Dec 24, 2008 - x3/2. 1 â x2. (ây1 y2. +. ây2 y1. ) . (2.16). 3The same index was computed in [21] using a different ..... S3/2 = x4Q1 . ...... Samsung Scholarship.
Preprint typeset in JHEP style - HYPER VERSION
CALT-68-2708
arXiv:0811.2855v2 [hep-th] 24 Dec 2008
Superconformal Indices for Orbifold Chern-Simons Theories
Jaehyung Choi1 , Sangmin Lee2 and Jaewon Song3 1
Department of Physics and Astronomy, SUNY, Stony Brook, NY 11794-3800, USA Department of Physics & Astronomy, Seoul National University, Seoul 151-747, Korea 3 California Institute of Technology 452-48, Pasadena, CA 91125, USA 2
Abstract: We calculate the superconformal indices of recently discovered threedimensional N = 4, 5 Chern-Simons-matter theories and compare them with the corresponding indices of supergravity on AdS4 times orbifolds of S 7 . We find perfect agreement in the large N and large k limit, provided that the twisted sector contributions at the fixed loci of the orbifolds are properly taken into account. We also discuss the index for the so-called “dual ABJM” proposal. Keywords: Superconformal index, Chern-Simons, M2-brane.
Contents 1. Introduction
1
2. Reviews 2.1 Superconformal index in three dimensions 2.2 N = 4, 5, 6 Chern-Simons matter theories 2.3 Indices for N = 6 ABJM theories
3 3 4 6
3. Indices for N = 5 theories 3.1 Field theory 3.2 Gravity
8 8 11
4. Indices for N = 4 theories 4.1 U -type 4.2 OSp-type 4.3 On the dual ABJM proposal
13 13 17 19
5. Conclusion
21
A. Matrix integrals and measure factors
23
B. Character formulas for SO(2r + 1) and SO(2r)
24
1. Introduction Recently, there has been a lot of progress in understanding the low energy effective theory of coincident M2-branes. Schwarz raised the possibility of using Chern-Simons theories to describe the superconformal theory on M2-branes [1]. The idea was first realized concretely by Bagger and Lambert [2] and Gustavsson [3] (BLG), where they constructed a Lagrangian for SU (2)×SU (2) Chern-Simons theory with N = 8 supersymmetry. The BLG theory had some peculiarities that appeared puzzling from the M2-brane point of view. First, it was not clear how to extend this theory to describe arbitrary number of M2-branes. Second, a string/M-theoretic derivation of the BLG theory was lacking. Soon afterwards, the BLG theory was followed by a variety of superconformal ChernSimons theories more clearly rooted in string/M-theory.1 In a type IIB string theory setup, Gaiotto and Witten [5] gave a general construction of N = 4 Chern-Simons theories with one type of hyper-multiplets, where the theories were shown to be classified by 1
In this paper, we will focus on relatively new N ≥ 4 theories only. For a nice summary of more conventional N ≤ 3 Chern-Simons theories, see e.g. [4] and references therein and thereto.
