where for all e ∈ EΓ , a = 1, . . . , vs(e) , b = 1, . . . , vt(e) : at(e),b − as(e),a + me ≫ 0

(3.5)

It implies that the period of the 1-cochain me on any closed 1-cycle in the quiver is non-trivial. The sign of the numerator in the third line of (3.4) is then equal to Y (−1)vs(e) e∈t−1 (i)

Collecting all the signs and using the equality in (2.2), we obtain: sign

Zλ′ = sign(qi )(−1)vi −1 Zλ

(3.6)

Thus, we shall assume qi ∈ (−1)vi −1 · R+ . Therefore, in the limit ǫ2 → 0 the sum over the set of all fixed points can be computed semi-classically by finding the dominant configuration, as was originally done in [21].


27

The important difference is that now, for finite ǫ1 , the profile of the dominant partition cannot be assumed to be a smooth function. Instead, the profile of partition λi,a shall be described by an infinite series of continuous variables (ξi,a,k ). This approach was suggested in [29–31]. An alternative approach is based on the Mayer expansion [20] and the ǫ2 → 0 analysis of the asymptotics of the gauge partition function expressed via the contour integral form of the ADHM integrals [1], leading to the TBA-like integral equations [5, 82]. It will be discussed elsewhere. 3.2. Entropy estimates. Let us now explain why finding the dominant configuration suffices for the evaluation of W(a, m; q, ǫ). Indeed, the argument similar to (3.2) may fail if the number N (ǫ2 ) of configurations λ whose actions Sλ fall within an order ǫ2 error in the vicinity of Sλ∗ , grows as eC/ǫ2 for some constant C, for ǫ2 → 0. This is how, for example, one may understand the phase transition in the Ising model, starting with the high temperature expansion, leading to the counting of random walks on the lattice [83]. Let us now explain why in our problem the entropy factor N (ǫ2 ) grows at most in a power-like fashion with 1/ǫ2 . Recall that the configurations λ in our problem are the multi-sets λi,a of partitions of integers, indexed by the vertices i ∈ IΓ of the quiver, and the colors a = 1, . . . , vi of the corresponding gauge group factor. Let us estimate the number of partitions, close to the given set λ = (λi,a) of partitions λi,a , such that the action Sλ is close to Sλ∗ . This number is equal to the number of boxes one can add, or remove from the Young diagrams of the all these partitions λi,a . For each row j = 1, . . . , λt1 we can add one box, λj 7→ λj + 1 if λj−1 > λj . Analogously we can remove one box λj 7→ λj − 1 if λj > λj+1 . At any rate, the number t of such modifications of a single partition is bounded above by 2λ1 , and for the whole configuration λ it is bounded above by X entropy eSλ (3.7) (λi,a )t1 ≤ exp K ′ i,a

′

for some constant K . Now let us estimate (λi,a)t1 . We shall show below, using the analysis of the limit shape equations in the form of the Eq. (4.10), that ǫ2 λi,a,k ∼ Ci,a,k qki ,

for

k −→ ∞

(3.8)

where A(ai,a + ǫk) Ci,a,k = × A′ (ai,a )   k Y Y Y  Pi (ai,a + ǫ(j − 1)) As(e) (ai,a + ǫ(j − 1) − me ) At(e) (ai,a + ǫj + me ) 2 A (a + ǫj) i i,a −1 −1 j=1 e∈t

(i)

e∈s

(i)

(3.9)

where Ai (x) =

vi Y

(x − ai,a)

a=1

28

NIKITA NEKRASOV, VASILY PESTUN, AND SAMSON SHATASHVILI ′′

Since for large k, Ci,a,k ∝ k K for some constant K ′′ which depends only on (a, m, ǫ), 2 so that we can conclude that (λi,a)t1 ∼ logǫ logqi entropy

eSλ

≤ ǫ−K 2

(3.10)

for some uniformly bounded constant K. Thus, the entropy factor is subleading compared to the exponential of the critical action exp(Sλ∗ /ǫ2 ). Remark. Note that the generating function of the leading asymptotics (3.9) ∞ X Ji,a (q) = Ci,a,k qk

(3.11)

k=1

is a generalized hypergeometric function, and can be identified with a vortex partition function [84] of some gauged linear sigma model (and the J-function of [85]). It would be interesting to understand the physics of this coincidence. 3.3. Analysis of the limit shape equations. In the limit q2 = 1 the functions Yi (ξ) (2.43) can be represented as the infinite product over the set Ξi (2.63) as 1

Yi (ξ) =

Y+ i (ξ)

where the Baxter function Q+ i (ξ) =

=

Y

2 Q+ i (q ξ)

(3.12)

1

−2 ξ) Q+ i (q

(1 − ξ ′/ξ).

(3.13)

ξ ′ ∈Ξi

The product (3.13) converges because of the set Ξi (2.63) is asymptotic to Ξ◦i : ◦ ξi,a,k → ξi,a,k ,

k→∞

(3.14)

where ˚ ξi,a,k = wi,aq k−1 by (2.74). Explicitly Y+ i (ξ)

:=

vi Y

(1 − q

− 21

1 ∞ Y 1 − q − 2 ξi,a,k+1/ξ

ξi,a,1/ξ)

1

1 − q 2 ξi,a,k /ξ

k=1

a=1

It is convenient to introduce the notation Y− i (ξ) for + Y− i (ξ) := Yi (ξ)

vi Y

a=1

1

q2ξ − wi,a

!

Y− i (ξ) =

(1 − q ξ/ξi,a,1)

a=1 1This

1 2

1 ∞ Y 1 − q 2 ξ/ξi,a,k+1 1

k=1

1 − q − 2 ξ/ξi,a,k

product formula follows from Y+ i (ξ)

=

Y− i (ξ)

1 vi ∞ Y 1 − q − 2 ξi,a,1 /ξ Y ξi,a,k+1 1 qξi,a,k 2 a=1 1 − q ξ/ξi,a,1

k=1

(3.15)

(3.16)

and from the asymptotics (3.14),(2.74) we see1 vi Y

!

!

!

(3.18)

(3.17)


The asymptotics of Y± i (ξ) functions are (c.f. (2.46) vi Y 1 + −vi −wi,a q − 2 , ξ→0: Yi (ξ) → ξ Y− i (ξ) → 1

29

a=1

ξ→∞:

Y+ i (ξ) → 1,

vi Y− i (ξ) → ξ

vi Y

−1 −wi,a q

a=1

Let Pi− (ξ)

wi Y

=

(1 −

P˜i+ (ξ) =

ξ/µfi,f),

w ˜i Y (1 − µ ˜fi,f /ξ)

1 2

(3.19)

(3.20)

f=1

f=1

be the fundamental (anti-fundamental) matter polynomials. The critical point equations are obtained by a simple variation of the exponents in the constituent factors of (2.55). Using ∂log ξ Li2 (ξ) = Li1 (ξ) = − log(1 − ξ)

(3.21)

we find the critical point equations on the dominant fixed point λ∗ encoded by the sets Ξi to be (3.22) exp(ıℓǫ2 ∂log ξi,a,k log Zλ∗ ) = 1 for each ξi,a,k denoted below as ξ. Explicitly, we find exp(ıℓǫ2 ∂log ξ log Zλ ) = 1 2

− qi (q ξ)

κi

Pi− (ξ)P˜i+(ξ) 1

1

+ −2 2 ξ) Y− i (q ξ)Yi (q Y Y −1 × Y− (µ ξ) Y+ e s(e) t(e) (µe ξ), e∈t−1 (i)

(3.23) ξ ∈ Ξi

e∈s−1 (i)

Therefore, the critical profiles (2.48) are the solutions to the following infinite system of difference equations in the variables ξi,a,k : For each i ∈ IΓ and for each ξ ∈ Ξi , 1

1

+ −2 2 Y− ξ) = i (q ξ)Yi (q 1 − qi (q 2 ξ)κi Pi− (ξ)P˜i+(ξ)

Y

e∈t−1 (i)

−1 Y− s(e) (µe ξ)

Y

Y+ t(e) (µe ξ),

ξ ∈ Ξi

e∈s−1 (i)

(3.24) It would appear that the left hand side of the equations (3.24) cannot be evaluated at − 21 1/2 ξ = ξi,a,k because of the first order pole in Y− ξ) at ξ → ξi,a,k . However, Y+ ξ) i (q i (q 1 + −1 2 ξ)Y (q 2 ξ) is nonhas a first order zero at ξ → ξi,a,k . Therefore the product Y− (q i i singular in the limit ξ → ξi,a,k . The equations (3.24) are formally equivalent to the algebraic trigonometric Bethe ansatzn equations for the gΓ -spin chain, with an infinite number of Bethe roots ξi,a,k , see e.g. [86–89]. These Bethe roots come in the Reggelike trajectories, or Bethe strings. The strings are labeled by the quiver nodes i and the individual colors a = 1, . . . , vi . Each string contains an infinite number of Bethe roots ξi,a,k labeled by k = 1, . . . , ∞. Recall, that the sequences (ξi,a,k ) parametrize the asymptotic profiles of the colored partitions (λi,a,k ).

30


We summarize: the critical profile equations of [21, 23] for the five dimensional gΓ quiver theory with ǫ1 = ǫ finite and ǫ2 = 0 are the Bethe equations for a formal gΓ XXZ spin chain. The number of Bethe roots is infinite with prescribed regular asymptotics at infinity: they have the slope given by ǫ and the intercepts equal to the Coulomb moduli a of the theory. 3.3.1. On-shell versus off-shell. Note that these Bethe equations should not be confused with those in (1.4) where one extremizes with respect to the Coulomb moduli a. Here we do no such minimization, so we are off-shell in that sense. In other words, in order to construct the off-shell superpotential, more precisely the universal twisted superpotential W(a, m, q; ǫ) we solve the Bethe equation for the Uq (gbΓ ) XXZ spin chain; in order to find the states in a proper Hilbert space, i.e. to go on-shell we need to solve another Bethe equation which follows from the extremization of W eff with expect to the Coulomb moduli a. For Ar theories the proposition was formulated with some restrictions in [29–31, 90]. The general ADE case was presented in [60, 61], the formal construction in the four dimensional case can also be found in [67]. − − + Remark. Since Y+ i (ξ) and Yi (ξ) as well as Pi (ξ) and Pi (ξ) differ only by a power of ξ and a constant factor, one can change an orientation of an edge in the quiver, or replace the fundamental by anti-fundamental matter and adjust the Chern-Simons level κi so that it compensates the change of the orientation in the equations (3.24); in this way such quiver theories are isomorphic.

Using (3.16) we can write the equations (3.24) only in terms of Y− i (ξ) as Y Y 1 − − 21 −1 2 Y− ξ) = −P− Y− Y− ξ ∈ Ξi i (ξ) i (q ξ)Yi (q s(e) (µe ξ) t(e) (µe ξ), e∈t−1 (i)

where

 vi Y Y 1 ξ κi 2 Pi− (ξ)P˜i+ (ξ)  P− − i (ξ) = −qi (q ξ) wi,a −1 a=1 e∈s

 Y wt(e),a  − 1 2µ ξ q e (i) a=1

(3.26)

ξ ∈ Ξi

(3.27)

vt(e)

or, equivalently, in term of Y+ i (ξ) as Y Y 1 + −1 + −1 + 2 ξ)Y (q 2 ξ) = −P (ξ) Y+ Y (µ ξ) (q Y+ e i i i t(e) (µe ξ), s(e) e∈t−1 (i)

(3.25)

e∈s−1 (i)

e∈s−1 (i)

where

 vi Y Y 1 wi,a κi 2 P+ − Pi− (ξ)P˜i+ (ξ)  i (ξ) = −qi (q ξ) qξ −1 a=1 e∈t

vs(e)

Y

(i) a=1

−

1 2

q ξ µe wt(e),a

! 

(3.28)

−1 The matter polynomials P− i (ξ) ∈ C[ξ, ξ ] are Laurent polynomials in ξ, with the following asymptotics at ξ → 0, ∞

ξ→0: ξ→∞:

−w ˜ i +κi +vi − P− i (ξ) ∼ ξ wi +κi +vi − P− i (ξ) ∼ ξ

P

j∈t(i)

vj

P

j∈t(i)

vj

(3.29)


31

0 Now pick the CS levels κi in such a way to set asymptotics P− i (ξ) ∼ ξ at ξ → 0 X κ− ˜i + vj (3.30) i = −vi + w j∈t(i)

Then κi =

κ− i

:

( 0 ξ → 0 : P− i (ξ) ∼ ξ w i +w ˜i ξ → ∞ : P− i (ξ) ∼ ξ

(3.31)

−1 Similarly, the matter polynomials P+ i (ξ) ∈ C[ξ, ξ ] have the following asymptotics at ξ → 0, ∞ P −w ˜ i +κi −vi + j∈s(i) vj ξ → 0 : P+ (ξ) ∼ ξ i P (3.32) wi +κi −vi + j∈s(i) vj ξ → ∞ : P+ (ξ) ∼ ξ i 0 and we can pick the CS levels κi in such a way to set asymptotics P+ i (ξ) ∼ ξ at ξ → ∞ X vj (3.33) κ+ i = vi − wi − j∈s(i)

then κi = κ+ i :

( −w ˜ i −wi ξ → 0 : P+ i (ξ) ∼ ξ 0 ξ → ∞ : P+ i (ξ) ∼ ξ

(3.34)

Notice that κ± i are precisely the boundary of the allowed range of CS couplings (2.4), + − and κi = κi for the quiver gauge theories in the conformal class (2.2). If the set of bifundamental masses µe is a trivial cocycle, i.e. if for any 1-cycle X X c= ce [e] ∈ C1 (γ), ∂c = ce ([t(e)] − [s(e)]) = 0 ∈ C0 (γ) e

e

Y

µcee = 1,

(3.35)

e∈cycle in γ

then we can find the set (˜ µi )i∈IΓ of compensators µ ˜i ∈ C× , such that (3.36)

µe = µ ˜s(e) /˜ µt(e)

This is possible for all acyclic quivers, that is for all cases except affine A-quiver. Then, if we set ˜ ± (ξ) = Y± (ξ/˜ ˜ ± (ξ) = P± (ξ/˜ Y µi ), P µi ) (3.37) i i i i the equations (3.25) (3.27) convert to ˜ i (q 21 ξ)Y ˜ i (q − 21 ξ) = −P ˜ i (ξ) Y

Y

˜ j (ξ), Y

ξ ∈ Ξi .

(3.38)

j∈s(t−1 (i))∪t(s−1 (i))

˜ i (ξ), P(ξ) ˜ ˜ ± (ξ), P ˜ ± (ξ). where Y denotes + or − form of Y i i The equations (3.38) are the q-version of the cross-cut equations of [23]. br quiver we consider its universal covering, the A∞ quiver, and the infinite set For A of functions Yi (ξ) for i ∈ Z with twisted periodicity mod r + 1.

32


br quiver with r + 1 nodes on a cycle, labeled 0, . . . , r, and (µe )e∈E Let Γ be A Γ representing a non-trivial class in H 1 (γ, C). Let us define µ ∈ C× by Y µe = µr+1 (3.39) e

Then (µe /µ)e represents a trivial cocycle, for which one can find the compensators, and define Yi′ as above. Now the equations (3.25)(3.27) convert to ˜ i (q − 21 ξ) = −P ˜ iY ˜ i−1 (µ−1 ξ)Y ˜ i+1 (µξ), ˜ i (q 21 ξ)Y Y

i ∈ Z/(r + 1)Z,

ξ ∈ Ξ′i

(3.40)

˜˜ Let us now define functions Y i (ξ) labeled by the vertices i ∈ Z on the universal cover ˜ Γ of γ ˜˜ i ˜′ Y (3.41) i (ξ) = Yi mod r+1 (µ ξ) i ∈ Z ˜˜ In terms of functions Y i (ξ) the equations (3.40) take the canonical form of equations (3.38) for A∞ quiver ˜˜ 21 ˜˜ − 21 ˜˜ ˜˜ ˜˜ Y ξ) = −P i (q ξ)Yi (q i Yi−1 (ξ)Yi+1 (ξ),

i ∈ Z,

ξ ∈ Ξi

(3.42)

˜ In any case, the limit shape equations are (3.38) associated to the universal cover Γ ˜ = Γ if Γ is of the finite ADE or affine D bE b type, and of the original quiver Γ, with Γ ˜ = A∞ if Γ = A br . In the A br case the functions Yi (ξ) are lifted to the cover with the Γ twisted r + 1-periodicity. In the language of quantum groups, such lifting corresponds µb to the relation between Uaff q ( glr+1 ) and Uq (gl∞ ) see [91], [92]. ˜ i (ξ) in the canonIn the subsequent sections we drop the symbol tilde on functions Y ical form (3.38), keeping in mind the change of variables (3.37). 4. Quantum geometry The set of algebraic equations on the set of Bethe roots (ξi,a,k ) in the theory of gΓ spin chains (known as the nested algebraic Bethe Ansatz equations in the case gΓ = slr ), one equation per each Bethe root, can be converted to a set of functional q-difference equations on functions Yi (ξ) (for elementary introduction to algebraic Bethe Ansatz see [93], for most generic case see [49, 94], and for solvable lattice models of statistical mechanics see [95]). The zeroes and poles of the functions Yi (ξ) encode information about Bethe roots. In the limit q → 1 the functional q-difference equations on Yi (ξ) reduces to the ordinary character equations determining the Seiberg-Witten curve in [23]. The functions Yi (x) of [23] provide the classical limit of their more complicated q-relatives Yi (x) of the present paper. For illustration we first consider the quiver gΓ = A1 = sl2 . The Bethe Ansatz equa± tions are simply (we can use here either plus or minus form of functions, Y± i (ξ), Pi (ξ)) 1

1

Y1 (q 2 ξ)Y1 (q − 2 ξ) = −P1 (ξ),

ξ ∈ Ξ1

(4.1)


33

1

with Ξ1 being the set of poles of Y1 (q 2 ξ), and we remind that ξ ∈ Ξ1 should be understood in the limit sense (see remark after (3.24)). Now consider the functional 1

P1 (q − 2 ξ) χ1 [Y1 (ξ)] = Y1 (ξ) + Y1 (q −1 ξ)

(4.2)

The function χ1 (ξ) is regular function everywhere on the multiplicative plane C× hξi provided the equations (4.1). Indeed, the poles which appear in the first term, Y1 (ξ), are precisely canceled by the second term due (4.1). Therefore, χ1 (ξ) is Laurent polynomial in C[ξ, ξ −1], i.e. a holomorphic function on the cylinder C× hxi . Now consider the asymptotics of χ1 (ξ) at ξ → 0 and ξ → ∞. We find that the degree in ξ of the first term and second term in χ1 (ξ) of (4.2) is − respectively (using Y− 1 (ξ), P1 (ξ) form) ξ → 0 : [ξ 0 ], ξ → ∞ : [ξ vi ],

[ξ 0 ] [ξ wi +w˜ i −vi ]

(4.3)

Therefore, at ξ → ∞ the first term dominates the second term by the positivity condition (2.1). We conclude that χ1 [Y1 (ξ), P1 (ξ)] = T1 (ξ)

(4.4)

where T1 (ξ) is polynomial in ξ of degree v1 , in which the highest and lowest coefficients in ξ are fixed from the known asymptotics of Yi (ξ) and Pi (ξ), i.e. in terms of masses and coupling constants. The functional q-difference equation (4.4) is quantum version of Seiberg-Witten for the A1 quiver theory, that is SU(v1 ) gauge theory with w1 and w ˜1 1 fundamentals and anti-fundamental. The Laurent polynomial χ1 in C[Y (ξq 2 Z )± ] given by (4.2) is the Frenkel-Reshetikhin q-character for Uaff q (A1 ) [49]. The v1 − 1 middle term coefficients in gauge polynomial T1 (ξ) play the role of the Coulomb moduli of the theory and are implicitly defined in terms of the SU(v1 ) Coulomb parameters a1,a . Note that in the four dimensional ℓ → 0 limit the Eq. (4.4) formally becomes identical to the Baxter equation for the A1 XXX spin chain. It is not yet a vacuum equation of the four dimensional N = 2 theory in the Ωǫ -background. As we discussed in the Introduction, the latter requires the second minimization with respect to the Coulomb moduli a and this corresponds to finding the proper solution of Baxter equation. We know the definition of proper from the integrable model side only in a few cases — in particular in the cases considered in [5]. Similar interpretation holds for the five dimensional A1 case (and XXZ spin chain), as well as for the arbitrary ADE or affine bD bE b quivers both in five and four dimensions. A The cancellation of poles between the two terms in (4.2) is the q-analogue of the invariance of the classical character used in [23] under crossing the cuts supporting the charge densities used to define the classical potentials Yi (ξ). In the classical limit q → 1 the strings of Bethe roots condense on a certain finite intervals Ii,a , which are interpreted as cuts of the analytic functions Yi (ξ). The remaining Bethe roots form ∗ ∗ the semi-infinite strings (ξi,a,k |k > ki,a ) for certain ki,a , and the classical limit their slope approaches a constant so that their contribution to Yi (ξ) defined by the product formula (3.12) cancels.

