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A Note on the Partition Function of ABJM theory on S3 2 1 0 2 n a J 0 Kazumi Okuyama 1 Department of Physics, Shinshu University, Matsumoto 390-8621, Japan ] h t [email protected] - p e h [ 3 v 5 5 5 We study the partition function Z of U(N) U(N) Chern-Simons matter theory 3 k × −k . (ABJM theory) on S3 which is recently obtained by the localization method. We evaluate 0 1 the eigenvalue integral in Z exactly for the N = 2 case. We find that Z has a different 1 1 dependence on k for even k and odd k. We comment on the possible implication of this : v result in the context of AdS/CFT correspondence. i X r a 1 1 Introduction In the seminal paper [1], the theory on the N coincident M2-branes on the orbifold R8/Z k was identified as the d = 3 = 6 U(N) U(N) Chern-Simons matter theory (ABJM k −k N × theory). Recently, the partition function Z of ABJM theory on S3 was obtained by N,k the localization method [2], and Z was given in the form of a matrix integral. The N,k behavior of Z has been analyzed previously [3, 4, 5, 6] in the ’t Hooft limit N,k N k,N , t = = fixed , (1.1) → ∞ k 3 and it was shown that the free energy F = logZ exhibits the correct N scaling as N,k 2 − predicted by the holographic dual gravity theory. The ABJM theory in the ’t Hooft limit is holographically dual to the type IIA theory on AdS CP3, which appears from the S1 4 × reduction of the M-theory on AdS S7/Z when k N1. 4 k 5 × ≫ However, if we are interested in the dynamics of M2-branes in the truly M-theory regime, or in the strong coupling regime of type IIA theory, we need to know the behavior ofZ atfinitek, sincetheIIAstringcouplingisinversely proportionaltok. Ofparticular N,k interest is the ABJM theory at k = 1, which is conjectured to describe the M2-branes on the flat eleven dimensional Minkowski space. Therefore we might want to develop a technique to analyze the partition function Z in the M-theory regime where N,k N , k = finite . (1.2) → ∞ This regime was studied in [7] by the saddle point method for the eigenvalue integral. In this paper we find the exact partition function Z for N = 2 with finite k by N,k performing the eigenvalue integral explicitly for the N = 2 case. We find that the result depends on the parity of k: k−1 k−1 1 1 s πs ( 1) Z = ( 1)s−1 tan2 + − 2 , (1.3) 2,oddk k − 2 − k k π s=1 (cid:18) (cid:19) X k−1 2 1 1 s πs Z = ( 1)s−1 tan2 . (1.4) 2,evenk k − 2 − k k s=1 (cid:18) (cid:19) X For both even k and odd k cases, the summation over s has a natural interpretation as the effect of Z orbifolding of R8/Z . k k This paper is organized as follows. In section 2, we first rewrite the partition function of ABJM theory on S3 in terms of the integrals associated with the cyclic permutations. Then we consider the grand partition function of ABJM theory, following the similar analysis of the matrix integrals which arise from the dimensional reduction of super Yang- Mills theories to 0-dimension [8, 9]. We also comment on the mirror description of the partition function of ABJM theory. In section 3, we compute the partition function of 2 U(2) U(2) ABJM theory and find that the result depends on the parity of k. In k −k × section 4, we speculate the possible implication of this result in the context of AdS/CFT correspondence. InAppendixAandB,wepresentthedetailsofthecalculationofintegrals used in section 3. 