–1–
an auxiliary Lie super-algebra. This construction was extended in [6] to include twisted hyper-multiplets, so that all N ≥ 4 theories can, in principle, fit into the Gaiotto-Witten classification. It was also pointed out in [6] that the Gaiotto-Witten setup can be related via T-duality to M2-branes probing orbifold geometries. Aharony, Bergman, Jafferis and Maldacena [7] (ABJM) then performed an in-depth study of an especially simple and instructive case of N = 6 theory with U (N ) × U (N ) gauge group. They gave convincing arguments that the Chern-Simons theory at level k is dual to M-theory on AdS4 × S 7 /Zk . The ABJM theory thus opened up a laboratory for testing the long-missing AdS4 /CFT3 duality and led to many exciting developments. More work on explicit construction and classification of N = 4, 5, 6 theories can be found in [8, 9, 10, 11, 12, 13, 14, 15]. The aim of this paper is to compute the superconformal index [16, 17] for some N = 4, 5 quiver Chern-Simons theories constructed in [6, 10] and further studied in [8, 9, 12], which are believed to be dual to M-theory on AdS4 times certain orbifolds of S 7 . The superconformal index, originally defined for 4-dimensional theories in [16, 17] counts the number of certain chiral operators. This index gets contributions only from short multiplets which cannot combine to form long multiplets as the parameters of the theory are varied. Just as for the Witten index [18], the superconformal index does not change under continuous change of parameters. The superconformal indices in 3, 5, 6 dimensions were constructed in [19, 21], and the index for the ABJM theory has been computed in [20, 21]. As in [20], we compute the superconformal indices for N = 4, 5 Chern-Simons theories in the large N and large k limit, where N is the rank of the gauge group and k is the Chern-Simons level. As explained in [7], in the large k limit, the circle fiber of S 7 shrinks. So, in our case, the compact space becomes an orbifold of CP3 . In the large N limit with ’t Hooft coupling λ = N/k fixed at a large value, the supergravity description is valid and we can compute the index by counting the Kaluza-Klein spectrum of the compact space. On the field theory side, since we have the effective coupling λ which can be taken to zero, we can compute the index in the free theory limit. However, when the orbifold has a non-trivial fixed locus, we have massless twisted sector contributions which survive the large N limit. This is an inherently stringy effect, so we cannot solely rely on supergravity description. Similar situation in four-dimensional quiver gauge theories has been discussed in [22]. We discuss how to figure out the twisted sector contribution to the index both in field theory and gravity side. After taking into account the twisted sector, we show that the indices in λ = 0 and λ = ∞ agree. The outline of this paper is as follows. In section 2, we briefly review the definition of the superconformal index, some salient features of the N = 4, 5 theories we study in later sections, and the computation of the index in ABJM theory [20]. In section 3 and 4, we compute the index for N = 5 and N = 4 theories in field theory as well as in gravity. Perfect agreement is found in the N = 5 without twisted sector contributions. Agreement is possible for N = 4 theories if and only if the twisted sector contributions are taken into account. In subsection 4.3, we also briefly discuss the “dual ABJM model” proposed in [23, 24]. In section 5, we conclude with some open questions. Some useful formulas are provided in the appendix.
–2–
2. Reviews 2.1 Superconformal index in three dimensions In many supersymmetric theories, there exist short multiplets which contain smaller number of states than ordinary long multiplets. Some of these short multiplets can be combined into long multiplets as the parameters of the theory are varied, but there are others that cannot. The superconformal index defined in [16, 17] receives contributions only from the latter states, so it is kept constant while the couplings of the theory change. In three dimensions, the bosonic subgroup of the superconformal group OSp(N |4) is SO(3, 2)×SO(N ), and its maximally compact subgroup is SO(2)×SO(3)×SO(N ), where N is the number of supersymmetries. Following [19, 20], we denote the eigenvalues of the Cartan generators of SO(2) × SO(3) × SO(N ) by 0 , j and hi (i = 1, · · · , [N /2]). In radial quantization (compactifying the theory on R × S 2 ), 0 and j are interpreted as energy and angular momentum. There are 4N real supercharges with 0 = ±1/2. In this paper, we will compute the three-dimensional version of the index of [16], h i h /2] I(x, {yi }) = Tr (−1)F x0 +j y1h2 · · · y[N[N/2]−1 . (2.1) The definition of the index singles out a particular supercharge Q with charges 0 = +1/2, j = −1/2, h1 = 1, hi = 0 (i ≥ 2). The superconformal algebra implies that {Q, Q† } = ∆ ≡ 0 − j − h1 .