34


Below we will use short-hand notations Yi,b ≡ Yi (q −b ξ) Pi,b ≡ Pi (q −b ξ)

(4.5)

For any (affine) simply-laced gΓ one can find recursively the q-characters χi [49, 96, 97] for i-th fundamental evaluation representation of Uaff q (gΓ ) as follows. Start from the highest weight χ˜i = Yi,0 + . . . (4.6) add a term to cancel the poles in the highest weight term Q Pi, 1 j∈s(t−1 (i))∪t(s−1 (i)) Yj, 1 2 2 χ ˜i = Yi,0 + + ... (4.7) Yi,1 This term introduces new poles from the functions Yj (ξ) of the nodes j linked with i. To cancel these poles, we add new monomials Yj,b (ξ) to (4.7) until all poles are canceled. The precise algorithm to compute the q-characters (4.7) for Uaff q (gΓ ) is given by Frenkel-Mukhin [96] for all finite dimensional Lie algebras gΓ . It is generalized for the general symmetrizable Kac-Moody Lie algebra gΓ in [97]. The symbols Yi,b (ξ) in the q-character are the current-weights encoding the (generalized) eigenvalues of an element of the maximal commuting subalgebra of Uaff q (gΓ ) on a (generalized) eigenspace in a representation of Uaff (g ), see [49, 94] and appendix for Γ q aff details. We give explicit formula for the current h(x) ∈ Uq (gΓ ) in (5.30), (5.12). The q-analogue of the iWeyl group WgΓ of [23] is the braid group B(WgΓ ), see e.g. [98]. We remind that as ξ crosses a classical cut of Yi (ξ) the Yi (ξ) is transformed by the reflection in WgΓ generated by i-th simple root [23]. The braid group B(WgΓ ) is freely build on generators Ti , i ∈ IΓ with the relations (for simply laced gΓ ) Ti Tj = Tj Ti , aij = 0 Ti Tj Ti = Tj Ti Tj , aij = −1.

(4.8)

The braid group B(WgΓ ) action on the weight space of Uaff q (gΓ ) generated by fundamental weights Yi is the natural analogue of the Weyl reflection Q Pi, 1 j∈s(t−1 (i))∪t(s−1 (i)) Yj, 1 2 2 Ti : Yi,0 7→ ; Yj 7→ Yj , j 6= i. (4.9) Yi,1 The central result of the present work is the following statement, which could be seen as the q-version of the cameral Seiberg-Witten curve of [23] for gΓ -quiver gauge theories. Proposition. The N = 2 gauge theory functions (Yi (ξ))i∈IΓ of gΓ -quiver gauge theory on R4ǫ,0 × S1ℓ (the Ω-background of [5]) defined by (2.43) satisfy a system of functional q = eıℓǫ difference equations {χ˜i {Yj (q Z ξ), Pj (q Z/2 ξ)}i∈IΓ = Ti (ξ) | i ∈ IΓ } (4.10) where χ ˜i is the twisted q-characters for the i-th fundamental module of Uaff q (gΓ ) (see [49, 97] and Appendix), and Ti (ξ) is a polynomial of degree vi with the highest and


35

lowest coefficient fixed by the masses and coupling constants of the theory, and the middle vi − 1 coefficients parametrizing the Coulomb branch. ◦ 4.1. The slope estimates. It follows from (4.10) that ξi,a,k approach ξi,a,k as k → ∞ exponentially fast (for |qi |< 1): ◦ ξi,a,k = ξi,a,k + Ci,a,k qki ,

k −→ ∞

(4.11)

where Ci,a,k all have a finite limit as q → 0, and behave at most power-like with k for ◦ large k. The idea is to study the small qi limit of (4.10) evaluated at ξ = ξi,a,k = wi,aq k−1 in five dimensions, or, at x = ai,a + ǫ(k − 1) in four dimensions. Let us stick with four dimensions. It is easy to see that Ci,a,k , k>1 Ci,a,k−1 Ci,a,1 k>1 Yi (ai,a) ∼ qi A′i (ai,a ) , Yj (ai,a + ǫ(k − 1)) ∼ Aj (ai,a + ǫ(k − 1)), j ∈ IΓ ,

Yi (ai,a + ǫ(k − 1)) ∼ qi Ai (ai,a + ǫ(k − 1))

j 6= i (4.12)

so that in the q-character χi it is the term proportional to qi will survive the q → 0 limit, all other terms being suppressed either by the explicit powers of q’s, or by the asymptotics of Yi itself, cf. (4.12). From this the Eq. (3.9) follows. 4.2. The convergence of q-characters. Because of the constraints (2.4) on the Chern-Simons couplings the highest term in (4.7) dominates the other terms in the limit ξ → ∞ (using − version of Pi (ξ), Yi (ξ) ) and ξ → 0 (using + version of Pi (ξ), Yi (ξ)). For gΓ -quiver theories in the conformal class (2.2) all terms have the same degree in ξ. For affine gΓ quiver, the q-character χ˜i is infinite convergent series in coupling constants (qj )j∈IΓ under the assumption (|qi |< 1)i∈IΓ . 4.3. The observables. In this section we record the formulae for the observables of the gauge theory which are encoded in the functions Y− i (ξ) obeying (4.10). For − the practical purposes, the expansion of functions Yi (ξ) in the instanton parameters qi can be conveniently obtained from (4.10) by representing the solution Y− i (ξ) in the continuous fraction form in terms of the polynomials Ti (ξ) and Pi (ξ) at ξ → ∞ starting from the zeroth order approximation (3.19) Y− i (ξ)

=ξ

vi

vi Y

a=1

−1 21 −wi,a q + ...

ξ→∞

(4.13)

The Chern character of the universal bundle Ei over the instanton moduli space corresponds to the observable Oi,n = trvi eıℓnφi (4.14) in the gauge theory where φi is the adjoint complex scalar of the N = 2 supersymmetric SU(vi ) gauge vector multiplet. The relation between (1.8) and (2.43) follows from the

36

NIKITA NEKRASOV, VASILY PESTUN, AND SAMSON SHATASHVILI 1

expansion of Yi (q − 2 ξ) at ξ = ∞ ıℓnφi

trvi e

=

I

1

ξ=∞

−2 ξ n d log Y− ξ) i (q

(4.15)

where the contour runs around ξ = ∞ and encloses all zeroes and poles of Yi . Therefore, the gauge theoretic definition of the functions Y± i (ξ) is: − 21 ξ) Y− i (q

=ξ

vi

vi Y

a=1

+ −1 −1 −wi,a Yi (q 2 ξ) =

=ξ

vi

vi Y

−1 −wi,a

a=1

exp =

vi Y

a=1

and

∞ X ξ −n

− trvi eıℓnφi n i=1

!

−1 −wi,a exp log detvi ξ − eıℓφi (4.16)

1

−2 ξ) = exp(hdetvi (1 − eıℓφi /ξ)i) Y+ i (q

(4.17)

4.4. Superpotential. Here we compute the q-derivatives of the twisted superpotential W(a, m; q, ǫ) = − lim ǫ2 log Z(a, m; q; ǫ1 = ǫ, ǫ2 ) ǫ2 →0

From (3.1),(2.73) we find Wi := ∂log qi W := −

X

ξ∈Ξi

Then, using the Baxter function

Q+ i (ξ; Ξ)

Wi = −

1 2πi

I

Ci

ξ log ˚ ξ crit

(4.18)

(4.19)

(3.13) we find

log ξ d log

Q+ i (ξ; Ξi ) ˚i ) Q+ (ξ; Ξ

(4.20)

i

in terms of the sets of Bethe roots Ξi = {ξi,a,k } (2.63) and ˚ Ξi = {ξi,a,k } (2.74), where ˚ the contour Ci encloses all points in Ξi and Ξi . In terms of the functions Y+ i (ξ) we find ∞ + + Qi (ξ) Y Yi (q −j ξ) = (4.21) ˚+ (ξ) ˚+ −j Q i j=0 Yi (q ξ)

For practical purposes of computation Wi in the q-expansion it is convenient to integrate by parts and represent the contour Ci as the difference of contours around ξ = ∞ and ξ = 0. Then we get I ∞ −j Y 1 dξ Y+ i (q ξ) Wi = log (4.22) + −j ˚ 2πı C∞ −C0 ξ Y (q ξ) j=0 i

Using the continuous fraction expression of Y+ i (ξ) from the system of q-characters the integrand can be expanded in the series in q with coefficients in the rational functions of ξ after which the contour integral is computed by taking the difference of coefficients at ξ −1 in the series expansion at ξ = 0 and at ξ = ∞.


37

4.5. Dual periods. Finally, we give the formula for the partial derivatives aD i,a = ∂ai,a W

(4.23)

which can be obtained from the variational limit shape problem Q Q − + ∞ −1 Y e∈t−1 (i) Ys(e) (µe ξi,a,k ) e∈s−1 (i) Yt(e) (µe ξi,a,k ) D exp ai,a = Q 1 + − − 21 2 ξi,a,k ) b6=a Yi,b (q ξi,a,k )Yi,b (q k=1

(4.24)

where we using the colored functions Y+ i,a (ξ) defined as factors in the product over a in (3.15) and 1 ∞ Y 1 1 − q 2 ξ/ξi,a,k+1 − 2 Yi,a (ξ) := (1 − q ξ/ξi,a,1) (4.25) − 21 ξ/ξi,a,k k=1 1 − q

so that (c.f. 3.12)

1

+ 2 Y− i,a (ξ) = (−q ξ/wi,a )Yi,a (ξ)

It would be interesting to compare the above computation of [99], [44].

(4.26) aD i,a

with the approach of

5. Quantum groups 5.1. Introduction. In [100, 101] Drinfeld and Jimbo introduced the notion of Quantum Groups generalizing the algebraic structures underlying quantum integrable systems known at that time after the work of Sklyanin, Takhtajan, Faddeev, Kulish, Reshetikhin and others [102–108]. Namely, for any symmetrizable Kac-Moody Lie algebra g and a complex parameter q ∈ C× the works [100, 101] associated the Hopf algebra Uq (g), also often called the quantum group, by q-deforming the Serre relations of g, this construction is often referred as first Drinfeld realization [100, 101]. Later Drinfeld found [47] that for an affine Kac-Moody Lie algebra b g, the algebra Uq (b g) can be obtained by a certain canonical quantum affinization process applied to finite-dimensional simple Lie algebra g, with the result being quantum affine algebra ∼ g) [50, 109, 110]. The construction [47] is known as second Drinfeld Uaff q (g) = Uq (b realization of quantum affine algebras, or Drinfeld loop realization, or Drinfeld currents realization. In fact, Drinfeld current realization of Uaff q (gΓ ) is defined for any symmetrizable KacMoody algebra gΓ [111, 112]. If gΓ = b g is an affine Kac-Moody Lie algebra, then the aff b g) is often called the g-quantum gΓ -quantum affine algebra Uq (gΓ ) = Uaff g) = Uq (b q (b toroidal algebra [58, 113], because of the presence of two loops inside. The gΓ -quantum affine algebra Uaff q (gΓ ) prominently appears in Nakajima’s work [112, 114, 115] on equivariant K-theory of quiver varieties associated to Γ. 5.1.1. Five dimensional version. In our work, gΓ -quantum affine algebra Uaff q (gΓ ), where gΓ is ADE or affine ADE, appears in the study of the 5d gΓ -quiver gauge theory on S1ℓ × R4ǫ1 ,ǫ2 in the limit ǫ1 = ǫ, ǫ2 = 0 [5] with quantization parameter q = eıℓǫ .

(5.1)

38


5.1.2. Four dimensional version. In the four-dimensional limit gΓ -quantum affine algebra Uaff q (gΓ ) contracts to gΓ -Yangian Yǫ (gΓ ) [47, 116–118]. The formulae in the present note are presented in terms of the multiplicative spectral parameter ξ = eıℓx

(5.2)

on Cx ∼ = C× for the 5d case of Uaff q (gΓ ) but are easily adapted to the 4d case of Yǫ (gΓ ). It is sufficient to take the limit ℓ → 0, with the multiplicative variables such as ξ = eiℓx and q = eiℓǫ sent to 1, but keeping additive variables x and ǫ finite. At the level of the quantum integrable systems, 4d limit of the 5d theory is the isotropic limit in which the trigonometric gΓ XXZ spin chain associated to the 5d theory becomes the rational gΓ XXX spin chain associated to the 4d theory; for classical Seiberg-Witten for gΓ = slr see [36]. 5.1.3. Six dimensional version. The lift of the gauge theory to six-dimensional spacetime R4ǫ,0 × T2ℓ,−ℓ/τp , where (ℓ, −ℓ/τp ) denote the periods of the compactification torus ell T2ℓ,−ℓ/τp , promotes Uaff q (gΓ ) to the gΓ -quantum elliptic algebra Uq,p (gΓ ) [119–121] with p denoting the multiplicative modulus of the compactification torus T2ℓ,−ℓ/τp p = e2πiτp

(5.3)

This is done perturbatively by summing over Kaluza-Klein modes [28, 122] and nonperturbatively by the study of elliptic genera of instanton moduli spaces [26, 68, 69, 123] ell replacing the quantum affine group Uaff q (gΓ ) by the gΓ -quantum elliptic algebra Uq,p (gΓ ) [121, 124]. The spectral additive parameter x lives on the elliptic curve (5.4) Cx = C/(Z ⊕ τp Z) ∼ = C× /pZ = Ep dual to the torus T2ℓ,−ℓ/τp and hence gets two periods 2π 2π , x→ x+ τp (5.5) ℓ ℓ see 2.1.1 for gauge theory discussion. The quantum affine algebra Uaff q (gΓ ) is promoted ell to quantum elliptic algebra Uq,p (gΓ ) [121, 124]. (See [119, 120] on earlier definitions, [125, 126] for elliptic KZ equation and [127] on the defining elliptic gamma function, [128] for connection to elliptic Wq,p (gΓ ) algebra.) For generalized Kac-Moody algebra gΓ with symmetric Cartan matrix the elliptic algebra Uell q,p (gΓ ) is defined using an elliptic version of Drinfeld currents realization in A.2.1. For gΓ of finite-dimensional ADE type see definition in [121], appendix A; keeping in mind that our quantization parameter q corresponds to q 2 in [121]. Elliptic version, as usual, is achieved by promoting factors 1 1 ξ 2 − ξ − 2 → θ1 (ξ; p) (5.6) where θ1 (ξ; p) is (2.82). In the world of quantum integrable systems the lift to the six dimensional theory corresponds to the deformation of the gΓ XXZ spin chain to the anisotropic gΓ XYZ spin chain with elliptic R-matrix. The classical limit ǫ = 0 of the associated integrable system [23] for gΓ of finite type is the gΓ -monopoles on S1 × Ep ∼ = T3 . The classical SW theory for gΓ = slr quiver, was studied in [37]. The classical limit ǫ = 0 of x→x+


39

the associated integrable system [23] for affine gΓ = gc Γ′ , where gΓ′ is finite-dimensional Q simple Lie algebra, is the moduli space of GΓ′ instantons on T 4 = Ep ×Eq for q = i qai i , where ai are Dynkin marks on Γ.

5.1.4. Dimension uniform description. To say uniformly, we deal with a quantum current algebra Uǫ gΓ (Cx ), where complex one-dimensional curve Cx is the domain of the additive spectral parameter x ∈ Cx , or loop variable in the definition of Uǫ gΓ (Cx ) by Drinfeld currents (ψi± (x), e± i (x))|i∈IΓ , see A.2.1, so that   Cx = C Yǫ (gΓ ), Uǫ gΓ (Cx ) = Uaff (5.7) Cx = C/Z ∼ = C× q (gΓ ),  Uell (g ), C = C/(Z ⊕ τ Z) ∼ C× /pZ = x p q,p Γ The representation theory of quantum algebras Uǫ gΓ (Cx ) is the topic of active research with a vast literature. We shall mention only some facts of direct relevance to what follows. The basic result of Chari and Presley [50, 110] is that the irreducible finite dimensional representations of Uaff q (gΓ ) for finite-dimensional simple Lie algebra gΓ of rank r, are classified by the r-tuples of polynomials, called Drinfeld polynomials. These Uaff q (gΓ )-modules can be described by the q-characters of Frenkel and Reshetikhin [49], such that the tensor product of representations corresponds to the product of q-characters. In the rational limit the q-characters reduce to the characters of the Yangian Yǫ (gΓ ) found in [48], see also [129]. Frenkel and Mukhin [96] gave combinatorial algorithm for the q-characters of Uaff q (gΓ ). Nakajima [112] realaff ized geometrically representation of Uq (gΓ ) on equivariant K-homology group of the quiver variety and in [130, 131] he defined non-commutative t-deformation of the qcharacter building on the geometrical methods of quiver varieties [15, 16, 132] and computed explicitly (q, t)-characters of Uaff q (g) for the g = Ar , Dr series in [77] and for g = E6 , E7 , E8 in [133]. Hernandez [97] constructed (q, t)-characters for gΓ -quantum affine algebra Uaff q (gΓ ) of generic symmetrizable Kac-Moody Lie algebra gΓ , developed representation theory in [134] and constructed quantum fusion tensor category in [135]; Chari et al [136, 137] constructed fundamental Uaff q (g) q-characters for classical series g = A, B, C, D using the action of the braid group B(Wg ) corresponding to the Weyl group Wg . More general category of infinite-dimensional representations for the Borel subalgebra Uaff q (gΓ ) was studied by Hernandez and Jimbo [138] following the work [38, 139] on the case gΓ = sl2 . The recent paper [94] interprets the Baxter functions for certain infinite-dimensional representations.

br the gΓ -theory 5.1.5. The twist mass for the A-type affine quivers. Recall that if gΓ = A 1 has mass parameter m associated to the H (Γ) that can not be removed by the shifts of scalars in the U(1) vector multiplets [23]. The additional equivariant parameter µ = eiℓm appears in H. Nakajima’s work [115] on K-theory of the quiver variety MΓ for br . In his picture the multiplicative group C× Γ of type A q , with defining character q, acts 2 in the K-theory of moduli space of sheaves on Chz3 ,z4 i /Γ by2 via the scaling symmetry of 2With

1/2

the identification qpresent = q of [115]

40


the base space: (z3 , z4 ) 7→ (z3 q 1/2 , z4 q 1/2 ). If Γ ⊂ SU(2) is cyclic Γ = Zr+1 , then there 2 −1 is another C× µ action on C commuting with Γ, namely (z3 , z4 ) → (µz3 , µ z4 ). Hence, for cyclic Γ the equivariant K-theory of moduli space of sheaves on C2 /Γ depends on two parameters (q, µ) for C× × C× action on C2 . Naturally, we expect that there b exists a µ-twisted version of Uaff q (glr+1 ). Indeed, such µ-twisted quantum toroidal µb algebra Uaff q ( glr+1 ) was constructed by Varagnolo and Vasserot [140] and it has two parameters q and µ. Our conventions correspond to the conventions of [140] as follows: µb µ-twisted quantum toroidal algebra Uaff q ( glr+1 ) is Uq,t (Z/nZ) of [140] for n[140] = r+1, 1 1 q[140] = q 2 µ and t[140] = q 2 µ−1 . b r+1 ) associated to A br quiver is a rational limit for µ = The affine Yangian Yǫ (m gl µb eıℓm of quantum toroidal algebra Uaff q ( glr+1 ), which admits additional deformation parameter that we identify with the twist mass m coupled to the cycle in H1 (Γ), see more on affine Yangian in [141–144]. br corresponds to the parameter of the The twist mass parameter m in H 1 (Γ) for Γ = A non-commutative deformation of Cx × Eq . Recall that the phase space of the associated integrable system is the moduli space of U(r + 1)-instantons on the non-commutative b∗ theory (i.e. N = 2∗ theory), the Cx × Eq [23]. The six dimensional case of gΓ = A 0 phase space of classical integrable system is the moduli space of non-commutative U(1) instantons on T 4 was studied in [123, 145, 146]. Our work implies that the quantization of the integrable system for the six dimensional theory is provided by the elliptic version µb of the gl1 -toroidal algebra, that we call Uell q,p ( gl1 ).