2 Structure of the Partition Function of ABJM The- ory on S3 2.1 Grand Partition Function of ABJM Theory Recently, by applying the localization method of [10], the partition function of general = 2 Chern-Simons matter theories on S3 with the gauge group G and the matter chiral N multiplet in a representation R R∗ was obtained in a form of matrix integral [2] 1 ⊕ 1 det (sinhπa) Z = dae−iπka2 Ad . (2.1) W det (coshπa) R | | Z Above, the integral of a is over the Cartan subalgebra of G, W is the order of the Weyl | | group of G, and k is the Chern-Simons coupling which is quantized to be an integer. Note that a originates from the constant mode of the real scalar field in the vector multiplet. Since the ABJM theory is the d = 3 U(N) U(N) Chern-Simons theory with k −k × bi-fundamental matter multiplets, its partition function on S3 is given by 1 Z = dNσdNσ∆(σ,σ)2eiπk(σ2−σ2) , (2.2) N,k e (N!)2 Z where σ2 is the shorthand for N σ2, and siemilarlyeσ2 = N σ2, and ∆(σ,σ) is given i=1 i i=1 i by P P sinhπ(σ σ )sinhπ(σ σ ) ∆(σ,σ) = i<j i − j e i − je . e (2.3) coshπ(σ σ ) Q i,j i − j e e Using the Cauchy identitye[11] Q e N 1 ∆(σ,σ) = ( 1)ρ , (2.4) − coshπ(σ σ ) i ρ(i) ρX∈SN Yi=1 − e the partition function is rewritten as e N 1 1 Z = dNσdNσeiπk(σ2−σ2) ( 1)ρ . (2.5) N,k e N! − coshπ(σ σ )coshπ(σ σ ) Z ρX∈SN Yi=1 i − i i − ρ(i) e 1 The partition function of the theory with matter multiplet in the non-self-conjugate representation e e was obtainedin [12, 13]. Note that (2.1)is valid only when the R-chargecarriedby the matter multiplet is 1/2. The partition function for the case of non-canonical R-charge q was also calculated in [12, 13]. See also [14] for a recent review on the localization technique in d=3 theories. 3 The sum over permutations can be simplified by noting that the integral depends only on the conjugacy class of permutation. The conjugacy class of permutation ρ is labeled by the cycle of length ℓ and the number d of such cycles contained in ρ ℓ [ρ] = [1d12d2 NdN] ℓdℓ , N = ℓd . (2.6) ℓ ··· ≡ hYℓ i Xℓ The number of elements in the conjugacy class [ρ] and the signature are given by N! #[ρ] = , ( 1)ρ = ( 1)Pℓdℓ(ℓ−1) . (2.7) ℓℓdℓdℓ! − − One can show that the integQral in (2.5) is decomposed into the integral associated with the cyclic permutation N 1 ( 1)ℓ−1A dℓ ℓ,k Z = − , (2.8) N,k d ! ℓ dℓ≥0,XPℓdℓ=NYℓ=1 ℓ (cid:20) (cid:21) where A denotes the integral coming from the cycle of length ℓ ℓ,k ℓ 1 A = dℓσdℓσeiπk(σ2−σ2) . (2.9) ℓ,k e coshπ(σ σ )coshπ(σ σ ) i i i i+1 Z i=1 − − Y e Here the mod-ℓ identification σ σ should be understood. ℓ+1 1 e e ≡ By introducing the chemical potential µ for N, the grand partition function is defined by e e ∞ (µ) = eµNZ . (2.10) k N,k Z N=0 X From (2.8) one can easily see that (µ) is exponentiated after summing over d ’s k ℓ Z ∞ ( 1)ℓ−1 (µ) = exp − eµℓA . (2.11) k ℓ,k Z ℓ " # ℓ=1 X Once we know the grand partition function, we can recover the fixed N partition function from the integral of (µ) by analytically continuing the chemical potential to a pure k Z imaginary value µ = iθ 2π dθ Z = e−iNθ (iθ) . (2.12) N,k k 2π Z Z0 It would be interesting to see whether the grand partition function of ABJM theory has a hidden integrable structure as in [9]. 4 2.2 Mirror Description of ABJM Theory By the mirror symmetry, the ABJM theory is dual to a theory without Chern-Simons term. More concretely, the mirror of ABJM theory is a U(N) super Yang-Mills theory with matter hypermultiplets in certain representations of U(N). As discussed in [11], the partition function on S3 is a useful tool to check this type of mirror symmetry. The key relation to prove the equality of partition functions of the original theory and its mirror is the following identity e2πixσ 1 dx = . (2.13) coshπx coshπσ Z Using this relation, the partition function of ABJM theory Z is rewritten as N,k Z = k2N dNσdNσdNxdNy ( 1)ρeiπkPNi=1[σi2−σei2+2xi(σi−σei)+2yi(σi−σeρ(i))] . (2.14) N,k N! − N coshπkx coshπky Z ρX∈SN i=1 i i e Q After doing the Gaussian integral for σ,σ and using the identity (2.13) again for the y-integral, (2.14) becomes e N N 1 1 Z = dx ( 1)ρ . (2.15) N,k i N! − coshπkx coshπ(x x ) i i ρ(i) Z Yi=1 ρX∈SN Yi=1 − Applying the Cauchy identity for the sum over permutations, we arrive at the mirror expression of the partition