(2.2)
The short multiplets contributing to the index satisfy ∆ = 0 and are annihilated by both Q and Q† . Note that these states can be interpreted as elements of Q-cohomology class. Since ∆ = QQ† + Q† Q, we can think of Q as analogous to the d operator in de Rham cohomology and ∆ to the Laplacian operator. On the field theory side, the index in the free theory limit can be computed using a matrix-integral formula [25], ! Z Y ∞ XX 1 n n n n (2.3) I(x, {yi }) = DUa exp fR (x , y1 , · · · y[N /2] )χR ({Ua }) , n a R n=1
where R denotes the representations of matter fields (“letters”), fR is the index computed over the single letters without restriction on gauge invariance, and χR is the group character. In the case of U (N ) gauge theory with bi-fundamental matter fields, χab ({Uc }) = TrUa TrUb† .
(2.4)
The index (2.3) enumerates all possible gauge-invariant multi-trace operators. The standard rule of Bose statistics (or equivalently the plethystic integral) relates the “multiparticle” index I to the “single-particle” index Isp through ! ∞ X 1 k k I(x, {yi }) = exp Isp (x , {yi }) . (2.5) k k=1
–3–
On the gravity side, the single particle index Isp can be computed from the KaluzaKlein spectrum of eleven-dimensional supergravity compactified on the internal sevenmanifold. For S 7 , Isp was computed in [19] using the known Kaluza-Klein spectrum in supergravity [26, 27]. For orbifolds of S 7 , the index counts states in untwisted and twisted sectors. The untwisted sector is simply the subset of the spectrum for S 7 that is invariant under the orbifold action. If the orbifold has fixed points, there may be additional twisted sector states localized at the fixed points. We will see that N = 5, 6 orbifolds have no twisted sector contributions, while N = 4 orbifolds do have such contributions. As usual, the index should not change as we vary the continuous parameter of the theory [18]. Contrary to the case of Yang-Mills theory, the effective coupling of the ChernSimons theory λ = N/k is discrete. But, when we take the large N limit, δλ = −λ2 /N becomes effectively continuous. So we can take the limit of ’t Hooft coupling λ = N/k to be zero, and then the theory becomes free. This means that we also have to take the Chern-Simons level k to be infinite, so k never enters into our calculation. In the following discussion, we take both large N and large k limit. 2.2 N = 4, 5, 6 Chern-Simons matter theories
In this section, we give a short summary of the N = 4, 5, 6 superconformal Chern-Simons matter theories we will study in later sections. Our notations closely follow those of [6, 10]. Our discussion will be brief, and we refer the reader to the original papers [5, 6, 7, 8, 9, 10, 11, 12, 13, 14] for details. The field theory computations in this paper will be done in free theory limit, so for most purposes, it is sufficient to recall the matter content and gauge symmetry of the theory. We proceed in descending order of number of supersymmetries. N = 6 ABJM theory The gauge group is U (M ) × U (N ). The theory makes sense at quantum level if and only if the rank of the gauge group and the Chern-Simons level k satisfy |M − N | ≤ k [29]. The matter content is summarized by the following table: ¯α ¯α Φα Φ Ψα Ψ ¯ ) (N, M ¯ ) (M, N ¯ ) (N, M ¯) U (M ) × U (N ) (M, N ¯ ¯ SO(6)R 4 4 4 4
(2.6)
To compare with the index computation of [20], we choose the convention for SO(6)R highest weights such that 4 representation have (h1 , h2 , h3 , h4 ) = (+ 21 , + 12 , + 12 , − 12 ). As explained in [7], the moduli space of vacua of this theory is (a symmetric product of) C4 /Zk . The scalars Φα can be thought of as coordinates on C4 . The orbifold action Zk , a residual discrete gauge symmetry on the Coulomb branch, acts on Φα as Φα → e2πi/k Φα . N = 5 theory The gauge group is O(M ) × Sp(2N ).2 Let us denote the generators of O(M ) and Sp(2M ) as Mab and Ma˙ b˙ , respectively. The invariant anti-symmetric tensor of Sp(2M ) is denoted by ωa˙ b˙ . The invariant tensor of Sp(4) = SO(5)R is denoted by Cαβ . We denote the bi-fundamental matter fields by Φaαa˙ , 2
Ψaαa˙ .