5.1.6. Quantum toroidal algebras. Quantum g-toroidal algebra was introduced in [113] for all simple Lie algebras g and in [147] it was shown that quantum glr+1 -toroidal algebra is Schur-Weyl dual to double affine Hecke algebra [59, 148]. The vertex representation at level 1 for slr+1 -toroidal was constructed in [149]. Representation theory of quantum toroidal algebras is based on non-symmetric MacDonald polynomials [150]. In [115] for g = ADE and [140] for g = A was shown that equivariant K-theory of Nakajima’s quiver variety [16] realizes representation of quantum g toroidal algebra. Another construction of representations of quantum toroidal algebra was given using Schur-Weyl duality to Dunkl-Cherednik representation [151] of DAHA [151]. In [152] [153] was constructed representation of quantum glr+1 -toroidal algebra in the q-Fock space (see [154] for quantum glr+1 -affine version). The isomorphism between the two constructions has been shown in [155]. The braid group and automorphisms of quantum glr+1 toroidal algebra was studied in [156]. Consider the algebra (C× )dµ , called the non-commutative multiplicative d-torus, gen±1 erated by the variables x±1 1 , . . . , xd subject to the relations xi xj = µij xj xi

(5.8)

where µij = µ−1 ji is the matrix of the non-commutativity c-valued parameters. For example, for d = 2, the algebra (C× )2µ is the non-commutative 2-torus, also known as ±1 algebra of the µ-difference operators on C× , generated by the two variables x±1 1 , x2 modulo relations x1 x2 = µx2 x1 . Let g be a finite-dimensional simple Lie algebra. The


41

paper [157] defined possibly µ-non-commutative (C× )dµ -toroidal g-algebras g[(C× )dµ ] and their central extensions. In particular, it was shown in [157] that only the A-type d-toroidal g-algebras allow non-trivial non-commutative deformation by parameter µ. The toroidal algebra Uaff g) associated to the affine quiver gauge theory gΓ = b g q (b × 2 is isomorphic to Uq g[(C )µ ], that is to the q-deformation of the universal enveloping algebra of the d = 2 µ-noncommutative toroidal algebra g[(C× )2µ ] of [157]; moreover, the non-commutativity parameter µ in H 1 (Γ) is possibly non-trivial only for the affine A-series, in agreement with our gauge theory considerations and the constructions of H. Nakajima. µb In this way, the simplest, gl1 d = 2 toroidal algebra, Uaff q ( gl1 ), with quantization parameter q and non-commutativity parameter µ was constructed and studied by Miki in [158] (this gl1 -toroidal algebra is called there Uq,γ with the parameter γ related to our µ, cf. Eqs. (3.18) - (3.24) of [158]). Morover, in [158] Miki has shown that the tensor product of n level 1 modules of gl1 toroidal algebra relates to the q-deformed W-algebra Wq (sln ) of [159, 160]. On the other hand, from Nakajima [16, 115, 132, 161] and related works [113, 140, 141, 147, 152, 162– µb 167], gl1 -toroidal algebra Uaff q ( gl1 ) at level n acts in the equivariant K-theory of the moduli space of framed torsion free rank n sheaves on C2q1 ,q2 with two equivariant 1 1 parameters q3 = q 2 µ and q4 = q 2 µ−1 . Hence, in [158], by relating the tensor product of n level 1 modules of gl1 -toroidal algebra to the q-deformed sln quantum W-algebra [128, 159, 160] Miki has shown (see also remark A.7 in [168]) a version of correspondence between equivariant rank n gauge theory on C2q3 ,q4 [19] and q-deformed sln -Toda theory that appeared later as an AGT conjecture [22, 169–171]. In the 4d, rational limit, the non-commutative 2-torus (C× )2µ , or the algebra of µ-difference operators, contracts to the Lie algebra of differential operators of onevariable, called d. Its central extension b d is also known as W1+∞ algebra [158]; and its quantization would produce the rational limit of gl1 -toroidal algebra, which is gl1 affine b 1 ). For explicit R-matrix for Yǫ (m gl b 1 ) see [172]. The tensor product Yangian Yǫ (m gl of n level 1 modules of gl1 affine Yangian relates to the ordinary W(sln )-algebra, the symmetry of the 2d conformal sln -Toda field theory; and since in the rational limit equivariant K-theory contracts to the equivariant cohomology, the ordinary W(sln ) algebra relates to the equivariant rank n gauge theory on C2ǫ3 ,ǫ4 (AGT conjecture [22, 169]) that has been proved in this form in [173–175] after [171, 176]. µb The gl1 toroidal algebra Uaff q ( gl1 ) appeared under different names in the literature. It was called quantum continuous gl∞ , or deformation of universal enveloping of qdifference operators in one-variable in [177, 178]; spherical elliptic Hall algebra or gl∞ DAHA in [179] and [165, 180]. In [181] the same algebra without Serre relations is called Ding-Iohara algebra [182] or trigonometric limit of Feigin-Odesskii [183] elliptic µb shuffle algebra (see also [184]). The elliptic deformation Uell q,p ( gl1 ) of the gl1 toroidal 1 1 µb −1 −1 2 2 algebra Uaff q ( gl1 ) is called A(q1 , q2 , q3 , p) in [168] with q1 = q µ, q2 = q µ , q3 = q and is equivalent without Serre relations to the Feigin-Odesskii elliptic shuffle algebra [183]. For details in equivalence of gln toroidal algebra and shuffle algebra see [185]. The elliptic deformation of gl1 toroidal algebra in a certain limit has been related to

42


the quantum cohomology of Hilbert scheme of points on C2 [186] in [168], and the relation to elliptic Macdonald operator was further studied in [187, 188]. The characters of quantum gl1 toroidal algebra are computed as a sum over plane partitions [189, 190]; and representation theory of quantum glr toroidal algebra is discussed in [191, 192]. 5.1.7. Other topics. The representation theory of quantum affine and toroidal algebras Uaff q (g) is connected to many exciting topics in geometry, algebra, and mathematical physics, e.g. the cluster algebras, the Y - and the T -systems, affine and double affine Hecke algebras [59], knots and three dimensional theories [193], quantum KnizhnikZamolodchikov-Smirnov equation [194–196], affine Toda integrable field theories and many other integrable systems, W -algebras [128, 197], the putative five dimensional version of the AGT [22] correspondence, Stokes multipliers for ordinary differential equations and CFT correlation functions [38, 39, 139, 198], holomorphic anomaly equation in the limit of [5] (see, e.g. [199]), geometric Langlands correspondence [200], spectrum of anomalous dimensions in N = 4 super Yang-Mills. We do not consider in the present note the above topics and the web of their interrelation leaving the discussion to the future. 5.2. Quantum group interpretation of the gauge theory results. There are two ways to think about the Coulomb moduli space M. In the first interpretation the expectation values of the observables (4.14) parametrize an element h(ξ) ∈ Uaff q (gΓ ) (in the maximal commutative subalgebra) which morally speaking is similar to picking a conjugacy class in a group (if Uaff q (gΓ ) were a group). For gauge groups of finite rank vi , this parametrization is clearly redundant because of the relations between Oi,n for high enough n. Conceptually, the elements h(ξ) in the maximal commutative subalgebra of Uaff q (gΓ ) parametrize the union of Coulomb moduli spaces over all ranks vi of the gauge groups and couplings qi . This interpretation in the classical limit q → 1 reduces to the construction in [23]. In the second interpretation the polynomial Ti (ξ) is an eigenvalue of the transfer matrix operator tVi ∈ Uq (b g) (where Vi is a fundamental Uaff q (g) module usually called an auxiliary space in the algebraic Bethe Ansatz) on an eigenstate |wi ∈ W where anaff other Uaff q (gΓ )-module W is the physical space of states of gΓ -spinchain. This Uq (gΓ )module W , defined by masses of the gauge theory, is usually not finite-dimensional, but the spectral problem in the Uaff q (gΓ )-module W can be converted to the spec˜ tral problem in a finite-dimensional Uaff q (gΓ )-module W for special discrete choices of Coulomb parameter ai,a as for example in [30, 31] (special electric ai,a ) or [5] (special magnetic ai,a ). For example, for gΓ = sl2 with w1 = 2v1 , as was advocated by Dorey et al [30, 31], one can split the set of masses M = {µ1,f } into two disjoint sets + M − = {µ− = {µ+ 1,a } and M 1,a } with a = 1, . . . , v and then choose Coulomb parame− ˜ ters w1,a = q na µ1,a for some integers na ∈ Z≥0 . Then Uaff q (sl2 )-module W is the tensor ˜ = Ws ,ζ ⊗ Ws ,ζ ⊗ . . . ⊗ Wsn ,ζn of the Uaff (sl2 ) evaluation Verma modules product W q 2 2 1 1 p − − + Wsa ,ζa at spectral parameter ζa = µa µa and highest weight sa = logq (µ+ a /µa ).


43

5.2.1. First interpretation of the gauge theory q-character equations. The conventions for the quantum affine algebras are summarized in the appendix A. ˚ First, we find an element ψ(ξ|Ξ) ∈ Uaff q (gΓ ) in the maximal commutative subalgebra aff of Uq (gΓ ) such that its evaluation in a finite-dimensional Uaff q (gΓ )-module V has the generalized eigenvalues (the symbols Yi,ζ ) equal to the gauge theory function Yi (ξ/ζ) Yi,ζ = Y+ evψ(ξ|Ξ) ˚ i (ξ/ζ)

(5.9)

Explicitly, comparing the evaluation definition (A.53) and the definition of the Y ˚ functions in the gauge theory (3.12) we conclude that such element ψ(ξ|Ξ) ∈ Uaff q (gΓ ) is given by Y Y ˚ ˚+ (ξ/ξ ′) ψ(ξ|Ξ) = ψ (5.10) i

i∈IΓ ξ ′ ∈Ξi

˚+ (ξ/ξ ′) is defined by (A.23). where ψ i For example, in the case gΓ = sl2 , the q-character of the fundamental module V1,q− 12 evaluated by the element (5.10) gives us 1

1

2 χq (V1,q− 12 ) = Y+ evψ(ξ|Ξ) ˚ i (q ξ) +

− 12 ξ) Y+ i (q

(5.11)

˚ Notice by comparing (4.16) with (5.10) that can represent the operator ψ(ξ|Ξ) ∈ as follows ! ∞ −n − n XX 2

ξ q ˚ trvi eıℓnφi hi,n (5.12) ψ(ξ|Ξ) = exp − [n] q n=1

Uaff q (gΓ )

i∈IΓ

For the gauge theory with fundamental matter we need to incorporate the extra factors of matter polynomials as in (4.2). We can do this by multiplying (5.10) by another element in Uaff q (gΓ ), which depends only on the masses and coupling constants. To find explicit expression for gΓ = sl2 , it is useful to notice that the equation 1

1

s(q − 2 ξ)s(q 2 ξ) = 1 − ξ −1

(5.13)

is solved by the function (defined as the expansion near ξ = ∞) ! ∞ X 1 ξ −n s(ξ) = exp − . n n n q− 2 + q 2 n=1

(5.14)

Define an operator ψ1∨ (ξ) ∈ Uaff q (sl2 ) as follows ∞ X ξ −n 1 ˚∨ (ξ) := exp − ψ h n 1 − n 1,n 2 + q 2 [n] q q n=1

!

(5.15)

˚1 (x), ψ ∨ (x) on a fundamental eigenvector The eigenvalues of the operators p1 (ξ), ψ 1 with generalized weight encoded by the Drinfeld polynomial P (ξ) = 1 − ξ −1 are given

44


by p1 (ξ)|vi = (1 − ξ −1 ) ˚∨ (ξ)|vi = s(ξ) ψ 1

(5.16)

1

˚1 (ξ)|vi = ψ

1 − q − 2 ξ −1 1

1 − q 2 ξ −1

Given a set M1 = {µf } ˚∨ (ξ|M) ∈ Uaff (sl2 ) of the masses for fundamental multiplets define the operator ψ 1 q Y ˚∨ (ξ/µf ) ˚∨ (ξ|M1 ) = ψ (5.17) ψ 1 1 µf ∈M1

with eigenvalue given by the function

Y

s11 (ξ|M1) =

(5.18)

s(ξ/µf)

µf ∈M1

that satisfies 1 1 s11 (ξq 2 |M)s11 (ξq − 2 |M) = ˚ P1 (ξ) ≡

Y

(1 − µf /ξ)

(5.19)

f

Then χq (V1,q− 21 ) character evaluated on the element h(ξ) ∈ Uaff q (sl2 ) 1 ˚1 (ξ|Ξ)/ψ ˚∨(ξ|M) h(ξ) = q− 2 h1,0 ψ 1

(5.20)

equals 1

− 12

1

Y1 (q 2 ξ)

1 2

s11 (ξq − 2 |M)

+q (5.21) 1 1 s11 (ξq 2 |M) Y1 (q − 2 ξ) Finally, define the twisted q-character χ˜1,ζ multiplying the q-character by a suitable scalar factor such the term corresponding to the highest weight vector in the fundamental module V1,ζ equals to Y1 (ξ/ζ) evh(ξ) χq (V1,q− 21 ) = q

(5.22)

χ˜1,ζ = c1 (ξ/ζ)χq (V1,ζ ) where 1

(5.23)

c1 (ξ) = q 2 s11 (ξ|M)

Hence, we obtain the first interpretation of the limit-shape equations (4.10) For gΓ = sl2 , the twisted character χ˜1,ζ evaluated on the element h(ξ) ∈ Uaff q (gΓ ) is a polynomial of degree v1 that we denote T1 (ξ) evh(ξ) χ ˜1,q−1/2 = T1 (ξ)

(⇒)

1

2 Y+ 1 (ξq ) +

P1 (ξ) 1 + Y1 (ξq − 2 )

= T1 (ξ)

(5.24)

This interpretation is completely in parallel with the construction of GΓ (Cx )-group element in [23] and becomes the one in the classical limit q → 1.


45

The equation (5.13) is the multiplicative q-version of the equation a·Λ∨ = Λ∨ relating the fundamental coweights and the simple coroots. The higher rank gΓ requires a generalization of (5.13). The multiplicative q-version of the equation X aij Λ∨j = αi∨ j

for the inverse Cartan matrix is 1

1

sik (q 2 ξ)sik (q − 2 ξ) Q = j:hiji=1 sjk (ξ)

( 1 − 1ξ , i = k 1, i 6= k

(5.25)

Then sik (ξ) is the q-multiplicative decomposition of the fundamental coweight Λ∨k over the basis of simple coroots αi∨ . Let a(q) be the q-Cartan matrix defined by replacing each entry of the Cartan matrix by its q-number, and a˜(q) its inverse: X a(q)ij a˜(q)jk = δik . (5.26) a(q)ij = [aij ]q , j

Then, expanding the defining equation (5.25) in power series at ξ = ∞ we find ! ∞ X ξ −n a ˜ij (q n ) (5.27) sij (ξ) = exp − n n=1 and for a set Mj = {µj,f } define

sij (ξ|Mj ) =

Y

sij (ξ/µf )

(5.28)

µf ∈Mj

˚∨ (ξ)i ∈ Uaff (gΓ ) Consequently, the generalization of (5.15) to the q-coweight operator ψ q is ! ∞ −n X ξ ∨ n ˚ (ξ) = exp − ψ a ˜ij (q )hj,n (5.29) i [n]q n=1 The generalization of the formula (5.20) to the element h(ξ) ∈ Uaff q (gΓ ) is Y −ã h ˚i (ξ|Ξi )/ψ ˚∨ (ξ|Mi ) qi ij j,0 ψ h(ξ) = i

(5.30)

i∈IΓ

+ where Ξi is the set of zeroes of Q+ i (ξ) and Mi is the set of zeroes of Pi (ξ). The gauge theory q-character equations are written using the twisted q-characters χ˜i,ζ for the fundamental Uaff q (gΓ ) modules Vi,ζ

χ˜i,ζ = ci (ξ/ζ)χq (Vi,ζ ) where the scalar factor is ci (ξ) =

Y

qa˜ij sij (ξ|Mj )

j

We summarize the first interpretation by the proposition

(5.31) (5.32)

46


Proposition. The element h(ξ) ∈ Uaff q (gΓ ) that encodes by (5.30 (5.12)) the gauge theory chiral ring generating functions Y+ i (ξ) satisfies the equations i ∈ IΓ

evh(ξ) χ˜i,ζ = Ti (ξ),

(5.33)

aff where χ ˜i,ζ is the twisted Uaff q (gΓ ) q-character of the fundamental Uq (gΓ )-module Vi,ζ defined by (5.31 (5.32)) and (A.54), and Ti (ξ) ∈ C[ξ −1 ] is a polynomial in ξ −1 of degree vi .

6. Examples and discussions 6.1. Finite Ar quiver. For gΓ = Ar the system of r q-difference equations (4.10) can be reduced to a single q-difference equation of order r + 1. This reduction is the q-analogue of the expansion of the characteristic polynomial of an sl(r + 1) matrix in the fundamental characters. Explicitly, we obtain r+1 X

[r+1−k] (−1)k Y1,k Tk,(k−1)/2

[k−j]

Pj,j/2 = 0

(6.1)

j=1

k=0

where

k−1 Y

[k] fi,j

=

k Y

fi,j+k−1,

f = Y, P

(6.2)

j=1

Now, in terms of Baxter functions (3.13) Yi,j =

Qi,j− 1 2

(6.3)

Qi,j+ 1 2

the q-determinant equation (6.1) reduces to the linear degree r +1 q-difference equation on Q1 (ξ). r+1 k−1 X Y [k−j] k (−1) Q1,k− 1 T k−1 (6.4) P j =0 2

k=0

k,

2

j=1

j,

2

The above is known as the first equation in the hierarchy of QT-system for the integrable Ar spin chain with inhomogeneous parameters encoded in the roots of Pi (ξ).

br−1 , we construct 6.2. A∞ quiver. For gΓ = A∞ , serving as the universal cover of A the q-character of quantum affine algebra Uaff q (A∞ ) using the same requirement (cancellation of poles) and from (3.42) arrive to the formula   Y ℓλ λj +j−1 XY i+λj −j+1, 2 P[[λj ]] j Y χi = (6.5) ℓ i−ℓλ , λ Y i−j+1, λ +j 2 j 2 λ j=1 i+λj −j,

2

where the sum is over all partitions λ. In (6.5) we omitted symbol˜˜compared to (3.42), and where k=i+λj −1 Y [[λj ]] (6.6) Pi,0 ≡ Pk, 1 (k−i) 2

k=i


47

By restricting the range of indices in (6.5) one naturally recovers the characters of all fundamental modules for Uaff q (gΓ ) for gΓ = Ar with a finite r. It is useful to express the formula (6.5) in term of the variables ti (ξ) defined by ti,0 :=

Yi,0 Yi−1, 1

(6.7)

2

which are the q-versions of the GL∞ eigenvalues. We find χi = Yi−ℓλ , 1 ℓλ 2

ℓλ XY λ

[[λ ]]

j Pi−j+1, 1 ti+λj −(j−1), 1 (λj +j−1) j 2

2

j=1

(6.8)

Now we define a non-commutative equivalent of the determinant generating function for all fundamental characters of A-type quivers. For j ∈ 12 + Z≥0 define →

j Pi,0

→ 1 2

≡ Pi,0 Pi+1, 1 · · · Pi+j− 1 , 1 (j− 1 )

←

2

2 2

2

(6.9)

← 1 2

j ≡ Pi,0 Pi−1, 1 . . . Pi−(j− 1 ), 1 (j− 1 ) Pi,0 2

2

2

2

where the polynomial Pi,0 has been factorized into a product of two polynomials ← 1 2

→ 1 2

(6.10)

Pi,0 = Pi,0 Pi,0

To get the canonical (minimal degree equation) for Ar theory we set i to be the vertex ← 1 2

with the maximal vi and split arbitrarily Pi,0 into the polynomials Pi,0 of degree vi −vi−1 ← 1 2

and Pi,0 of degree vi − vi+1 (c.f. the degree profile in section 7.1.1 of [23]). Then we consider the operator ←

→

(6.11)

Θi = (Θi Yi,0 Θi ) where ← Y

←

Θi =

1−D

− 21

←

j Pi,0

j∈ 21 +Z≥0 →

Θi =

→ Y

Yi−j− 1 ,− 1 + j 2

4

2

Yi−j+ 1 , 1 + j 2 4

1 2

→

j 1 − D Pi,0

j∈ 12 +Z≥0

2

Yi+j+ 1 ,− 1 + j 2

4

2

Yi+j− 1 , 1 + j 2 4

D

2

D

1 2

− 21

!