function of ABJM theory 1 N sinh2π(x x ) Z = dx i<j i − j . (2.16) N,k i N! coshπkx coshπ(x x ) Z i=1 i Q i i,j i − j Y Q Q From this, we can read off the matter content of the mirror of ABJM theory. When k = 1, the mirror theory is the U(N) super Yang-Mills theory with one adjoint and one fundamental hypermultiplets, where the factors 1/ coshπ(x x ) and 1/ coshπx i,j i− j i i in (2.16) are the 1-loop determinant of those hypermultiplets, respectively [11]. When Q Q k 2 it is not clear whether the factor 1/ coshkπx can be interpreted as the 1-loop ≥ i i determinant of hypermultiplet in some representation R. In particular it is different from Q the 1-loop determinant of hypermultiplet in the kth symmetric product of fundamental representations. The grand partition function of the mirror theory of ABJM theory has the same form as (2.11), and the contribution from the cycle of length ℓ in the mirror description is given by ℓ 1 A = dℓx . (2.17) ℓ,k coshπkx coshπ(x x ) i i i+1 Z i=1 − Y 5 3 Partition function of U(2) U(2) ABJM theory k k × − In this section, we study the partition function Z of U(2) U(2) ABJM theory. 2,k k −k × Since this model is conjecture to describe the dynamics of two M2-branes on R8/Z , we k expect that some informationof the two-bodyinteraction of M2-branes is contained in the partition function Z . Therefore, the study of the partition function of U(2) U(2) 2,k k −k × theory would be a modest first step toward the understanding of the still mysterious multiple M2-brane dynamics.2 Here we evaluate the eigenvalue integral of Z in (2.5) explicitly for the N = 2 case. N,k To do that, we first rewrite Z as a combination of the integral A coming from the 2,k ℓ,k cyclic permutation of length ℓ as shown in (2.8) 1 Z = (A )2 A . (3.1) 2,k 1,k 2,k 2 − h i Although A is originally written as an integral over four variables (2.9), after some 2,k computation this four-variable integral can be reduced to a single variable integral. We find that A and A are given by (see Appendix A for details) 1,k 2,k 1 A = , (3.2) 1,k k ∞ 2λ 1 ∞ 2λ sinh2πλ A = dλ = dλ . (3.3) 2,k sinhπkλcosh2πλ k2 − sinhπkλcosh2πλ Z−∞ Z−∞ Plugging this into (3.1), we obtain ∞ λ sinh2πλ Z = dλ . (3.4) 2,k sinhπkλcosh2πλ Z−∞ Note that λ is related to the original variables (up to permutation) as λ = σ σ . (3.5) 1 1 − As explained in Appendix B, the remaining λ-integral can be evaluated by picking up the e residues of the poles of 1 and 1 . It turns out that the result depends on the sinhπkλ cosh2πλ parity of k k−1 k−1 1 1 s πs ( 1) Z = ( 1)s−1 tan2 + − 2 , (3.6) 2,oddk k − 2 − k k π s=1 (cid:18) (cid:19) X k−1 2 1 1 s πs Z = ( 1)s−1 tan2 . (3.7) 2,evenk k − 2 − k k s=1 (cid:18) (cid:19) X 2Inaslightlydifferentcontext,theexactevaluationofthepartitionfunctionofU(2)IIBmatrixmodel was reported in [15, 16]. 6 In the above expression of Z , the s = k term should be understood as the limit 2,evenk 2 1 1 s 2 πs ( 1)k−1 lim ( 1)s−1 tan2 = − 2 . (3.8) s→k k − 2 − k k kπ2 2 (cid:18) (cid:19) Let us consider the physical interpretation of this result (3.7). For both even k and odd k cases, the sum over s comes from the poles at sinhπkλ = 0. It is natural to interpret this sum as the effect of the Z orbifolding of R8/Z . On the other hand, k k k−1 the second term (−1) 2 in Z comes from the pole at coshπλ = 0. This pole π 2,oddk corresponds to the zero of the 1-loop determinant of the bi-fundamental hypermultiplet, so it represents a singularity on the space of vector multiplet scalar fields where one of the bi-fundamental hypermultiplet becomes massless. However, thelocationof thesingularity is at the imaginary value of the scalar field i σ σ = , (3.9) 1 1 − 2 and hence this singularity is not realized in the physical theory. We should also mention e that the poles coming from the 1 factor do not correspond to the zeros