We use the notation in which Sp(2N ) has rank N .
–4–
(2.7)
They obey the reality condition of the form ˙ ¯ αaa Φ = (Φaαa˙ )† = δab ωa˙ b˙ C αβ Φbβb , ˙
(2.8)
and similarly for the fermions. As explained in [10], the moduli space of vacua of this theory is C4 /Dk where Dk is the binary dihedral group with 4k elements. The dihedral group is generated by α : Φα → eπi/k Φα ,
¯β . β : Φα → Cαβ Φ
(2.9)
˙ doublet indices for the SU (2)L × SU (2)R RN = 4 quiver theories We use (α, β; α, ˙ β) ˙ symmetry group. We denote the invariant tensors by αβ , α˙ β˙ and their inverses by αβ , α˙ β such that αγ γβ = δ αβ , α˙ γ˙ γ˙ β˙ = δ α˙β˙ . The hyper-multiplets are denoted by (qα , ψα˙ ) and the twisted hyper-multiplets by (˜ qα˙ , ψ˜α ). A doublet of SU (2)L has the SO(4) highest weight (h1 , h2 ) = ( 21 , 21 ) and a doublet of SU (2)R has ( 21 , − 12 ). We will consider two types of N = 4 quiver gauge theories : the U (M |N )-type and the OSp(M |N )-type [6] or more briefly U -type and OSp-type. The U -type quivers can be viewed as an extension of the N = 6 theory. It consists of a product of 2m U (Ni ) gauge groups. The ranks can be different in general, but for simplicity we will assume that they ¯ ) of are all equal, so the gauge group is U (N )2m . There are m hypers (qαi , ψαi˙ ) in (N, N i i ˜ ¯ U (N )2i−1 × U (N )2i and m twisted hypers (˜ qα˙ , ψα ) in (N, N ) of U (N )2i × U (N )2i+1 . The † β hermitian conjugates (¯ qα = αβ (q ) , etc.) belong to the same R-symmetry representation, but form anti-bi-fundamental representations under the gauge groups. The matter content of the U (M |N )-type quiver theory is summarized in Fig. 1(a). The OSp-type quivers can be regarded as an extension of the N = 5 theory. The quiver consists of an alternating series of m factors of O(Mi ) and Sp(Ni ). For simplicity, we assume that the gauge group is [O(2N ) × Sp(2N )]m . There are m hypers (qαi , ψαi˙ ) in bi-fundamental of O(2N )i × Sp(2N )i and m twisted hypers (˜ qαi˙ , ψ˜αi ) in bi-fundamentals of Sp(2N )i × O(2N )i+1 . The hermitian conjugates obey a reality condition similar to (2.8) ˙ with C αβ replaced by αβ or α˙ β . The matter content of the OSp-type quiver theory is summarized in Fig. 1(b).
qiα
q˜iα˙−1
q˜iα˙
qiα+1
q¯˜iα˙
q¯iα+1
(a) q¯˜iα˙−1 q¯iα U ( N )2i−1 U ( N )2i
(b)
qiα
q˜iα˙−1 O(2N )i
U ( N )2i+1
qiα+1
q˜iα˙ Sp(2N )i
O(2N )i+1
Figure 1: Matter content of (a) U -type and (b) OSp-type quiver theories.
The moduli space of vacua of the U -type theories were studied in [8] and [12]. When the Chern-Simons level k is unity, the solution to F-term and D-term conditions gives the
–5–
moduli space of vacua (C2 /Zm )2 , where the generators of Zm act on coordinates of C4 as (z1 , z2 , z3 , z4 ) ∼ (ωm z1 , ωm z2 , z3 , z4 ) ∼ (z1 , z2 , ωm z3 , ωm z4 ) (ωm ≡ e2πi/m ) .