! (6.12)

and D is the shift operator D = q ξ∂ξ

(6.13)

48


so that Dfi,j D −1 = fi,j−1 where f = Y, P. The notation for the non-commutative products over an ordered index set J is → Y fj = . . . f• f•+1 . . . j∈J ← Y

(6.14)

fj = . . . f•+1 f• . . .

j∈J

The operator Θ, like the A∞ determinant or lattice theta-function, is the generating function in the operator variable D of all fundamental q-characters: ! ← −k Y X j− 21 k (−1) Pi, 1 −k−j χi+k,− k D k Θi = j=1

k0

!

χi+k,− k D k (6.15) 2

Our gauge theory equations state that the characters χi (ξ) are the (Laurent) polynomials Ti (ξ) in ξ. We call Ti (ξ) the Coulomb polynomials. After substitution χi (ξ) = Ti (ξ) the operator version of the spectral curve is Θi |Qi (ξ)i = 0

(6.16)

b∗r quiver using periodicity modulo r + 1 Remark. If A∞ q-characters is specialized to A than the polynomials Pi (ξ) are of degree zero, i.e. Pi (ξ) = q′′i where qi is a certain complex number made from coupling constants qi , masses µe and q using (3.26)(3.37). The q-characters χi are defined for i ∈ Z, but there are only r + 1 functionally independent ones. 6.2.1. Details on the A1 quiver. The operator Θ1 is ! ← Θ1 =

1−D

− 21

1 2

P1,0 Y 1 1 D

− 12

1, 2

and its expansion is

← 1 2

Θ1 = P1, 1 D 2

−1

1

→ 1 2

1 − D 2 P1,0 Y 1 1 D

Y1,0

1, 2

→ 1 2

+ χ1,0 − P1,− 1 D

1 2

!

(6.17)

(6.18)

2

In the equation Θ1 |Q1 i = 0

(6.19)

we recognize the Baxter equation → 1 2

← 1 2

− P1, 1 Q1,1 + T1,0 Q1,0 − P1,− 1 Q1,−1 = 0 2

(6.20)

2

→ 1 2

← 1 2

where the polynomials P1, 1 , P1,− 1 and T1,0 would be conventionally called A, D and T 2

2

Baxter polynomials. The equation (6.20) is the second order linear difference equation


D

(0)

(1)

49

E

of two independent solutions. We choose one solution that it with a basis Q1 , Q1 is annihilated the last factor of (6.17). Then we find Q1,− 1 2

=

Q1,+ 1

Yi,+ 1 2

(6.21)

→ 1 2

P1,0

2

so that Q1 is Q1 of (3.13) up to a factor of γ

→ 1

.

2 q,P1,0

Here the function γq;P (ξ) is a generalization of q-Gamma function, it is determined by the polynomial P(ξ) as the solution of the q-difference equation 1

γq,P (q 2 ξ) γq,P (q

− 12

ξ)

=

1 P(ξ)

(6.22)

6.2.2. Details on the A3 = D3 and Ar quivers. The new feature of the Ar quivers with r ≥ 3 is the presence of the interior nodes. To be specific, consider the A3 quiver with the maximal rank at the middle node i = 2. The q-character χi,0 can be displayed by the Yi,0 i

Pi, 1

2

Yi−1, 1 Yi+1, 1 2

2

Yi,1

i+1 Yi+1, 1

diagram

Pi, 1 Pi−1,1 Y 2

i−1

Yi−1, 1

Pi, 1 Pi+1,1 Y

2

3 i−1, 2

2

i+1

2

i+1, 3 2

i−1 Y Pi, 1 Pi−1,1 Pi+1,1 Y 3i,1 Yi+1, 3 2 i−1, 2

2

i

Pi, 1 Pi, 3 Pi−1,1 Pi+1,1 Y1i,2 2

2

where each node denotes a term in the character and each arrow with a label j denotes the q-Weyl reflection in the quiver node j. The character χ2 for A3 theory is also the fundamental character of the vector representation for the D3 theory. The q characters χi−1, 1 and χi+1,− 1 are 2

2

50


Yi+1,− 1

Yi−1, 1 2

2

i−1

Pi−1,1 Y

i+1

Yi,1

Pi+1,0 Y

3 i−1, 2

i

Pi−1,1 Pi, 3

Yi,0 i+1, 1 2

i Yi+1, 3

2

Pi+1,0 Pi, 1

2

Yi,2

Yi−1, 1 2

Yi,1

2

i+1

i−1

Pi−1,1 Pi, 3 Pi+1,2 Y 2

1

Pi+1,0 Pi, 1 Pi−1,1 Y

i+1, 5 2

2

1 3 i−1, 2

Above we have defined the q-determinant operator Θi ! ! ← ← 3

1

1

1

1 2 D− 2 1 − D − 2 Pi,0 Yi−1,1

Θi =

1

2 1 − D − 2 Pi,0

Yi−1,0 − 1 D 2 Yi, 1 2

→ 1 2

1 2

1 − D Pi,0

Yi+1,0 D Yi, 1

1 2

2

!

Yi,0 1 2

→ 3 2

1 − D Pi,0

1 Yi+1,1

D

1 2

!

(6.23)

and its expansion (6.15) over the fundamental q-characters is ← 3 2 i, 21

← 1 2 i, 23

Θi = P P D

−2

← 1 2 i, 12

− P χi−1, 1 D 2

−1

→ 1 2

→ 1 2

→ 3 2

+ χi,0 − Pi,− 1 χi+1,− 1 D + Pi,− 1 Pi,− 3 D 2 2

2

2

(6.24)

2

The operator Θi annihilates the function Qi (ξ) D

(s)

Let Qi

E

s∈(0,1,2,3) (3) and let Qi

theory, satisfies equation

Θi |Qi i = 0

(6.25)

be the basis of solutions for the D-spectral equation (6.16) for A3 (3)

be solution annihilated by the last factor in the (6.23). Then Qi (3) Qi, 1 2

→ 3 2

− Pi,0

1

(3)

Yi+1,1

(6.26)

Qi,− 1 = 0 2

as we can see from the last factor of (6.23). For a generic Ar quiver the corresponding equation on Qr is → 1 1 +r−i 2 Q 1 =0 Qr, 1 − Pi,0 (6.27) 2 Yr, 1 (1+r−i) r,− 2 2

or, equivalently Qr,− 1 2

Qr,+ 1 2

=

Yr, 1 (1+r−i) 2

(6.28)

→ 1 +r−i 2

Pi,0

→ 1 +r−i 2

in the We conclude, that up to a product of γq factors coming from the Pi,0 above equation and a simple shift, the function Qr (ξ) is the function Qr (ξ) (see (3.13))


51

associated to the right terminal node r of the linear quiver Qr = γ

→ 1 +r−i 2 q,Pi,0

(6.29)

Qr, 1 (1+r−i) 2

6.3. From Baxter equation to Y -functions from D-Wronskian. . Here we will recover relation to the Y and Q functions of (3.13) from the solutions to the D-spectral equation (6.16), see [38, 89] for details. Define operator Lk to be the product of k right factors in the Θi . For k ≤ r − i + 1 we find ! r−i+ 21 → Y Y 1 1 1 1 1 i+j+ ,− + j j 2 4 2 D2 Lk = (6.30) 1 − D 2 Pi,0 1 1 1 Y i+j− , + j 1 j=r−i+ 2 −(k−1)

D E (r−j;k) Let Qi

j∈{0,1,...,k−1}

2 4

2

be the basis in the kernel of Lk (r−j;k)

k−1 ker Lk = ⊕j=0 CQi D E (r−j;k) Let Wk be the D-Wronskian built on functions Qi

(6.31) j∈{0,1,...,k−1}

(r−j;k) Wk = det Qi,j ′ j∈{0,1,...,k−1}

(6.32)

j ′ ∈{0,1,...,k−1}

Expand Lk in the powers of D Lk = 1 +

k−1 X

cj D j +

j=1

+

(−1)

k

Yr−k+1, 1 (r−i−k+1) 2

 

r−i+ 12

Y

j=r−i+ 23 −k



→

j  D k (6.33) Pi,−j+(r−i−(k−1)) (r−j;k)

where cj are certain polynomials in Y and P. The equations (Lk Qi = 0)j∈{0,1,...,k−1} together with determinant definition (6.32) implies     r−i+ 21 → Y 1 j   D  Wr−k+1 = 0 1 − Pi,−j+r−i−k+1 (6.34) Yr−k+1, 1 (r−i−k+1) 3 2

j=r−i+ 2 −k

so that finally

Yr−(k−1), 1 (r−i−(k−1)) Wr−(k−1),−1 2 =Q → r−i+ 21 Wr−(k−1),0 j P 3 j=r−i+ −k i,−j+(r−i−(k−1)) 2

(6.35)

52


For k > r − i + 1 we find k−1−r+i Y

Lk = (−1)r−i−k+1 Yr−k+1,− 1 (r−i−k+1) 2

← j− 21

Pi,k−r+i−j− 1 2

j=1

r−i X

+

!

cj D j +

j=r−i−k+2



r−i+ 21

Y

(−1)r−i+1  (r−j;k)

The equations (Lk Qi

D r−i−k+1 +

j= 21

→



j  Pi,−j D r−i+1 (6.36)

= 0)j∈{0,1,...,k−1} and the determinant definition (6.32) imply

Wr−k+1,k−r+i−2 = Wr−k+1,k−r+i−1

Qk+i−r−1

Yr−k+1,− 1 (r−i−k+1)

j=1

2

Qr−i+ 21 j= 21

→

j Pi,−j

←

j− 21

Pi,k− 1 −r+i−j 2

!

(6.37)

or, equivalently,

Wr−k+1,−1 = Wr−k+1,0

Yr−k+1, 1 (r−i−k+1) 2

Qr−i+ 21 j= 21

Qk−r−1+i j=1

→

←

j− 21

Pi,−j+ 1

j Pi,−j+r−i−k+1

2

!

(6.38)

This formula (6.35) and (6.38) together with the D-Wronskian definition (6.32) allows us to compute all functions Yj from the complete basis in the ker Θi .

b∗0 quiver. Consider the type II* theory corresponding to the A b0 quiver, i.e. the 6.4. A ∗ N = 2 theory with the gauge group SU(N), where N = v0 . There is one fundamental q-character in this case: χ(ξ) = Y(ξ)

X λ

=

X λ

q

|λ|

q|λ| Q

Q

Y Y(σ µξ)Y(σ µ−1 ξ) 1

1

∈∂+ λ

∈∂− λ

1

Y(σ q 2 ξ)Y(σ q − 2 ξ)

∈λ

Y(σ q 2 ξ)

Y(σ q

− 21

ξ)

,

(6.39)


53

Where for = (i, j) ∈ λ, 1−i−j

σ = µj−i q 2 , and X 1 1 σ q 2 + q − 2 − µ − µ−1 = 1− ∈λ

X

=

X

1

σ q 2 −

∈∂+ λ

(6.40) 1

σ q − 2

∈∂− λ

The outer and inner boundaries ∂± λ correspond to the generators and relations of the monomial ideal Iλ corresponding to the partition λ. Let us now present the quantization of the curve R(t, x) = 0 found in [23] for this theory. Introduce the notation (not to be confused with (4.5)): yi,j = Y(ξq −

i+j 2

µj−i ),

i, j ∈ Z

(6.41)

Then (6.39) can be rewritten in the form: χ=

X

|λ|

q yℓλ ,0

j=1

λ

Now let us introduce the operator

ℓλ Y yj−1,λj

(6.42)

yj,λj

U = q −ξ∂ξ

(6.43)

Uyi,j U −1 = yi+1,j+1

(6.44)

such that Introduce also the notations y−1,j gj = , y0,j

g˜j =

yj,−1 , yj,0

Let us also introduce the shift operators −ξ∂ξ ± 21 , U± = µq

j∈Z

U+ = UU−

(6.45)

(6.46)

which commute with U. The quantities gj , g˜j can be written also in the following form: gj = U−−j g0 U−j ,

g˜j = U+j g˜0 U+−j

(6.47)

Using U, g, g˜ we can present χ in a more suggestive form: χ=

X λ

=

|λ|

q yℓλ ,0

X

ℓλ Y

U i gλi −i U −i

i=1

q|λ| yℓλ ,0 Ugλ1 −1 Ugλ2 −2 . . . Ugλℓλ −ℓλ U −ℓλ

λ

=

X λ

q

ℓ2 λ 2

−→ Yℓλ

1

qλi −i+ 2 Ugλi −i

i=1

= CoeffU 0 Θ(ξ, q; U)

!

y0,−ℓλ U −ℓλ

(6.48)

54


with the q-vertex operator Θ, (6.49)

Θ(ξ, q; U) = Θ− Y(ξ)Θ+ whose factors are given by the ordered infinite products: ←− Y r Θ− = 1 + q Ugr− 1 2

r∈Z≥0 + 12 −→ Y

Θ+ =

r∈Z≥0 + 12

The notations

←− Y

r

1 + q g˜r− 1 U 2

2

(6.50)

−→ Y

,

for the noncommutative products read as follows: ←− −→ Y Y ar = . . . a 5 a 3 a 1 , 2

−1

2

r∈Z≥0 + 12

˜ 3 a˜ 5 . . . a˜r = a ˜1a 2

2

2

(6.51)

r∈Z≥0 + 21

The vertex operator Θ is the noncommutative analogue of the theta-product: X n2 Y (1 + tqr ) 1 + t−1 qr (6.52) ϑ(t; q) ≡ q 2 tn = φ(q) r∈Z≥0 + 12

n∈Z

Just like the theta-function ϑ(t; q), the operator Θ can be expanded as a U-series: X n2 Θ(ξ, q; U) = q 2 χn (ξ) U n (6.53) n∈Z

where

1

χn (ξ) = χ(ξµn− ), µ± = µq ± 2 ˜ Now let us define the function Q(ξ) by the property: 1 1 −1 ˜ =0 ˜ ˜ 1 + q 2 Ug0 y0,0 Q ⇔ q 2 Q(ξq )Y(ξµ− ) = −Y(ξ)Q(ξ)

(6.54)

(6.55)

It is related to the Q-function

Y(ξ) = in an obvious manner. Define

Q(ξ) Q(ξq −1 )

˜ (ξ) Q(ξ) = Θ−1 Q +

(6.56)

(6.57)

Then Θ(ξ, q; U)Q(ξ) = 0, which is an infinite order linear difference equation: X n2 (6.58) q 2 χ(ξµn− )Q(ξq −n ) = 0 n∈Z

which is an analogue of the Hirota difference equation [201]. We discuss the fourdimensional limit of the equation (6.58) in the section below.


55

7. Four dimensional limit and the opers. 7.1. The A1 case. The q-character equation (4.2), up to non-essential ǫ shifts in the variable x, assumes the following form in the four dimensional limit: ˚ P(x) ǫ = (1 + q)T (x) Y x+ +q (7.1) 2 Y x − 2ǫ

with the degree Nf = 2N monic polynomial ˚ P(x) and degree N monic polynomial T (x). ˚ The roots of polynomial P(x) are the masses of the fundamental hypermultiplets, while the polynomial T (x) T (x) = xN + u1 xN −1 + u2 xN −2 + . . . + uN

(7.2)

determines the vacuum of the theory. Now let us factorize the fundamental polynomial ˚ P(x) = A(x − ǫ)D(x), (7.3) where A(x), D(x) are degree N monic polynomials. Obviously, there are many representations like (7.3). Define Baxter function Q(x) to be an entire function of x, such → 1 2

that (c.f. with (6.21) where the partial matter polynomial A(x) corresponds to P1,0 (ξ)) ǫ Q x + 2ǫ (7.4) Y(x) = A x − 2 Q x − 2ǫ

Comparing with (3.12) we find

A(x)

Q(x + ǫ) Q(x + ǫ) = Q(x) Q(x)

(7.5)

therefore

Q(x) ΓA;ǫ (x) where for a monic polynomial A(x) the ΓA;ǫ (x) function satisfies Q(x) =

ΓA;ǫ (x + ǫ) = A(x)ΓA;ǫ (x)

(7.6)

(7.7)

Up to an ǫ-periodic function the ΓA;ǫ (x) function is a product of ordinary Γǫ -functions N Y

Γǫ (x − mf )

(7.8)

f =1

its inverse is an entire function with zeroes at mf − ǫZ≥0 . The zeroes ξa,i of Q(x) behave as ξa,i = aa + ǫ(i − 1) + xa,i , xa,i ∝ qi → 0, i → ∞ (7.9) The factor ΓA;ǫ (x) adds a string of zeroes going in the opposite direction mf − kǫ,

k≥0

(7.10)

If ma − aa ∈ ǫZ≥0 , then an infinite string of zeroes ma + ǫZ going in both directions can be removed from Q(x) by factoring out the periodic function sin πǫ (x − ma ), this is the case studied in [30, 31], in which Q(x) becomes polynomial.

56


The equation (7.1) becomes the celebrated T Q-relation (c.f. eq. (6.20): A(x)Q(x + ǫ) + qD(x)Q(x − ǫ) = (1 + q)T (x)Q(x) Now define: Ψ(t) =

X

Q(x)t−x/ǫ

(7.11) (7.12)

x∈Γ

where Γ ⊂ C is some lattice, invariant under the shifts by ǫ: Γ + ǫ = Γ. Then (7.11) implies: A(−ǫt∂t )t + qD(−ǫt∂t )t−1 − (1 + q)T (−ǫt∂t ) Ψ(t) = 0 (7.13) i.e. Ψ is a solution of the N-order differential equation with 4 regular singularities t = 0, q, 1, ∞. The monodromy of (7.13) around t = 0 and t = ∞ has N distinct eigenvalues ± e2πımi /ǫ , i = 1, . . . , N, where m± are the roots of D(x) and A(x), respectively. The monodromy of (7.13) around t = q, 1 has N − 1 eigenvalues equal to 1, and one ± non-trivial eigenvalue e2πıµ /ǫ , respectively, where µ± can be computed from the U(1) Coulomb moduli u1 (7.2) and m± i . 7.2. Other examples. 7.2.1. q-characters for the A2 theory. We have two Y-functions and two matter polynomials P1,2 (x), − Pi (x) = (x − m+ (7.14) i )(x − mi ) . The equations determining the vacua are: P1 (x − ǫ)P2 (x − 32 ǫ) ǫ P1 (x − ǫ)Y2 (x − ǫ) Y1 x − + q1 = + q1 q2 2 Y2 (x − 2ǫ) Y1 (x − 23 ǫ) ǫ = (1 + q1 + q1 q2 )T1 x − 2 (7.15) P1 x − 2ǫ P2 (x) P2 (x)Y1 (x) ǫ + q1 q2 + q2 = Y2 x + 2 Y1 (x − ǫ) Y2 x − 2ǫ ǫ = (1 + q2 + q1 q2 )T2 x + 2 where T1,2 (x) are the monic degree 2 polynomials: Ti (x) = x2 − ui,1x − ui,2,

i = 1, 2

(7.16)

The parameters ui,1 are the U(1) Coulomb moduli, they are determined by the masses of the fundamental hypermultiplets and the mass of the bi-fundamental hypermultiplet. Shift the argument of the first equation in (7.15) as x 7→ x + ǫ/2, multiply the result by q2 P2 (x)/Y2 x − 2ǫ and subtract the result from the second equation in (7.15) to obtain P2 (x)T1 (x) ǫ − + (1 + q1 + q1 q2 )q2 Y2 x + 2 Y2 x − 2ǫ (7.17) ǫ ǫ 2 P1 x − 2 P2 (x − ǫ)P2 (x) = (1 + q2 + q1 q2 )T2 x + − q1 q2 2 Y2 x − 3ǫ2 Y2 x − 2ǫ


Now, write (cf. (6.28))

ǫ Q(x + 2ǫ ) Y2 (x) = P2 x + 2 Q(x − 2ǫ )

57

(7.18)

to reduce (7.17) to the linear equation (cf. (6.15), (6.16)): P2 (x + ǫ)Q(x + ǫ) − (1 + q2 + q1 q2 )T2 (x + 21 ǫ)Q(x) + q2 (1 + q1 + q1 q2 )T1 (x)Q(x − ǫ) − q1 q22 P1 (x − 21 ǫ)Q(x − 2ǫ) = 0 (7.19) Now let us make a Fourier transform: Ψ(t) =

X

Q(x)t−x/ǫ

(7.20)

x

where the sum goes over some lattice x ∈ a + Zǫ. Now the substitution: eǫDx 7→ t

x 7→ −ǫtDt ,

(7.21)

maps the Eq. (7.19) to the second order differential equation, obeyed by the function Ψ(t) 3 t P2 (−ǫtDt ) − (1 + q2 + q1 q2 )t2 T2 (−ǫtDt − 2ǫ )+ (7.22) +tq2 (1 + q1 + q1 q2 )T1 (−ǫtDt − ǫ) − q1 q22 P1 (−ǫtDt − 23 ǫ)) Ψ(t) = 0 The symbol of the left hand side vanishes at the zeroes of

t3 − (1 + q2 + q1 q2 )t2 + q2 (1 + q1 + q1 q2 )t − q1 q22 = (t − 1)(t − q2 )(t − q1 q2 ) (7.23) There are, then, five regular singularities, at t = 0, q1 q2 , q2 , 1, ∞. Near t = 0 the dominant term is q1 q22 P1 (−ǫtDt − 23 ǫ), the eigenfunctions of the monodromy of the Eq. (7.22) around t = 0 are ±

3

± ψ(0) ∼ t−m1 /ǫ− 2 (1 + O(t))

(7.24)

and the corresponding eigenvalues are ±

3

e−2πı(m1 /ǫ+ 2 )

(7.25)

Analogously, near t = ∞ the dominant term is t3 P2 (−ǫtDt ), and the eigenfunctions are ± ± ψ(∞) ∼ t−m2 /ǫ 1 + O(t−1) (7.26) with the corresponding eigenvalues

±

e−2πım2 /ǫ

(7.27)

Near the points t = zi , z1 = q1 q2 , z2 = q2 , z3 = 1 the equation (7.22) has one singular solution (t − zi )−µi /ǫ (7.28) and one non-singular solution. The monodromy around these points has, therefore, one trivial eigenvalue 1 and one non-trivial eigenvalue e−2πıµi /ǫ ,

i = 1, 2, 3

(7.29)

58


The exponents µi are determined by the U(1) Coulomb moduli and the masses, and in fact can be used as convenient parameters instead of u1,1 , u2,1, subject to the obvious relation 3 X − + − µi = m+ (7.30) 2 + m2 − m1 − m1 − 3ǫ i=1

The redefinition

Ψ(t) = t

−m+ −m− −3ǫ 1 1 2ǫ

3 Y µi (t − zi )− 2ǫ χ(t)

(7.31)

i=1

maps (7.22) to the sl2 oper (c.f. [202–208]) 2 2 ǫ ∂t + T (t) χ = 0

(7.32)

where

T =

3 X i=0

ci ∆i + (t − zi )2 t − zi

z0 = 0, z1 = q1 q2 , z2 = q2 , z3 = 1, z4 = ∞

(7.33)

Remark. Observe that the equation obeyed by Ψ(t) is the ǫ2 → 0 limit of the equation obeyed by the degenerate conformal block of AN −1 Toda conformal field theory, with b2 = ǫ2 /ǫ1 , corresponding to the 5-punctured sphere. According to the AGT correspondence this is a Z-partition function of the linear quiver theory of type A2 . It should be interesting to understand the relation between Ψ(t) and Z(a, m; q, t, ǫ) in more general context. See [209] for some developments in this direction. b0 case. We claim that in the four dimensional limit the equation 7.2.2. Type II* A (6.58) can be transformed to an interesting differential equation. Recall that our master equations state that χ(ξ) = T (ξ) where T (ξ) is a (Laurent) polynomial of degree N for the v0 = N theory. In the four dimensional limit, T is a degree N polynomial in x (recall that ξ = eıℓx ): 1 T (x) = xN + u1 xN −1 + . . . + uN (7.34) φ(q) where we used the standard notation ∞ Y φ(q) = (1 − qn ), n=1

X 1 q|λ| = φ(q) λ

The equation (6.58) adapts to, again in the four dimensional limit, X n2 T (x + nm− )Q(x − nǫ)q 2 = 0 , n∈Z

where

m± = m ±

ǫ 2

(7.35)


Now let us perform the Fourier transform: Ψ(t) = ϑ(t; q)−

m+ ǫ

u1

t− Nǫ

X

x

t ǫ Q(x)

59

(7.36)

x∈a+Zǫ

assuming this sum converges. Then (7.35) becomes the differential equation: DΨ(t) = 0 where

u1 1 X n2 n 2 D= q t T ǫt∂t − + (n − t∂t logϑ(t; q))m+ ϑ(t; q) n∈Z N N

= (ǫt∂t ) + v2 (t)(ǫt∂t )

N −2

(7.37)

+ . . . + vN (t)

is the N-order differential operator on the torus Eq = C× /qZ with one puncture, which 1 is at t = −q 2 in our normalization. The coefficients vi (t) are q-periodic, vi (qt) = vi (t), 1 and have the i’th order pole at t = −q 2 . In line with [8, 9] we should view D as the pglN -oper. Example. Let us now consider the small N cases. For N = 1 the Eq. (7.37) reads: ǫt∂t Ψ = 0 =⇒ Ψ(t) = const

(7.38)

For N = 2 the Eq. (7.37) is equivalent to:

(ǫt∂t )2 Ψ(t) + m+ m− (t∂t )2 logϑ(t; q) + u2 − u21 /4 Ψ(t) = 0

(7.39)

7.2.3. D and E quivers. For the gΓ = Dr , the quantum determinant equation, or the linear QT-equation like (6.1) was found in [210], but for the exceptional series E6 , E7 , E8 the linearization problem seems to be open. The Er -series (q, t)-characters were computed by Nakajima [133] on supercomputer using sophisticated optimization. Appendix A. Quantum affine algebras A.1. Quantum affine algebra. In this section we collect the conventions and the definitions related to quantum affine algebras [101, 116] and the q-characters [49]. A.1.1. Conventions. Literature differs in the conventions for the q-parameter of quantum groups. The parameter q of the present note relates to the definitions of qcharacters by Frenkel and Reshetikhin [49] and Drinfeld [116] and Jimbo [101] and [211] 2 eıℓǫ1 = qpresent = q[116] = eh[116] = t4[101] = q[49] (A.1) where ℓ is the circumference of the circle S1 in the compactification of the five dimen4 ×S1 sional theory on the twisted bundle R^ hǫ1 ,ǫ2 ;ℓi . Our multiplicative spectral parameter ξ corresponds to the variable z of Nakajima [131] which corresponds to the variable u−1 of Frenkel and Reshetikhin [49], see more details in the appendix ξpresent = u−1 (A.2) [49] = z[130] We mostly use the conventions of [49, 109, 131], with the dictionary of conventions (A.1)(A.2).

60


Let aij = 2δij − Iij , where Iij is the incidence matrix, be the Cartan matrix gΓ . The indices i, j ∈ IΓ label the nodes of the Dynkin diagram Γ. We restrict our definitions to simply laced ADE Lie algebras gΓ . Let (αi )i∈IΓ denote the simple roots. Then aij = (αi , αj ) where (, ) is the standard bilinear form on the dual Cartan algebra in normalization where the squared length simple roots is 2. By [n]q we denote a q-number, [n]q =

n

n

1

1

q 2 − q− 2 q 2 − q− 2

(A.3)

,

by [n! ]q the q-factorial, and by

s

k q

[n! ]q = [n]q [n − 1]q . . . [1]q ,

(A.4)

the q-binomial coefficient: [s! ]q s = k q [(s − k)! ]q [k! ]q

(A.5)

A.1.2. Chevalley-Drinfeld-Jimbo construction. . Let g be a generalized Kac-Moody algebra with symmetrizable Cartan matrix a ˜ij . In particular, g could be a finitedimensional simple Lie algebra of affine Kac-Moody Lie algebra. Then we can define the quantum algebra Uq (g) by a q-deformation of Serre relation for Chevalley generators of g [50, 101, 116, 211]. If gΓ is finite-dimensional, then applying such construction for g = gbΓ we obtain quantum affine algebra Uq (g) = Uq (gbΓ ) ∼ = Uaff q (gΓ ). To simplify presentation, we assume that g is simply laced (A.6)

aij = aji

The algebra Uq (g) is an associative algebra generated by the Chevalley generators − (hi , e+ i , ei )i∈Ig , called the coroots, the raising and the lowering elements, respectively, where i runs over the set of nodes Ig on Dynkin graph of g. The relations are the q-deformation of the Chevalley relations3 hi

hi

1

−2 − = q 2 aij e± [e+ hi hj = hj hi , q 2 e± j q j , i , ej ] = δij [hi ]q 1−aij X 1−aij −r ± ± r r 1 − aij (e± ej (ei ) = 0, i 6= j (−1) i ) r q r=0

(A.7)

The co-multiplication ∆ : Uq (g) → Uq (g)4, the antipod γ : Uq (g) → Uq (g) and the counit ε : Uq (g) → C are given by hi

hi

hi

hi

+ + 2 ⊗ e , ∆(e+ i ) = ei ⊗ 1 + q i

∆(q 2 ) = q 2 ⊗ q 2 , hi

+ −2 γ(e+ , i ) = −ei q

ε(e± i ) = 0,

hi

− 2 γ(e− i ) = −q ei ,

hi

− −2 ∆(e− + 1 ⊗ e+ i ) = ei ⊗ q i

q hi = 1

hi

ε(q 2 ) = 1 (A.8)

3Here

1

1

1

q 2 hi = ki |[50] = (ki2 )[101, 211] = (e 2 h¯ hi )|[116] and q 2 = q[50] 4Different sign choices are possible and different conventions exist in the literature


61

The Borel subalgebra Uq (g± ) ⊂ Uq (g) is a subalgebra generated by the generators (hi , e± i ) in this realization (A.7). A.2. Drinfeld current realization. In the Drinfeld current realization [47, 109], applicable for arbitrary symmetrizable Kac-Moody gΓ , the quantum affine algebra Uaff q (gΓ ) is generated by the loop coroots (hi,n )i∈IΓ ,n∈Z , loop raising and lowering generators (x± i,n )i∈IΓ ,n∈Z , the center c and the energy d, where i ∈ IΓ runs over the nodes of gΓ , with commutation relations ([49, 50, 134])5 1 ∓c|n|/4 ± [hi,n6=0 , e± ej,n+m j,m ] = ± [naij ]q q n 1 [hi,0 , hj,0 ] = 0, [hi,n6=0 , hj,m] = δn+m,0 [naij ]q [nc]q n (n−m)c/4 + −(n−m)c/4 − q ψi,n+m − q ψi,n+m − [e+ 1 1 i,n , ej,m ] = δij − q2 − q 2 ± [hi,0 , e± j,n ] = ±aij ej,m ,

1

1

± ± 2 aij ± ± ± ± ± e± ei,n ej,m+1 = q ± 2 aij e± i,n+1 ej,m − q j,m ei,n+1 − ej,m+1 ei,n a XX ± ± ± ± k a e± (−1) i,nσ(1) . . . ei,nσ(k) ej,m ei,nσ(k+1) . . . ei,nσ(a) = 0, k q σ∈S a

(A.9) (A.10) (A.11) (A.12)

(i 6= j, a = 1 − aij )

k=0

± where (ψi,n )i∈IΓ ,n∈Z≥0 are loop modes 6 of the current ψi± (ξ) ∈ Uaff q (gΓ ) ! ∞ ∞ X X 1 1 1 ± ∓n hi,±n ξ ∓n ψi± (ξ) := ψi,n ξ := q ± 2 hi exp ±(q 2 − q − 2 )

(A.13)

(A.14)

n=1

n=0

The level zero specialization Uaff q (gΓ )|c=0 is a specialization to the center c = 0. At c = 0 the operators (hi,n )i∈IΓ ,n∈Z generate the commutative subalgebra of Uaff q (gΓ )|c=0 , ± aff and the currents ψi (ξ) are commutative. In the limit q → 1, the Uq (gΓ )|c=0 becomes algebra U(LgΓ ) where LgΓ is the loop algebra of gΓ .

A.2.1. Drinfeld currents. We can write the defining relations of Uaff q (gΓ ) in terms of ± currents xi (ξ), see [134] and [121], X −n e± (ξ) = e± (A.15) i i,n ξ n∈Z

In terms of the currents (ψi± (ξ), e± i (ξ))i∈IΓ , the relations (A.9), (A.10), (A.11) are 7 respectively packed into 5Here

1

1

q 2 = q[49, 50, 134] and q 2 c = c[50] and hi,n q −c|n|/4 = ai,n |[121] −1 −1 6Here ξ = z [121] = u[49, 50] = z[134] 7It is convenient to compute [log ψ + (ξ), e± (ξ ′ )] from (A.9) and then use the fact that for any i j operators h, x, the commutator [h, x] = λx, where λ is a number, implies eh xe−h = eλ x. If operators a, b commute as [a, b] = λ, where λ is a number, then ea eb e−a = eλ eb

62


1

′ ψi♯ (ξ)e± j (ξ )

=q

1−

♯ ′ e± j (ξ )ψi (ξ), ± 21 aij ∓♯ 4c ′ q q ξ /ξ

1

ψi+ (ξ)ψj− (ξ ′ )

=

c

1 − q ∓ 2 aij q ∓♯ 4 ξ ′ /ξ

± 12 aij

1

1

(1 − q

− ′ [e+ i (ξ), ej (ξ )] =

δi,j 1 2

q −q

q

1 c 2

1 a 2 ij

ξ ′ /ξ)(1

1 q − 2 c ξ ′/ξ)

ψj− (ξ ′)ψi+ (ξ);

(A.17)

−q ± ± ′ ψi (ξ)ψj (ξ ) = ψj± (ξ ′)ψi± (ξ) c

− 21

(A.16)

1

(1 − q 2 aij q 2 c ξ ′ /ξ)(1 − q − 2 aij q − 2 c ξ ′/ξ) − 21 aij

♯ ∈ {+, −}

c

(A.18) c

c

ψi+ (ξ ′ q 4 )δ(q 2 ξ ′ /ξ) − ψi− (ξ ′ q − 4 )δ(q − 2 ξ ′ /ξ) 1

± ′ e± i (ξ)ej (ξ )

=q

± 21 aij

1 − q ∓ 2 aij ξ ′ /ξ 1−

′ ± e± j (ξ )ei (ξ) ± 21 aij ′ ξ /ξ q

where δ(z) :=

X

(A.19) (A.20)

zn

(A.21)

n∈Z

and

a ± ± ′ ± ± Symξ1 ,...,ξa e± (−1) i (ξ1 ) . . . ei (ξk )ej (ξ )ei (ξk+1 ) . . . ei (ξa ) = 0, k q k=0

a X

k

(i 6= j, a = 1 − aij )

We will also use the current ˚± (ξ) ψ i

:=

∞ X

˚± ξ ∓n ψ i,n

1 2

:= exp ±(q − q

− 12

)

n=0

and

∞ X hi,±n n=1

ψi±

∈ Uq (gΓ ) and

∞ X n=1

p± i (ξ) := exp − The currents

(A.22)

p± i

[n]q

ξ ∓n

!

hi,±n ξ

∓n

!

(A.23)

(A.24)

∈ Uq (gΓ ) are related 1

ψi± (ξ)

=q

± 21 hi

2 p± i (q ξ) 1

−2 p± ξ) i (q

.

(A.25)

A.2.2. Yangian and elliptic version. Here we define the algebras Yǫ (g), Uaff q (g) and ell Uq,p (g) uniformly called Uǫ gΓ (Cx ) for a Kac-Moody Lie algebra gΓ with symmetric Cartan matrix aij using Drinfeld currents on Cx , where Cx = C for Yangian Yǫ (g), 2π Z for quantum affine Uaff Cx = C/ 2π q (g), and Cx = C/( ℓ )(Z + τp Z) for quantum ℓ elliptic Uell q,p (g). Instead of multiplicative variable ξ = eiℓx

(A.26)

we use additive variable x ∈ Cx in the domain Cx which might have 0, 1, or 2 periods.