(2.10)
For k > 1, as in the N = 6 theory, the residual discrete gauge symmetry induces further orbifolding by (z1 , z2 , z3 , z4 ) ∼ ωk (z1 , z2 , z3 , z4 ) .
(2.11)
Note that all the orbifold actions (2.10, 2.11) can be generated by two (as opposed to three) generators. In the simple case where m and k are relatively prime, we may choose the independent generators to be (ωm , ωm , 1, 1) and (ωmk , ωmk , ωmk , ωmk ) ,
(2.12)
and say that the moduli space is C4 /(Zm × Zmk ). Similarly, the moduli space of the OSptype theories can be shown to be (C2 /Zm )2 /Dk ' C4 /(Zm × Dmk ). The orbifold action Zm is the same as (2.10) and the action of Dk is the same as in the N = 5 theory. For later purposes, let us review some details of the supersymmetry algebra of the N = 4 theories. We denote the supercharges by Qαα˙ and write their components as Q±± . The matter fields can be written as q± , ψ± and so on. In this notation, the special supercharge involved in the definition of the superconformal index is Q++ . The supersymmetry transformation rule includes h i i i i i i i+1 i+1 i i i Q++ , q+ = 0, Q−+ , q+ = ψ+ , Q++ , ψ˜+ = q˜+ q+ q¯+ − q¯+ q+ q˜+ . (2.13) and similar relations obtained by hermitian conjugation and/or exchange of hypers with twisted hypers. 2.3 Indices for N = 6 ABJM theories
We now review the computation of the index of the N = 6 theory following [20]. 3 We will be slightly more general and allow the gauge group to be U (M ) × U (N ). The compact bosonic subgroup of the superconformal group is SO(2) × SO(3) × SO(6), and the index is given by I(x, y1 , y2 ) = Tr (−1)F x0 +j y1h2 y2h3 , (2.14) where hi ’s are the second and third Cartan charges of SO(6). The single letter index for bi-fundamental and anti-bi-fundamental matter fields are r r y1 y2 x3/2 1 x1/2 √ + − y1 y2 + √ , (2.15) f12 = 1 − x2 y2 y1 1 − x2 y1 y2 r r x1/2 1 x3/2 y1 y2 √ f21 = y1 y2 + √ − + . (2.16) 2 2 1−x y1 y2 1−x y2 y1 3
The same index was computed in [21] using a different method of evaluating the integral (2.17). The same reference also gives general formulas for the index for superconformal algebra OSp(2N |4). See also a related work [28] where the indices of SO/Sp gauge theories (in four dimensions) were computed, which complements the method we develop in the next section.
–6–
On the field theory side, the superconformal index is given by Z ∞ XX 1 I = DU1 DU2 exp fab (xn , y1n , y2n )Tr(Uan )Tr(Ub†n ) . n
(2.17)
a,b n=1
The only difference from [20] is that now U2 is an element of U (M ). Now, we make the change of variables in a standard way in large N computations, ρn =
1 TrU1n , N
χn =
1 TrU2n . M
(2.18)
Then the measure is given by (a derivation of this measure factor is given in appendix A) ! ∞ X ρn ρ−n Y DU1 = dρn exp −N 2 , (2.19) n n n=1 ! ∞ Y X χn χ−n . (2.20) DU2 = dχn exp −M 2 n n n=1
Substituting this, and writing M = N + m, we get I=
Y n
N2 X 1 T dρn dχn exp − C Mn C n 2 n n n
! .