63

The basic (quasi)-periodic function s(x), with 0, 1 or 2 periods would be given by   x, Cx = C , Cx = C/ 2π Z s(x) = 2ℓ sin ℓx (A.27) 2 ℓ  iℓx  θ1 (e ;p) , C = C/ 2π (Z + τ Z) x p ℓ ∂x θ1 (eiℓx ;p)|x=0

where θ1 (ξ; p) for p = e2πiτp is defined in (2.82). We have the hierarchy of degenerations: elliptic → trigonometric → linear p→0

ℓ→0

sC/ 2π (Z+τp Z) (x) −→ sC/ 2π Z (x) −→ sC (x) ℓ

ℓ

(A.28)

Then we would simply replace the defining relations (A.9)(A.10)(A.11)(A.12) by ′ ψi♯ (ξ)e± j (ξ )

s(x − x′ ± 21 aij ǫ ± ♯ 4c ǫ) ± ′ ♯ = e (ξ )ψi (ξ), s(x − x′ ∓ 12 aij ǫ ± ♯ 4c ǫ) j

ψi+ (ξ)ψj− (ξ ′ )

♯ ∈ {+, −}

s(x − x′ − 21 aij ǫ − 12 cǫ)s(x − x′ + 21 aij ǫ + 21 cǫ) − ′ + ψ (ξ )ψi (ξ); = s(x − x′ + 21 aij ǫ − 12 cǫ)s(x − x′ − 12 aij ǫ + 21 cǫ) j ψi± (ξ)ψj± (ξ ′ ) = ψj± (ξ ′ )ψi± (ξ)

− ′ [e+ i (ξ), ej (ξ )] =

δij ǫ + ′ c ψ (x + 4 ǫ)δ(x − x′ − 2c ǫ) − ψi− (x′ − 4c ǫ)δ(x − x′ + 2c ǫ) s(ǫ) i

(A.29) (A.30) (A.31) (A.32)

± ′ e± i (ξ)ej (ξ ) =

s(x − x′ ± 21 aij ǫ) ± ′ ± e (ξ )ei (ξ) s(x − x′ ∓ 12 aij ǫ) j

(A.33)

See more on elliptic currents in [212]. A.3. Level zero representations. A.3.1. Level zero representations and generalized weights. For illustration, we first con× sider a fundamental evaluation module L1,q− 21 µ for Uaff q (sl2 ). Let µ ∈ C be an evaluation parameter, and |ω0i, |ω1 i be basis vectors in L1,q− 12 µ so that |ω0 i is the highest vector. In this basis the currents ψ(ξ), e± (ξ) are represented by matrices ! 1 1−µq −1 /ξ 2 q 0 1−µ/ξ ψ ± (ξ) = (A.34) 1 0 q − 2 1−µq/ξ 1−µ/ξ ±

and

0 δ(µ/ξ) e (ξ) = , 0 0 +

−

e (ξ) =

0 0 δ(µ/ξ) 0

(A.35)

where symbols f (ξ)± for a function f (ξ) denotes expansions near ξ = ∞ and ξ = 0 respectively X X fn ξ n (A.36) fn ξ −n , f (ξ)− = f (ξ)+ = n≥n+

n≥n−

64


The key property of these expansions is localization to the poles, such as X X 1 1 − = ξ −n + ξ n = δ(ξ) 1 − 1/ξ + 1 − 1/ξ − n≥0 n>0

(A.37)

For example, checking the commutation relation (A.20), in the right hand side we find the matrix elements such as −1 −1 X 1 1 − µq 1 1 − µq 1 1 1 1 /ξ /ξ q2 − q2 = (q 2 − q − 2 ) (µ/ξ)n = (q 2 − q − 2 )δ(µ/ξ) 1 − µ/ξ 1 − µ/ξ + − n∈Z (A.38) so that indeed 1 1 δ(ξ/ξ ′)δ(µ/ξ) q 2 − q − 2 0 1 0 + − ′ ′ = [e (ξ), e (ξ )] = δ(µ/ξ)δ(µ/ξ ) = 1 1 1 1 0 −1 0 q− 2 − q 2 q 2 − q− 2 1 ′ + − (A.39) = 1 1 δ(ξ/ξ )(ψ (ξ) − ψ (ξ)) q 2 − q− 2 The diagonal elements in (A.34) are called the current weights or loop weights of an operator ψ(ξ). The highest weight of the evaluation representation L1,µq− 12 is encoded by the Drinfeld polynomial 1

P1 = (1 − µq − 2 /ξ) The operator ψ(ξ) ∈ Uaff q (sl2 ) acts by 1 − µq −1 /ξ |ω0 i 1 − µ/ξ +1 − 21 1 − µq /ξ |ω1 i ψ(ξ)|ω1i = q 1 − µ/ξ 1

ψ(ξ)|ω0 i = q 2

(A.40)

Hence, by definition, the q-character of the Uaff q (sl2 ) fundamental module V1,aq − 12 is χq [V1,µq− 12 ] = Y1,µq− 21 +

1 Y1,µq 12

(A.41)

Notice that the sum of eigenvalues of ψ(ξ) does not encode as much information about the Uaff q (sl2 ) module V1,µq − 21 as it turns out to be simply 1

1

trV ψ1 (z) = q 2 + q − 2

(A.42)

± In terms of the current modes ψ1,n , e± 1,n for the same example of the evaluation 1 aff representation L1,q− 21 µ of Uq (sl2 ) with evaluation parameter µq − 2 we find the explicit


65

action by the Uaff q (sl2 ) generators on V1,µq − 21 is given by n e− 1,n |ω0 i = µ |ω1 i

n e+ 1,n |ω1 i = µ |ω0 i

[n]q |ω0 i (n > 0), h1,0 |ω0i = |ω0 i (A.43) n [2n]q n [n]q − )|ω1 i (n > 0), h1,0 |ω1i = −|ω1 i h1,n |ω1 i = µn (q − 2 n n or equivalently by the matrices − n 0 0 + n 0 1 e1,n = µ e1,n = µ 1 0 0 0 ! 1−q −n 0 µn q1/2 1 0 −1/2 −q h1,n6=0 = h1,0 = 1−q n 0 −1 0 n q 1/2 −q −1/2 ±  1 n 1 q 2 (1−µq −1 /ξ) −1 0 2 − q 2) µ (q 0 ± 1−µ/ξ ±  ⇒ ψ1,n∈±Z>0 = ± ψ (ξ) =  1 1 1 q − 2 (1−µq/ξ) 0 µn (q 2 − q − 2 ) 0 1−µ/ξ (A.44) which obviously satisfy the commutation relations (A.9)(A.10)(A.11)(A.12) of Uaff q (sl2 ) at c = 1 [2n]q 1 − + − [h1,n6=0 , e± em+n [e+ (A.45) 1 (ψ1,m+n − ψ1,m+n ) 1,m ] = 1,m , e1,n ] = 1 − n q2 − q 2 n

h1,n |ω0 i = µn q − 2

A.3.2. Level zero fundamental highest weight modules. The basic level zero fundamental representations of quantum affine algebra Uaff q (g) are highest weight modules Li,µ , labeled by a node i ∈ Ig of Dynkin diagram of g and an evaluation parameter µ ∈ C× . ± A module Li,µ is a highest weight Uaff q (g)-module with the current weights of (ψi )i∈Ig on the highest vector |ωi,µ i given by  1, j 6= i ± 1 −2 (A.46) ψj (ξ)|ωi,µi = 1 q 2 1−µq 1 /ξ , j = i 1−µq 2 /ξ

The fundamental module Li,µ is a quantum affine algebra analogue of the i-th fundamental highest weight evaluation module at ξ = µ for the loop algebra g((ξ)).

A.4. Loop weights. Following [50],[49] we call a vector |wi ∈ W , for a Uaff q (gΓ )± aff module W , a generalized eigenvector of the operators ψi (ξ) ∈ Uq (gΓ ) with the gen∓ eralized eigenvalues Ψ± i (ξ) ∈ C[[ξ ]] if there exists a positive integer n, such that n (ψi± (ξ) − Ψ± i (ξ)) |wi = 0 for all i ∈ IΓ .

(A.47)

in other words it belongs to the Jordan block of uniformly bounded size n for all 1 − h −1h 2 i and q 2 i since ψi± (ξ)’s. The values Ψ+ i (∞) and Ψi (0) are the eigenvalues of q 1

q 2 hi |wi = Ψ+ i (∞)|wi 1

q − 2 hi |wi = Ψ− i (0)|wi.

(A.48)

66


All integrable finite-dimensional modules of Uaff q (gΓ ) with highest weight have been classified [50] (theorem 3.3). The generalized eigenvalue Ψ± (ξ) of the operator ψ ± (ξ) on the highest vector |vi of an integrable finite-dimensional module V of Uaff q (gΓ ) always has the form !± 1 2 ξ) 1 P (q i deg Pi 2 (A.49) Ψ± 1 i (ξ) = q Pi (q − 2 ξ) Q Pi −1 with Pi (∞) = 1 and ()± where Pi+ (ξ) := deg k=1 (1 − ξk /ξ) is a polynomial in ξ denotes the expansion at ξ = ∞ and ξ = 0 respectively. The collection of polynomials Pi (ξ)i∈IΓ , defining an integrable highest weight module of Uaff q (gΓ ) is called Drinfeld polynomial. The equation (A.24) implies that the highest vector |v0 i in the i-th fundamental representation with Drinfeld polynomial Pj (ξ) = 1 for j 6= i and Pi (ξ) = 1 − 1/ξ, such that the eigenvalue of p± j (ξ) on |v0 i is equal to Pj (ξ), is a common eigenvector of the generators hj,n with the corresponding eigenvalues   0, j 6= i (A.50) hj,n = 1, j = i, n = 0   [n]q , j = i, n 6= 0 n

E. Frenkel and N. Reshetikhin [49] have shown that for any finite-dimensional Uaff q (gΓ ) ± module V the generalized eigenvalues of the operators ψi (ξ) ∈ Uq (gbΓ ) always have the form Y (A.51) evψ(ξ) (Yi,ξk )± ξk

where evaluation on ψ(ξ) is defined multiplicatively by 1

evψ(ξ) (Yi,ζ ) := q

1 2

1 − q − 2 ζ/ξ 1

1 − q 2 ζ/ξ

(A.52)

˚ Similarly, for stripped element ψ(ξ) 1

evψ(ξ) ˚ (Yi,ζ ) :=

1 − q − 2 ζ/ξ 1

1 − q 2 ζ/ξ

(A.53)

The q-character χq (V ) of Uq (gbΓ ) finite-dimensional module V is defined as a sum of dim V monomials of the form X Y (Yi,ξi )±1 (A.54) χq (V ) = |vi i,ξi ∈Ξ|vi

one monomial for each generalized eigenvector |vi in Uaff q (gΓ ) module V for the comaff muting set of operators ψi (ξ) ∈ Uq (gΓ ), such that each such monomial encodes the generalized eigenvalue of the operator Y ψ(ξ) = ψi∈IΓ (ξ) ∈ Uq (gbΓ ) (A.55) i

by the evaluation (A.53).


67

A.5. Universal R-matrix. Abstractly, quantum affine algebra A = Uaff q (gΓ ) is a Hopf algebra, and hence is equipped with the comultiplication operation ∆ : A → A ⊗ A. The comultiplication operation is what allows to define tensor products of representations for associative algebras. If ρi : A → End(Vi ) are two representations, then the tensor product ρ : A → End(V1 ) ⊗ End(V2 ) is defined by x 7→ ρ1 ⊗ ρ2 (∆(x)). If an abstract Hopf algebra A is co-commutative, then A-modules V1 ⊗ V2 and V2 ⊗ V1 are naturally isomorphic, but there is no natural reason for such isomorphism if A is not co-commutative. Quantum affine algebras have extra structure in addition to the structure of generic Hopf algebras, called quasi-triangular structure that ensures that tensor products V1 ⊗ V2 and V2 ⊗ V1 are isomorphic. Such isomorphism is not a simple permutation of the factors, but is given by a certain linear map that depends on representations V1 , V2 . Namely, by definition, a quasi-triangular structure on a Hopf algebra A is an element R ∈ A ⊗ A, called universal R-matrix such that R∆(x) = ∆op (x)R (∆ ⊗ IdA )(R) = R13 R23 = ai ⊗ aj ⊗ bi bj

(A.56)

(IdA ⊗ ∆)(R) = R13 R12 = ai aj ⊗ bj ⊗ bi P P P where x ∈ A and if R = a ⊗b then R = a ⊗b ⊗1, R = i i 12 i i 13 i i i ai ⊗1⊗bi , R23 = P op denotes the co-multiplication in the opposite order. The map i 1 ⊗ ai ⊗ bi , and ∆ RV′ 1 ,V2 = P RV1 ,V2 , where P : V1 ⊗ V2 → V2 ⊗ V1 is a permutation of factors, gives isomorphism RV′ 1 ,V2 : V1 ⊗ V2 → V2 ⊗ V1 . The relations (A.56) imply that the universal R-matrix R ∈ A ⊗ A satisfies universal Yang-Baxter equation R12 R13 R23 = R23 R13 R12

(A.57)

The axioms on R-matrix ensure that the category of representations of A is a braided tensor category (which means this tensor category is equipped with a commutativity isomorphism σV1 ,V2 : V1 ⊗ V2 → V2 ⊗ V1 , an associativity isomorphism αV1 ,V2 ,V3 : (V1 ⊗ V2 ) ⊗ V3 → V1 ⊗ (V2 ⊗ V3 ) and unit morphisms λV : 1 ⊗ V → V, ρV : V ⊗ 1 → V , which satisfy certain compatibility axioms. For more details see [213]). For any finite-dimensional Hopf algebra A, Drinfeld constructed [116] structure of quasi-triangular Hopf algebra on Drinfeld double D(A) = A ⊗ A∗op If ai is a basis in A and ai is a dual basis in A∗ then D(A) is a quasi-triangular Hopf algebra with the R-matrix X (A.58) R= (ai ⊗ 1A∗ ) ⊗ (1A ⊗ ai ) i

The quasi-triangular structure on Uaff q (gΓ ) (for gΓ of finite type) can be found using + aff the fact that Uq (gΓ ) is almost Drinfeld double of it Borel subalgebra Uaff q (gΓ ) [213, ± 214]. The structure of Hopf algebra on Uaff q (gΓ ) naturally induces the structure of ± ∗ Hopf algebra on its dual Uaff q (gΓ ) . Moreover, V. Drinfeld has shown [116] that there − aff + ∗op exists an isomorphism of Hopf algebras θ : Uaff such that for h in q (gΓ ) → Uq (gΓ ) h h the extended Cartan subalgebra of gbΓ we have θ(q ) = q if we identify the extended

68


Cartan subalgebra8 with its dual using the standard Cartan-Killing quadratic form. The isomorphism θ provides a non-degenerate bilinear form (called Drinfeld pairing) + aff − Uaff q (gΓ ) ⊗ Uq (gΓ ) → C. Using Drinfeld pairing we present the Drinfeld double of the + Borel Uaff q (gΓ ) as + aff + aff − D(Uaff (A.59) q (gΓ )) = Uq (gΓ ) ⊗ Uq (gΓ ) + h h The Drinfeld double D(Uaff q (gΓ )) contains two-sided ideal H generated by q ⊗1−1⊗q + for h in the extended Cartan of gbΓ . One can check that the map f : D(Uaff q (gΓ ))/H → aff + Uaff q (gΓ ) is isomorphism. The pushforward by f of the universal R-matrix on D(Uq (gΓ ))/H will give a universal R-matrix on Uaff q (gΓ ). An explicit formula can be found in [215].

A.6. Dual interpretation. The q-character χq (V ) for a representation V of Uaff q (gΓ ) can be interpreted not only as a formal expression encoding generalized eigenvalues aff of diagonal generators of Uaff q (gΓ ) but also as a certain element of Uq (gΓ ), in fact, that is how the q-character was originally introduced in [49]. The tensor product Uq (gbΓ ) ⊗ Uq (gbΓ ) contains a special element R known as the universal R-matrix [49]. Let (V, πV ) denote a finite-dimensional representation of Uq (gbΓ ). Let V (ξ) denote the twist of representation V by spectral parameter ξ. Define the transfer matrix tV (ξ) associated to V as an element of Uaff q (gΓ ) given by the trace over V : tV (ξ) = trV q ρ (πV (ξ) ⊗ id)R

(A.60)

R = R+ R0 R− T

(A.61)

P˜ ˜ where ρ = h ˜ij hj . i with hi = a The explicit formula for universal R-matrix is given in [216] (eq 42), [217] where

X n a˜ij (q n )hi,n ⊗ hi,−n , R0 = exp −(q 1/2 − q −1/2 ) [n] q n>0

T =q

!

(A.62)

− 12 a ˜ij hi ⊗hj

± ∓ c and the factors R± ∈ Uaff q (gΓ ) ⊗ Uq (gΓ ). The q-character χq (V (ξ)) is essentially the diagonal projection of tV (ξ), i.e.

χq (V (ξ)) = trV (q ρ ΓV (πV ⊗ 1)(T ))

X n ΓV = exp −(q 1/2 − q −1/2 ) a˜ij (q n )ξ −n πV (hi,n ) ⊗ hj,−n [n]q n>0

Now we introduce special elements Ybi,a of Uaff q (gΓ ) by definition Ybi,ζ = q (ρ,ωi ) q

hi − 12 ˜

exp −(q 1/2 − q −1/2 )

where ωi are fundamental weights. 8If

X n>0

˜ i,−n ζ n ξ −n h

!

!

(A.63)

(A.64)

gΓ of finite type has rank r, the extended Cartan subalgebra of gc Γ has rank r+2 and is generated by the coroots of gΓ , the center and the energy element


69

The q-character is a sum of dim V terms. Each term in χq (V (ξ)) corresponds to an eigenvector of V . The eigenvalue of a given eigenvector of V is encoded by Drinfeld polynomial loop-weights which can be factorized into a product of basic monomials (P ) = (1, . . . , 1, 1 − ζ/ξ, 1, . . . , 1) with 1 − ζ/ξ, say, in i-th position. Each such factor produces a factor Ybi,ζ in the given term in the q-character associated with this eigenvalue.

A.6.1. Twisted q-characters. For our purposes it is natural to consider the twisted P˜ transfer matrix defined by the formula similar to the (A.60) in which q hi is replaced Q h˜ i : by q− i Y ˜ hi tq,V (ξ) = trV (ξ) q− (πV (ξ) ⊗ id)R (A.65) i i∈i

and then set

Ybi (ξ) = q

− 21 ˜ hi

1 2

exp −(q − q

− 21

)

X n>0

˜ i,−n ξ −n h

!

(A.66)

The q-twisted q-character is a sum of monomials associated to eigenvectors. Each ± monomial is a product of factors Ybi,a encoding the generalized eigenvalues with an additional qi -dependent factor precisely like in (4.7) for the quiver theory without fundamental matter, with Pi = qi . For example, the q-twisted q-character associated to the A1 quiver and the fundamental evaluation module is 1 1 1 tq,L1 = q− 2 Yb1 (ξ) + q 2 (A.67) Yb1 (q −1 ξ) Proposition. The equations

q-twisted q-characters χi = c-number polynomials Ti (ξ)

(A.68)

can be interpreted as follows: there exists a special eigenvector9 of the Uaff q (gΓ ) elements Ybi (ξ) with the generalized eigenvalues equal to our c-number functions Yi (ξ) and the eigenvalue of the q-twisted q-character viewed as an element of Uaff q (gΓ ) equal to the polynomial Ti (ξ). It looks like our formulation is T -Q dual (in a sense of [201]) to the results of the recent paper [94] on the category O of representations of the quantum affine algebras, at least for the finite dimensional gΓ . References [1] G. W. Moore, N. Nekrasov, and S. Shatashvili, “Integrating over Higgs branches,” Commun. Math. Phys. 209 (2000) 97–121, arXiv:hep-th/9712241. [2] A. A. Gerasimov and S. L. Shatashvili, “Higgs Bundles, Gauge Theories and Quantum Groups,” Commun.Math.Phys. 277 (2008) 323–367, arXiv:hep-th/0609024 [hep-th]. [3] A. A. Gerasimov and S. L. Shatashvili, “Two-dimensional Gauge Theories and Quantum Integrable Systems,” arXiv:0711.1472 [hep-th]. 9It

is not clear whether this eigenvector belongs to a natural Uaff q (gΓ )-module though

70


[4] N. Nekrasov and S. Shatashvili, “Bethe Ansatz and supersymmetric vacua,” AIP Conf. Proc. 1134 (2009) 154–169. [5] N. A. Nekrasov and S. L. Shatashvili, “Quantization of Integrable Systems and Four Dimensional Gauge Theories,” arXiv:0908.4052 [hep-th]. [6] N. A. Nekrasov and S. L. Shatashvili, “Quantum integrability and supersymmetric vacua,” Prog. Theor. Phys. Suppl. 177 (2009) 105–119, arXiv:0901.4748 [hep-th]. [7] N. A. Nekrasov and S. L. Shatashvili, “Supersymmetric vacua and Bethe ansatz,” Nucl. Phys. Proc. Suppl. 192-193 (2009) 91–112, arXiv:0901.4744 [hep-th]. [8] N. Nekrasov and E. Witten, “The Omega Deformation, Branes, Integrability, and Liouville Theory,” JHEP 09 (2010) 092, arXiv:1002.0888 [hep-th]. [9] N. Nekrasov, A. Rosly, and S. Shatashvili, “Darboux coordinates, Yang-Yang functional, and gauge theory,” Nucl.Phys.Proc.Suppl. 216 (2011) 69–93, arXiv:1103.3919 [hep-th]. [10] E. Witten, “Gauge theories and integrable lattice models,” Nucl.Phys. B322 (1989) 629. [11] A. Gorsky and N. Nekrasov, “Relativistic Calogero-Moser model as gauged WZW theory,” Nucl. Phys. B436 (1995) 582–608, arXiv:hep-th/9401017. [12] A. Gorsky and N. Nekrasov, “Hamiltonian systems of Calogero type and two-dimensional Yang-Mills theory,” Nucl. Phys. B414 (1994) 213–238, arXiv:hep-th/9304047. [13] A. Gorsky and N. Nekrasov, “Elliptic Calogero-Moser system from two-dimensional current algebra,” arXiv:hep-th/9401021. [14] N. Nekrasov, “On the BPS/CFT correspondence,” Feb. 3, 2004. http://www.science.uva.nl/research/itf/strings/stringseminar2003-4.html. lecture at the string theory group seminar, University of Amsterdam. [15] H. Nakajima, “Gauge theory on resolutions of simple singularities and simple Lie algebras,” Internat. Math. Res. Notices no. 2, (1994) 61–74. http://dx.doi.org/10.1155/S1073792894000085. [16] H. Nakajima, “Quiver varieties and Kac-Moody algebras,” Duke Math. J. 91 no. 3, (1998) 515–560. http://dx.doi.org/10.1215/S0012-7094-98-09120-7. [17] C.Vafa and E.Witten, “A Strong coupling test of S duality,” Nucl.Phys. B431 (1994) 3–77, arXiv:hep-th/9408074. [18] A. Losev, G. W. Moore, N. Nekrasov, and S. Shatashvili, “Four-dimensional avatars of two-dimensional RCFT,” Nucl. Phys. Proc. Suppl. 46 (1996) 130–145, arXiv:hep-th/9509151. [19] N. A. Nekrasov, “Seiberg-Witten prepotential from instanton counting,” Adv.Theor.Math.Phys. 7 (2004) 831–864, arXiv:hep-th/0206161 [hep-th]. To Arkady Vainshtein on his 60th anniversary. [20] A. S. Losev, A. Marshakov, and N. A. Nekrasov, “Small instantons, little strings and free fermions,” arXiv:hep-th/0302191 [hep-th].