Here, CnT = (χn ρn χ−n ρ−n ) and 0 0 (1 + α)2 −(1 + α)f21;n 0 0 −(1 + α)f12;n 1 Mn = , 2 (1 + α) −(1 + α)f12;n 0 0 −(1 + α)f21;n 1 0 0
(2.21)
(2.22)
with α = m/N and fab;n ≡ fab (xn , y1n , y2n ). The overall normalization of the Gaussian integral is fixed by requiring that I(x = 0, yi = 0) = 1. The final result is I = Isp
Y Y (1 + α)2 √ = (1 − detMn n n
xn y1n
(1 − x2n )2 , n )(1 − xyn )(1 − xn y1n )(1 − xn y2n ) 2
x 1 1 x 2 = + + + − . y1 − x 1 − xy1 y2 − x 1 − xy2 1 − x2
(2.23)
Note that the result is independent of α as we take the large N limit. This agrees with the observations on the gravity dual [29]. The only difference comes from the torsion flux or in type IIA description, NS-NS 2-form flux through CP1 ⊂ CP3 . Since the torsion does not affect the classical equations of motion, we expect exactly the same set of graviton states as in U (N ) × U (N ) theory. The torsion flux makes difference in the baryonic spectrum, but they do not contribute to the index in this limit. In [29], it was also shown that this theory is superconformal only if |M − N | ≤ k. But in our field theory calculation, we are taking limit of λ = N/k → 0. Therefore, we do not expect any sign of inconsistency for |M − N | large. In view of these observations, we will neglect the differences in the ranks of the gauge group in the following sections.
–7–
3. Indices for N = 5 theories In this section, we calculate the index of the N = 5 O(2N ) × Sp(2N ) theory of [10]. The bosonic subgroup of the superconformal group is SO(3, 2) × SO(5), so we define the index to be h i I = Tr (−1)F x0 +j y h2 . (3.1) where h2 is the second Cartan charge of SO(5). 3.1 Field theory We begin with the formula for the index of a free field theory with two gauge groups and bi-fundamental matter fields only, ! Z ∞ X 1 n n n n I = DADB exp f (x , y )Tr(A )Tr(B ) , (3.2) n n=1
For the N = 5 theory, we take A ∈ SO(M ) and B ∈ Sp(2N ). The function f denotes the index of ∆ = 0 “letters”. The letters contributing to f are summarized in the following table: type bosons fermions derivatives
operators Φ Ψ ∂
0 1 2
1 1
j 0
SO(5) ( 12 , 12 ) ( 12 , 12 ) (0, 0)
1 2
1
Table 1: The ∆ = 0 letters of the N = 5 theory.
Contrary to the N = 6 case, there is no distinction between bi-fundamental and antibi-fundamental matter fields. We also lose a Cartan generator and, consequently, the 4 and ¯ 4 representations of SO(6) become the same representation of SO(5). It follows that, to obtain the index over the letters of the N = 5 theory, we can simply set y2 = 1 in the N = 6 result (2.15) or (2.16). The result is 1 3 1 1 1 1 x2 x2 x2 √ √ √ f (x, y) = y+√ − y+√ = y+√ . (3.3) 1 − x2 y 1 − x2 y 1+x y Integration measure SO(2N ) :
Any element A ∈ SO(2N ) can be block-diagonalized into the form ! N M cos αi − sin αi A= (|αi | ≤ π) . sin αi cos αi i=1
(3.4)
In this basis, the Harr measure over the SO(2N ) group manifold is (see appendix A) DA =
N Y i=1
dαi
Y i 0 without loss of generality and restrict the domain of ρ(θ) to be [0, π], so that Z π ρ(θ)dθ = N. (3.7) 0
Now, instead of integrating over αi ’s we can integrate over the Fourier modes of ρ : Z π X 1 1 ρn cos(nθ) , ρ(θ) cos(nθ)dθ . (3.8) ρ(θ) = ρ0 + ρn = 2 π 2 0 n≥1
The normalization of ρn is chosen such that4 Z X n Tr(A ) = 2 cos(nαi ) = 2 dθρ(θ) cos(nθ) = ρn .
(3.9)
i
We can rewrite the measure factor as Y X α + α α − α i j i j 2 DA = dαi exp log sin2 + log sin 2 2 i i