71

[21] N. Nekrasov and A. Okounkov, “Seiberg-Witten theory and random partitions,” arXiv:hep-th/0306238 [hep-th]. [22] L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville Correlation Functions from Four-dimensional Gauge Theories,” Lett. Math. Phys. 91 (2010) 167–197, arXiv:0906.3219 [hep-th]. [23] N. Nekrasov and V. Pestun, “Seiberg-Witten geometry of N = 2 quiver gauge theories,” arXiv:1211.2240 [hep-th]. [24] N. Seiberg and E. Witten, “Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD,” Nucl. Phys. B431 (1994) 484–550, hep-th/9408099. [25] N. Seiberg and E. Witten, “Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory,” Nucl. Phys. B426 (1994) 19–52, hep-th/9407087. [26] A. Losev, N. Nekrasov, and S. L. Shatashvili, “Issues in topological gauge theory,” Nucl. Phys. B534 (1998) 549–611, arXiv:hep-th/9711108. [27] A. Losev, N. Nekrasov, and S. L. Shatashvili, “Testing Seiberg-Witten solution,” arXiv:hep-th/9801061. [28] N. Nekrasov, “Five dimensional gauge theories and relativistic integrable systems,” Nucl. Phys. B531 (1998) 323–344, arXiv:hep-th/9609219. [29] R. Poghossian, “Deforming SW curve,” JHEP 1104 (2011) 033, arXiv:1006.4822 [hep-th]. [30] N. Dorey, S. Lee, and T. J. Hollowood, “Quantization of Integrable Systems and a 2d/4d Duality,” JHEP 1110 (2011) 077, arXiv:1103.5726 [hep-th]. [31] H.-Y. Chen, N. Dorey, T. J. Hollowood, and S. Lee, “A New 2d/4d Duality via Integrability,” JHEP 1109 (2011) 040, arXiv:1104.3021 [hep-th]. [32] L. Bao, E. Pomoni, M. Taki, and F. Yagi, “M5-Branes, Toric Diagrams and Gauge Theory Duality,” JHEP 1204 (2012) 105, arXiv:1112.5228 [hep-th]. [33] A. Mironov, A. Morozov, B. Runov, Y. Zenkevich, and A. Zotov, “Spectral Duality Between Heisenberg Chain and Gaudin Model,” arXiv:1206.6349 [hep-th]. [34] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, “Integrability and Seiberg-Witten exact solution,” Phys.Lett. B355 (1995) 466–474, arXiv:hep-th/9505035 [hep-th]. [35] A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov, “N=2 supersymmetric QCD and integrable spin chains: Rational case Nf ≤ 2Nc ,” Phys.Lett. B380 (1996) 75–80, arXiv:hep-th/9603140 [hep-th]. [36] A. Gorsky, S. Gukov, and A. Mironov, “Multiscale N=2 SUSY field theories, integrable systems and their stringy / brane origin. 1.,” Nucl.Phys. B517 (1998) 409–461, arXiv:hep-th/9707120 [hep-th]. [37] A. Gorsky, S. Gukov, and A. Mironov, “SUSY field theories, integrable systems and their stringy / brane origin. 2.,” Nucl.Phys. B518 (1998) 689–713, arXiv:hep-th/9710239 [hep-th]. [38] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, “Integrable structure of conformal field theory. 3. The Yang-Baxter relation,”

72


Commun.Math.Phys. 200 (1999) 297–324, arXiv:hep-th/9805008 [hep-th]. [39] P. Dorey, C. Dunning, and R. Tateo, “The ODE/IM Correspondence,” J.Phys. A40 (2007) R205, arXiv:hep-th/0703066 [hep-th]. [40] A. Gerasimov, S. Kharchev, D. Lebedev, and S. Oblezin, “On a class of representations of the Yangian and moduli space of monopoles,” Commun.Math.Phys. 260 (2005) 511–525. [41] A. Gerasimov, S. Kharchev, D. Lebedev, and S. Oblezin, “On a Class of Representations of Quantum Groups,” ArXiv Mathematics e-prints (Jan., 2005) , arXiv:math/0501473. [42] D. Galakhov, A. Mironov, A. Morozov, A. Smirnov, A. Mironov, et al., “Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation,” Theor.Math.Phys. 172 (2012) 939–962, arXiv:1104.2589 [hep-th]. [43] A. Mironov and A. Morozov, “Nekrasov Functions and Exact Bohr-Zommerfeld Integrals,” JHEP 1004 (2010) 040, arXiv:0910.5670 [hep-th]. [44] A. Mironov and A. Morozov, “Nekrasov Functions from Exact BS Periods: The Case of SU(N),” J.Phys. A43 (2010) 195401, arXiv:0911.2396 [hep-th]. [45] J. Teschner, “Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I,” Adv.Theor.Math.Phys. 15 (2011) 471–564, arXiv:1005.2846 [hep-th]. [46] K. Muneyuki, T.-S. Tai, N. Yonezawa, and R. Yoshioka, “Baxter’s T-Q equation, SU(N)/SU(2)N −3 correspondence and Ω-deformed Seiberg-Witten prepotential,” JHEP 1109 (2011) 125, arXiv:1107.3756 [hep-th]. [47] V. G. Drinfeld, “A new realization of Yangians and of quantum affine algebras,” Dokl. Akad. Nauk SSSR 296 no. 1, (1987) 13–17. [48] H. Knight, “Spectra of tensor products of finite-dimensional representations of Yangians,” J. Algebra 174 no. 1, (1995) 187–196. http://dx.doi.org/10.1006/jabr.1995.1123. [49] E. Frenkel and N. Reshetikhin, “The q-characters of representations of quantum affine algebras and deformations of W-algebras,” Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998) 248 (1999 http://dx.doi.org/10.1090/conm/248/03823. [50] V. Chari and A. Pressley, “Quantum affine algebras and their representations,” hep-th/9411145. [51] E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys. 121 (1989) 351. [52] A. Kirillov and N. Y. Reshetikhin, “Representations of the algebra uq (sl2 ), orthogonal polynomials and invariants of links,”. http://www.worldscientific.com/worldscibooks/10.1142/1056. [53] N. Reshetikhin and V. Turaev, “Invariants of three manifolds via link polynomials and quantum groups,” Invent.Math. 103 (1991) 547–597. [54] N. Y. Reshetikhin and V. Turaev, “Ribbon graphs and their invariants derived from quantum groups,” Commun.Math.Phys. 127 (1990) 1–26. [55] E. Witten, “Fivebranes and Knots,” arXiv:1101.3216 [hep-th].


73

[56] N. Nekrasov, H. Ooguri, and C. Vafa, “S-duality and topological strings,” JHEP 10 (2004) 009, arXiv:hep-th/0403167. [57] N. Nekrasov, “Z-theory: Chasing M-F-Theory,” Comptes Rendus Physique 6 (2005) 261–269. [58] D. Hernandez, “Quantum toroidal algebras and their representations,” Selecta Math. (N.S.) 14 no. 3-4, (2009) 701–725, abs/0801.2397. http://dx.doi.org/10.1007/s00029-009-0502-4. [59] I. Cherednik, “Introduction to double Hecke algebras,” ArXiv Mathematics e-prints (Apr., 2004) , arXiv:math/0404307. [60] talk by V. Pestun, “Integrable systems for 4d N=2 ADE quiver theories from instanton counting,” Aug 2012. http://people.physik.hu-berlin.de/~ahoop/pestun.pdf. [61] talk by V. Pestun, “Supersymmetric Four-dimensional Quiver Gauge Theories and Quantum ADE Spin Chains,” Sep 2012. http://scgp.stonybrook.edu/archives/4709. [62] talk by N. Nekrasov, “Seiberg-Witten geometry of N=2 quiver theories, and quantization,” May 2012. http://brahms.mth.kcl.ac.uk/cgi-bin/main.pl?action=seminars&id=1108. [63] talk by N. Nekrasov, “Seiberg-Witten Geometry of N=2 Superconformal Theories, and ADE Bundles on Curves,” March 2012. http://media.scgp.stonybrook.edu/video/video.php?f=20120510_1_qtp.mp4. [64] talk by S.Shatashvili, “Gauge theory angle at integrability,” May 2012. http://www.kcl.ac.uk/nms/depts/mathematics/research/theorphysics/pastevents/strin [65] talk by S.Shatashvili, “Integrability and quantization,” February 2013. https://indico.desy.de/contributionDisplay.py?contribId=16&confId=6969. [66] talk by S.Shatashvili, “Integrability and supersymmetric vacua (IV),” July 2013. http://cdsagenda5.ictp.trieste.it/full_display.php?ida=a13168. [67] F. Fucito, J. F. Morales, and D. R. Pacifici, “Deformed Seiberg-Witten Curves for ADE Quivers,” arXiv:1210.3580 [hep-th]. [68] N. Nekrasov, “Four dimensional holomorphic theories,” Princeton University PhD. thesis (1996) . [69] L. Baulieu, A. Losev, and N. Nekrasov, “Chern-Simons and twisted supersymmetry in various dimensions,” Nucl. Phys. B522 (1998) 82–104, arXiv:hep-th/9707174. [70] J. H. Schwarz, “Anomaly - free supersymmetric models in six-dimensions,” Phys.Lett. B371 (1996) 223–230, arXiv:hep-th/9512053 [hep-th]. [71] J. Blum and K. Intriligator, “New phases of string theory and 6-D RG fixed points via branes at orbifold singularities,” Nucl.Phys. B506 (1997) 199–222, arXiv:hep-th/9705044. [72] J. D. Blum and K. A. Intriligator, “New phases of string theory and 6-D RG fixed points via branes at orbifold singularities,” Nucl.Phys. B506 (1997) 199–222, arXiv:hep-th/9705044 [hep-th]. [73] N. Seiberg, “Nontrivial fixed points of the renormalization group in six-dimensions,” Phys.Lett. B390 (1997) 169–171,

74


arXiv:hep-th/9609161 [hep-th]. [74] V. G. Kac, Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third ed., 1990. http://dx.doi.org/10.1017/CBO9780511626234. [75] H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces. AMS, 1999. AMS University Lecture Series, ISBN 0-8218-1956-9. [76] H. Nakajima and K. Yoshioka, “Lectures on Instanton Counting,” ArXiv Mathematics e-prints (Nov., 2003) , math/0311058. [77] H. Nakajima, “t-analogs of q-characters of quantum affine algebras of type An , Dn ,”. http://dx.doi.org/10.1090/conm/325/05669. [78] S. Shadchin, “On certain aspects of string theory/gauge theory correspondence,” arXiv:hep-th/0502180 [hep-th]. Ph.D. Thesis. [79] V. Pestun, “Localization of gauge theory on a four-sphere and supersymmetric Wilson loops,” Commun.Math.Phys. 313 (2012) 71–129, arXiv:0712.2824 [hep-th]. [80] N. Nekrasov and A. S. Schwarz, “Instantons on noncommutative R4 and (2, 0)-superconformal six dimensional theory,” Commun. Math. Phys. 198 (1998) 689–703, arXiv:hep-th/9802068. [81] M. Atiyah, N. J. Hitchin, V. Drinfeld, and Y. Manin, “Construction of Instantons,” Phys.Lett. A65 (1978) 185–187. [82] K. Kozlowski and J. Teschner, “TBA for the Toda chain,” arXiv:1006.2906 [math-ph]. [83] A. M. Polyakov, Gauge fields and strings. Harwood, Chur, Switzerland, 1987. [84] S. Shadchin, “Status report on the instanton counting,” SIGMA 2 (2006) 008, arXiv:hep-th/0601167. [85] A. Givental, “A mirror theorem for toric complete intersections,” arXiv (Jan., 1997) 1016, alg-geom/9701016. [86] V. Bazhanov and N. Reshetikhin, “Restricted Solid on Solid Models Connected With Simply Based Algebras and Conformal Field Theory,” J.Phys. A23 (1990) 1477. [87] A. Kirillov and N. Y. Reshetikhin, “Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum,” J.Phys. A20 (1987) 1565–1585. [88] A. Kuniba and J. Suzuki, “Analytic Bethe Ansatz for fundamental representations of Yangians,” Commun.Math.Phys. 173 (1995) 225–264, arXiv:hep-th/9406180 [hep-th]. [89] A. Kuniba, T. Nakanishi, and J. Suzuki, “T-systems and Y-systems in integrable systems,” J.Phys. A44 (2011) 103001, arXiv:1010.1344 [hep-th]. [90] F. Fucito, J. Morales, D. R. Pacifici, and R. Poghossian, “Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves,” JHEP 1105 (2011) 098, arXiv:1103.4495 [hep-th]. b ∞ ) and applications,” [91] D. Hernandez, “The algebra Uq (sl J. Algebra 329 (2011) 147–162. http://dx.doi.org/10.1016/j.jalgebra.2010.04.002.


75

d [92] E. Frenkel and E. Mukhin, “The Hopf algebra Rep Uq gl ∞ ,” Selecta Math. (N.S.) 8 no. 4, (2002) 537–635. http://dx.doi.org/10.1007/PL00012603. [93] L. Faddeev, “How algebraic Bethe ansatz works for integrable model,” arXiv:hep-th/9605187 [hep-th]. [94] E. Frenkel and D. Hernandez, “Baxter’s Relations and Spectra of Quantum Integrable Models,” ArXiv e-prints (Aug., 2013) , arXiv:1308.3444 [math.QA]. [95] R. J. Baxter, Exactly solved models in statistical mechanics, vol. 1 of Ser. Adv. Statist. Mech. World Sci. Publishing, Singapore, 1985. [96] E. Frenkel and E. Mukhin, “Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras,” Comm. Math. Phys. 216 no. 1, (2001) 23–57. http://dx.doi.org/10.1007/s002200000323. [97] D. Hernandez, “The t-analogs of q-characters at roots of unity for quantum affine algebras and beyond,” J. Algebra 279 no. 2, (2004) 514–557. http://dx.doi.org/10.1016/j.jalgebra.2004.02.022. [98] P. Etingof and A. Varchenko, “Dynamical Weyl groups and applications,” ArXiv Mathematics e-prints (Oct., 2000) , arXiv:math/0011001. [99] F. A. Smirnov, “Baxter Equations and Deformation of Abelian Differentials,” International Journal of Modern Physics A 19 (2004) 396–417, math-ph/0302014. [100] V. G. Drinfeld, “Hopf algebras and the quantum Yang-Baxter equation,” Dokl. Akad. Nauk SSSR 283 no. 5, (1985) 1060–1064. [101] M. Jimbo, “A q-difference analogue of U(g) and the Yang-Baxter equation,” Lett. Math. Phys. 10 no. 1, (1985) 63–69. http://dx.doi.org/10.1007/BF00704588. [102] E. Sklyanin and L. Faddeev, “Quantum Mechanical Approach to Completely Integrable Field Theory Models,” Sov.Phys.Dokl. 23 (1978) 902–904. [103] E. K. Sklyanin, L. A. Takhtajan, and L. D. Faddeev, “Quantum inverse problem method. I,” Teoret. Mat. Fiz. 40 no. 2, (1979) 194–220. [104] L. Takhtajan and L. Faddeev, “The Quantum method of the inverse problem and the Heisenberg XYZ model,” Russ.Math.Surveys 34 (1979) 11–68. [105] E. Sklyanin, “Quantum version of the method of inverse scattering problem,” J.Sov.Math. 19 (1982) 1546–1596. [106] E. Sklyanin, “Some algebraic structures connected with the Yang-Baxter equation,” Funct.Anal.Appl. 16 (1982) 263–270. [107] L. Faddeev, N. Y. Reshetikhin, and L. Takhtajan, “Quantization of Lie Groups and Lie Algebras,” Leningrad Math.J. 1 (1990) 193–225. [108] P. Kulish and N. Y. Reshetikhin, “Quantum linear problem for the Sine-Gordon equation and higher representation,” J.Sov.Math. 23 (1983) 2435–2441. [109] J. Beck, “Braid group action and quantum affine algebras,” Comm. Math. Phys. 165 no. 3, (1994) 555–568. http://projecteuclid.org/getRecord?id=euclid.cmp/1104271413.

76


[110] V. Chari and A. Pressley, “Quantum affine algebras,” Comm. Math. Phys. 142 no. 2, (1991) 261–283. http://projecteuclid.org/getRecord?id=euclid.cmp/1104248585. [111] N. Jing, “Quantum Kac-Moody algebras and vertex representations,” Lett. Math. Phys. 44 no. 4, (1998) 261–271. http://dx.doi.org/10.1023/A:1007493921464. [112] H. Nakajima, “Quiver varieties and finite-dimensional representations of quantum affine algebras,” J. Amer. Math. Soc. 14 no. 1, (2001) 145–238. http://dx.doi.org/10.1090/S0894-0347-00-00353-2. ´ Vasserot, “Langlands reciprocity for [113] V. Ginzburg, M. Kapranov, and E. algebraic surfaces,” Math. Res. Lett. 2 no. 2, (1995) 147–160, q-alg/9502013. http://dx.doi.org/10.4310/MRL.1995.v2.n2.a4. [114] H. Nakajima, “Quiver varieties and quantum affine algebras [translation of S¯ ugaku 52 (2000), no. 4, 337–359; mr1802956],” Sugaku Expositions 19 no. 1, (2006) 53–78. Sugaku Expositions. [115] H. Nakajima, “Geometric construction of representations of affine algebras,” math/0212401. [116] V. G. Drinfeld, “Quantum groups,” Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) (1987) 798–820. [117] S. Gautam and V. Toledano Laredo, “Yangians and quantum loop algebras,” Selecta Math. (N.S.) 19 no. 2, (2013) 271–336, 1012.3687. http://dx.doi.org/10.1007/s00029-012-0114-2. [118] N. Guay and X. Ma, “From quantum loop algebras to Yangians,” J. Lond. Math. Soc. (2) 86 no. 3, (2012) 683–700. http://dx.doi.org/10.1112/jlms/jds021. [119] G. Felder and A. Varchenko, “On representations of the elliptic quantum group Eτ,η (sl2 ),” Comm. Math. Phys. 181 no. 3, (1996) 741–761. http://projecteuclid.org/getRecord?id=euclid.cmp/1104287911. [120] G. Felder, “Elliptic quantum groups,” XIth International Congress of Mathematical Physics (Paris, 1994) (1995) 211–218. c2 ): [121] M. Jimbo, H. Konno, S. Odake, and J. Shiraishi, “Elliptic algebra Uq,p (sl Drinfeld currents and vertex operators,” Comm. Math. Phys. 199 no. 3, (1999) 605–647, math/9802002. http://dx.doi.org/10.1007/s002200050514. [122] A. Marshakov and A. Mironov, “5-d and 6-d supersymmetric gauge theories: Prepotentials from integrable systems,” Nucl.Phys. B518 (1998) 59–91, arXiv:hep-th/9711156 [hep-th]. [123] T. J. Hollowood, A. Iqbal, and C. Vafa, “Matrix models, geometric engineering and elliptic genera,” JHEP 0803 (2008) 069, arXiv:hep-th/0310272 [hep-th]. c2 ),”. [124] H. Konno, “The elliptic quantum group Uq,p (sl http://www.mis.hiroshima-u.ac.jp/~konno/RIMS09.pdf. [125] G. Felder and A. Varchenko, “Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations,”


77

arXiv:hep-th/9502165 [hep-th]. [126] G. Felder, V. Tarasov, and A. Varchenko, “Solutions of the elliptic qKZB equations and Bethe ansatz. I,” q-alg/9606005. [127] G. Felder and A. Varchenko, “The elliptic gamma function and SL(3, Z) ⋉ Z3 ,” Adv. Math. 156 no. 1, (2000) 44–76, math/9907061. http://dx.doi.org/10.1006/aima.2000.1951. [128] B. Feigin and E. Frenkel, “Quantum W -algebras and elliptic algebras,” Comm. Math. Phys. 178 no. 3, (1996) 653–678, arXiv:q-alg/9508009 [q-alg]. http://projecteuclid.org/getRecord?id=euclid.cmp/1104286770. [129] V. Chari and A. Pressley, “Yangians: their representations and characters,” Acta Appl. Math. 44 no. 1-2, (1996) 39–58. http://dx.doi.org/10.1007/BF00116515. Representations of Lie groups, Lie algebras and their quantum analogues. [130] H. Nakajima, “T -analogue of the q-characters of finite dimensional representations of quantum affine algebras,”. http://dx.doi.org/10.1142/9789812810007_0009. [131] H. Nakajima, “Quiver varieties and t-analogs of q-characters of quantum affine algebras,” Ann. of Math. (2) 160 no. 3, (2004) 1057–1097. http://dx.doi.org/10.4007/annals.2004.160.1057. [132] H. Nakajima, “Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras,” Duke Math. J. 76 no. 2, (1994) 365–416. http://dx.doi.org/10.1215/S0012-7094-94-07613-8. [133] H. Nakajima, “t-analogs of q-characters of quantum affine algebras of type E6 , E7 , E8 ,”. http://dx.doi.org/10.1007/978-0-8176-4697-4_10. [134] D. Hernandez, “Representations of quantum affinizations and fusion product,” Transform. Groups 10 no. 2, (2005) 163–200. http://dx.doi.org/10.1007/s00031-005-1005-9. [135] D. Hernandez, “Drinfeld coproduct, quantum fusion tensor category and applications,” Proc. Lond. Math. Soc. (3) 95 no. 3, (2007) 567–608. http://dx.doi.org/10.1112/plms/pdm017. [136] V. Chari, “Braid group actions and tensor products,” Int. Math. Res. Not. no. 7, (2002) 357–382. http://dx.doi.org/10.1155/S107379280210612X. [137] V. Chari and A. A. Moura, “Characters and blocks for finite-dimensional representations of quantum affine algebras,” Int. Math. Res. Not. no. 5, (2005) 257–298. http://dx.doi.org/10.1155/IMRN.2005.257. [138] D. Hernandez and M. Jimbo, “Asymptotic representations and Drinfeld rational fractions,” ArXiv e-prints (Apr., 2011) , arXiv:1104.1891 [math.QA]. [139] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, “Integrable structure of conformal field theory. 2. Q operator and DDV equation,” Commun.Math.Phys. 190 (1997) 247–278, arXiv:hep-th/9604044 [hep-th]. [140] M. Varagnolo and E. Vasserot, “On the K-theory of the cyclic quiver variety,” Internat. Math. Res. Notices no. 18, (1999) 1005–1028, math/9902091.

78


http://dx.doi.org/10.1155/S1073792899000525. [141] M. Varagnolo, “Quiver varieties and Yangians,” Lett. Math. Phys. 53 no. 4, (2000) 273–283, math/0005277. http://dx.doi.org/10.1023/A:1007674020905. [142] N. Guay, “Cherednik algebras and Yangians,” Int. Math. Res. Not. no. 57, (2005) 3551–3593. http://dx.doi.org/10.1155/IMRN.2005.3551. [143] N. Guay, “Affine Yangians and deformed double current algebras in type A,” Adv. Math. 211 no. 2, (2007) 436–484. http://dx.doi.org/10.1016/j.aim.2006.08.007. [144] N. Guay, “Quantum algebras and quivers,” Selecta Math. (N.S.) 14 no. 3-4, (2009) 667–700. http://dx.doi.org/10.1007/s00029-009-0496-y. [145] H. Braden, A. Gorsky, A. Odessky, and V. Rubtsov, “Double elliptic dynamical systems from generalized Mukai-Sklyanin algebras,” Nucl.Phys. B633 (2002) 414–442, arXiv:hep-th/0111066 [hep-th]. [146] H. W. Braden and T. J. Hollowood, “The Curve of compactified 6-D gauge theories and integrable systems,” JHEP 0312 (2003) 023, arXiv:hep-th/0311024 [hep-th]. [147] M. Varagnolo and E. Vasserot, “Schur duality in the toroidal setting,” Comm. Math. Phys. 182 no. 2, (1996) 469–483, q-alg/9506026. http://projecteuclid.org/getRecord?id=euclid.cmp/1104288156. [148] I. Cherednik, “Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators,” Internat. Math. Res. Notices no. 9, (1992) 171–180. http://dx.doi.org/10.1155/S1073792892000199. [149] Y. Saito, “Quantum toroidal algebras and their vertex representations,” Publ. Res. Inst. Math. Sci. 34 no. 2, (1998) 155–177, q-alg/9611030. http://dx.doi.org/10.2977/prims/1195144759. [150] I. Cherednik, “Nonsymmetric Macdonald polynomials,” Internat. Math. Res. Notices no. 10, (1995) 483–515. http://dx.doi.org/10.1155/S1073792895000341. [151] I. Cherednik, “Double affine Hecke algebras and Macdonald’s conjectures,” Ann. of Math. (2) 141 no. 1, (1995) 191–216. http://dx.doi.org/10.2307/2118632. [152] M. Varagnolo and E. Vasserot, “Double-loop algebras and the Fock space,” Invent. Math. 133 no. 1, (1998) 133–159, q-alg/9612035. http://dx.doi.org/10.1007/s002220050242. [153] Y. Saito, K. Takemura, and D. Uglov, “Toroidal actions on level 1 modules of b n ),” Transform. Groups 3 no. 1, (1998) 75–102, q-alg/9702024. Uq (sl http://dx.doi.org/10.1007/BF01237841. [154] M. Kashiwara, T. Miwa, and E. Stern, “Decomposition of q-deformed Fock spaces,” Selecta Math. (N.S.) 1 no. 4, (1995) 787–805, q-alg/9508006. http://dx.doi.org/10.1007/BF01587910.


79

[155] K. Nagao, “K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials,” Osaka J. Math. 46 no. 3, (2009) 877–907, 0709.1767. http://projecteuclid.org/getRecord?id=euclid.ojm/1256564211. [156] K. Miki, “Toroidal braid group action and an automorphism of toroidal algebra Uq (sln+1,tor ) (n ≥ 2),” Lett. Math. Phys. 47 no. 4, (1999) 365–378. http://dx.doi.org/10.1023/A:1007556926350. [157] S. Berman, Y. Gao, and Y. S. Krylyuk, “Quantum tori and the structure of elliptic quasi-simple Lie algebras,” J. Funct. Anal. 135 no. 2, (1996) 339–389. http://dx.doi.org/10.1006/jfan.1996.0013. [158] K. Miki, “A (q, γ) analog of the W1+∞ algebra,” J. Math. Phys. 48 no. 12, (2007) 123520, 35. http://dx.doi.org/10.1063/1.2823979. [159] H. Awata, H. Kubo, S. Odake, and J. Shiraishi, “Quantum WN algebras and Macdonald polynomials,” Comm. Math. Phys. 179 no. 2, (1996) 401–416. http://projecteuclid.org/getRecord?id=euclid.cmp/1104286998. [160] J. Shiraishi, H. Kubo, H. Awata, and S. Odake, “A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions,” Lett. Math. Phys. 38 no. 1, (1996) 33–51. http://dx.doi.org/10.1007/BF00398297. [161] H. Nakajima, “Heisenberg algebra and Hilbert schemes of points on projective surfaces,” Ann. of Math. (2) 145 no. 2, (1997) 379–388, alg-geom/9507012. http://dx.doi.org/10.2307/2951818. ´ Vasserot, “Langlands reciprocity for affine quantum groups [162] V. Ginzburg and E. of type An ,” Internat. Math. Res. Notices no. 3, (1993) 67–85. http://dx.doi.org/10.1155/S1073792893000078. [163] I. Grojnowski, “Instantons and affine algebras. I. The Hilbert scheme and vertex operators,” Math. Res. Lett. 3 no. 2, (1996) 275–291, alg-geom/9506020. http://dx.doi.org/10.4310/MRL.1996.v3.n2.a12. [164] V. Baranovsky, “Moduli of sheaves on surfaces and action of the oscillator algebra,” J. Differential Geom. 55 no. 2, (2000) 193–227. http://projecteuclid.org/getRecord?id=euclid.jdg/1090340878. [165] O. Schiffmann and E. Vasserot, “The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A2 ,” ArXiv e-prints (May, 2009) , arXiv:0905.2555 [math.QA]. [166] E. Carlsson and A. Okounkov, “Exts and vertex operators,” Duke Math. J. 161 no. 9, (2012) 1797–1815, 0801.2565. http://dx.doi.org/10.1215/00127094-1593380. [167] E. Carlsson, N. Nekrasov, and A. Okounkov, “Five dimensional gauge theories and vertex operators,” ArXiv e-prints (Aug., 2013) , arXiv:1308.2465 [math.RT]. [168] B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida, “A commutative algebra on degenerate CP1 and Macdonald polynomials,” J. Math. Phys. 50 no. 9, (2009) 095215, 42, 0904.2291. http://dx.doi.org/10.1063/1.3192773.

80


[169] N. Wyllard, “A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories,” JHEP 0911 (2009) 002, arXiv:0907.2189 [hep-th]. [170] H. Awata and Y. Yamada, “Five-dimensional AGT Conjecture and the Deformed Virasoro ’lgebra,” JHEP 1001 (2010) 125, arXiv:0910.4431 [hep-th]. [171] V. Fateev and A. Litvinov, “Integrable structure, W-symmetry and AGT relation,” JHEP 1201 (2012) 051, arXiv:1109.4042 [hep-th]. [172] A. Smirnov, “On the Instanton R-matrix,” ArXiv e-prints (Feb., 2013) , arXiv:1302.0799 [math.AG]. [173] H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi, et al., “Notes on Ding-Iohara algebra and AGT conjecture,” arXiv:1106.4088 [math-ph]. [174] D. Maulik and A. Okounkov, “Quantum Groups and Quantum Cohomology,” ArXiv e-prints (Nov., 2012) , arXiv:1211.1287 [math.AG]. [175] O. Schiffmann and E. Vasserot, “Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on Aˆ2,” ArXiv e-prints (Feb., 2012) , arXiv:1202.2756 [math.QA]. [176] V. A. Alba, V. A. Fateev, A. V. Litvinov, and G. M. Tarnopolskiy, “On combinatorial expansion of the conformal blocks arising from AGT conjecture,” Lett.Math.Phys. 98 (2011) 33–64, arXiv:1012.1312 [hep-th]. [177] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, “Quantum continuous gl∞ : semiinfinite construction of representations,” Kyoto J. Math. 51 no. 2, (2011) 337–364, 1002.3100. http://dx.doi.org/10.1215/21562261-1214375. [178] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, “Quantum continuous gl∞ : tensor products of Fock modules and Wn -characters,” Kyoto J. Math. 51 no. 2, (2011) 365–392, 1002.3113. http://dx.doi.org/10.1007/BF01237841. [179] I. Burban and O. Schiffmann, “On the Hall algebra of an elliptic curve, I,” Duke Math. J. 161 no. 7, (2012) 1171–1231, math/0505148. http://dx.doi.org/10.1215/00127094-1593263. [180] O. Schiffmann, “Drinfeld realization of the elliptic Hall algebra,” J. Algebraic Combin. 35 no. 2, (2012) 237–262, 1004.2575. http://dx.doi.org/10.1007/s10801-011-0302-8. [181] B. L. Feigin and A. I. Tsymbaliuk, “Equivariant K-theory of Hilbert schemes via shuffle algebra,” Kyoto J. Math. 51 no. 4, (2011) 831–854, 0904.1679. http://dx.doi.org/10.1215/21562261-1424875. [182] J.-t. Ding and K. Iohara, “Generalization and deformation of Drinfeld quantum affine algebras,” Lett.Math.Phys. 41 (1997) 181–193. [183] B. Feigin and A. Odesskii, “A family of elliptic algebras,” Internat. Math. Res. Notices no. 11, (1997) 531–539. http://dx.doi.org/10.1155/S1073792897000354. [184] B. Enriquez, “On correlation functions of Drinfeld currents and shuffle algebras,” Transform. Groups 5 no. 2, (2000) 111–120, math/9809036.


81

http://dx.doi.org/10.1007/BF01236465. [185] A. Negut, “An Isomorphism between the Quantum Toroidal and Shuffle Algebras, and a Conjecture of Kuznetsov,” ArXiv e-prints (Feb., 2013) , arXiv:1302.6202 [math.RT]. [186] A. Okounkov and R. Pandharipande, “Quantum cohomology of the Hilbert scheme of points in the plane,” Invent. Math. 179 no. 3, (2010) 523–557, math/0411210. http://dx.doi.org/10.1007/s00222-009-0223-5. [187] Y. Saito, “Elliptic Ding-Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator,” ArXiv e-prints (Jan., 2013) , arXiv:1301.4912 [math.QA]. [188] Y. Saito, “Elliptic Ding-Iohara Algebra and Commutative Families of the Elliptic Macdonald Operator,” ArXiv e-prints (Sept., 2013) , arXiv:1309.7094 [math.QA]. [189] B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, “Quantum toroidal gl1 -algebra: plane partitions,” Kyoto J. Math. 52 no. 3, (2012) 621–659, 1110.5310. c c1 : [190] G. S. Mutafyan and B. L. Fe˘ıgin, “The quantum toroidal algebra gl calculation of the characters of some representations as generating functions of plane partitions,” Funktsional. Anal. i Prilozhen. 47 no. 1, (2013) 62–76. http://dx.doi.org/10.1007/s10688-013-0006-z. [191] B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, “Representations of quantum toroidal gln ,” J. Algebra 380 (2013) 78–108, 1204.5378. http://dx.doi.org/10.1016/j.jalgebra.2012.12.029. [192] B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, “Branching rules for quantum toroidal gl(n),” arXiv:1309.2147 [math.QA]. [193] H. Fuji, S. Gukov, and P. Sulkowski, “Super-A-polynomial for knots and BPS states,” Nucl.Phys. B867 (2013) 506–546, arXiv:1205.1515 [hep-th]. [194] F. A. Smirnov, “Dynamical symmetries of massive integrable models, 1. Form-factor bootstrap equations as a special case of deformed KnizhnikZamolodchikov equations,” Int.J.Mod.Phys. A71B (1992) 813–837. [195] F. Smirnov, “Form-factors in completely integrable models of quantum field theory,” Adv.Ser.Math.Phys. 14 (1992) 1–208. [196] I. B. Frenkel and N. Y. Reshetikhin, “Quantum affine algebras and holonomic difference equations,” Comm. Math. Phys. 146 no. 1, (1992) 1–60. http://projecteuclid.org/getRecord?id=euclid.cmp/1104249974. [197] B. Feigin, E. Frenkel, and N. Reshetikhin, “Gaudin model, Bethe ansatz and correlation functions at the critical level,” Commun.Math.Phys. 166 (1994) 27–62, arXiv:hep-th/9402022 [hep-th]. [198] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, “Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz,” Commun.Math.Phys. 177 (1996) 381–398, arXiv:hep-th/9412229 [hep-th]. [199] M.-x. Huang, “On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit,” JHEP 1206 (2012) 152, arXiv:1205.3652 [hep-th].

82


[200] A. Chervov and D. Talalaev, “Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence,” arXiv:hep-th/0604128 [hep-th]. [201] I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, “Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations,” Commun.Math.Phys. 188 (1997) 267–304, arXiv:hep-th/9604080 [hep-th]. [202] A. Beilinson and V. Drinfeld, “Opers,” ArXiv Mathematics e-prints (Jan., 2005) , math/0501398. [203] A. A. Beilinson and V. G. Drinfeld, “Quantization of Hitchin’s fibration and Langlands’ program,”. http://dx.doi.org/10.1007/978-94-017-0693-3_1. [204] A. A. Beilinson and V. G. Drinfeld, “Quantization of hitchin’s integrable system and hecke eigensheaves,”. http://www.math.uchicago.edu/~mitya/langlands/QuantizationHitchin.pdf. [205] E. Frenkel, “Affine algebras, Langlands duality and Bethe ansatz,” arXiv:q-alg/9506003 [q-alg]. [206] E. Frenkel and N. Reshetikhin, “Quantum affine algebras and deformations of the Virasoro and W-algebras,” Comm. Math. Phys. 178 no. 1, (1996) 237–264, q-alg/9505025. http://projecteuclid.org/getRecord?id=euclid.cmp/1104286563. [207] E. Frenkel, “Opers on the projective line, flag manifolds and Bethe Ansatz,” arXiv:math/0308269 [math-qa]. [208] E. Frenkel, “Affine Kac-Moody algebras, integrable systems and their deformations,” ArXiv Mathematics e-prints (May, 2003) , math/0305216. [209] A. Litvinov, S. Lukyanov, N. Nekrasov, and A. Zamolodchikov, “Classical conformal blocks and Painlevé VI,” arXiv:arXiv:1309.4700 [hep-th]. [210] A. Kuniba, M. Okado, J. Suzuki, and Y. Yamada, “Difference L operators related to q-characters,” J. Phys. A 35 no. 6, (2002) 1415–1435. http://dx.doi.org/10.1088/0305-4470/35/6/307. [211] M. Jimbo, “Introduction to the Yang-Baxter equation,” in Braid group, knot theory and statistical mechanics, vol. 9 of Adv. Ser. Math. Phys., pp. 111–134. World Sci. Publ., Teaneck, NJ, 1989. http://dx.doi.org/10.1142/9789812798350_0005. [212] B. Enriquez, “Quantum currents realization of the elliptic quantum groups Eτ,η (sl2 ),” in Calogero-Moser-Sutherland models (Montréal, QC, 1997), CRM Ser. Math. Phys., pp. 161–176. Springer, New York, 2000. [213] P. I. Etingof, I. B. Frenkel, and A. A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, vol. 58 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998. [214] T. Tanisaki, “Killing forms, Harish-Chandra isomorphisms, and universal R-matrices for quantum algebras,”. [215] V. N. Tolstoy and S. M. Khoroshkin, “The universal r-matrix for quantum untwisted affine lie algebras,” Funct. Anal. Appl. 26 (1992) 69–71. http://dx.doi.org/10.1007/BF01077085.


[216] S. M. Khoroshkin and V. N. Tolstoy, “On Drinfel′ d’s realization of quantum affine algebras,” J. Geom. Phys. 11 no. 1-4, (1993) 445–452. http://dx.doi.org/10.1016/0393-0440(93)90070-U. Infinite-dimensional geometry in physics (Karpacz, 1992). [217] S. Khoroshkin and V. Tolstoi, “Twisting of quantum (super)algebras: Connection of Drinfeld’s and Cartan-Weyl realizations for quantum affine algebras,” arXiv:hep-th/9404036 [hep-th]. Simons Center for Geometry and Physics, Stony Brook, USA on leave of absence from

Institut des Hautes Etudes Scientifiques, France, Institute of Theoretical and Experimental Physics, Russia, Institute for Information Transmission Problems, Lab. 5, Russia E-mail address: [email protected] Institute for Advanced Study, Princeton, USA E-mail address: [email protected] School of Mathematics, Trinity College Dublin, Ireland Hamilton Mathematics Institute, TCD, Dublin 2, Ireland Chaire Louis Michel, Institut des Hautes Etudes Scientifiques, France, on leave of absence from

Euler International Mathematical Institute, St. Petersburg, Russia Institute for Information Transmission Problems, Lab. 5, Russia E-mail address: [email protected]

83