of the 1-loop sinhπkλ determinant of the hypermultiplets in the original ABJM theory. Those poles effectively show up only after integrating out some of the variables σ ,σ , which are coupled via the i i Chern-Simons term eπik(σ2−σ2). e From (3.3), we see that A is positive. Therefore, we fined the inequality 3 2,k 1 Z < (Z )2 (3.13) 2,k 1,k 2 where Z = A = 1 is the partition function of U(1) U(1) theory. From this 1,k 1,k k k × −k inequality (3.13), it is tempting to draw a conclusion that the binding energy of two M2- branes is negative and M2-branes tend to dissociate into a configuration of two separated 3 The normalization of the partition function in [3] is different from ours by the factor of 2 in the 1-loop determinant. Namely, the partition function in [3] is related to ours by the replacement sinh → 2sinh,cosh 2cosh → 2 Z(DMP) = 1 dNσdNσeiπk(σ2−σe2) i<j2sinhπ(σi−σj)·2sinhπ(σi−σj) . (3.10) N,k (N!)2 2coshπ(σ σ ) Z "Q i,j i− j # e e One can easily see that the differeence between Z(DMP) anQd ours is just the eoverallfactor 2−2N N,k Z(DMP) =2−2NZ(ours) . (3.11) N,k N,k However, this factor drops out when taking the ratio of (Z )2 and Z 1,k 2,k (Z(DMP))2 (Z(ours))2 1,k 1,k = . (3.12) Z(DMP) Z(ours) 2,k 2,k Therefore, the statement Z < 1(Z )2 has a physical meaning regardless of the normalization we 2,k 2 1,k choose. 7 M2-branes. However, we think this is not the correct interpretation. When the ABJM theory is put on S3, the bi-fundamental matter multiplets acquire a mass term from the coupling to the curvature of S3, and hence the moduli space corresponding to the freely moving M2-branes on R8/Z is lifted. Therefore, the free energy of ABJM theory on k S3 is not a suitable measure of the binding energy of M2-branes on flat R1,2 R8/Z . k × Rather, the partition function on S3 is a natural quantity to consider in the context of the Euclidean version of AdS/CFT duality, where S3 appears as the boundary of Euclidean AdS . In the next concluding section we discuss a possible implication of our result in 4 the context of AdS /CFT duality. 4 3 4 Discussions As discussed in [1], the ABJM theory is dual to the M-theory on AdS S7/Z with the 4 k × metric R2 ds2 = ds2 +R2ds2 , (4.1) 4 AdS4 S7/Zk where the radius of curvature R is given by 6 R = 32π2kN . (4.2) l (cid:18) p(cid:19) The classical d = 11 supergravity description is valid when the radius of S7/Z is much k larger than the eleven-dimensional Planck length l p R l k5 N . (4.3) p ≪ k → ≪ In particular, the large N limit of ABJM theory with k fixed to a finite integer is in the regime of (4.3). On the ABJM theory side, it seems that the even/odd k difference of the behavior of the partition function Z persists for N > 2. This is because, in the integral of A in N,k ℓ,k (A.6), the pole of the form 1 related to the Z orbifolding appears also for general sinhπkλ k ℓ > 2 in the same way as A by integrating out some of the variables in σ ,σ coupled 2,k i i through the Chern-Simons term, and the remaining integral over λ depends on the parity of k. Since the partition function Z is written as a combination of A (2.8)e, Z also N,k ℓ,k N,k depends on the parity of k, unless some miraculous cancellation happens. But we think that is unlikely and the dependence on the parity of k is not an artifact of Z but the 2,k general property of Z for all N 2. N,k ≥ If we believe in the duality between the ABJM theory and M-theory on AdS 4 × S7/Z , this difference of even/odd k must be encoded in the M-theory dual, perhaps k in a very subtle way. However, so far there is no known indication of this difference in the supergravity approximation of M-theory on AdS S7/Z . Even if we take into 4 k × 8 account of the wrapped brane configuration in this background, the bulk theory seems to be insensitive to the parity of k. In fact, the BPS configuration of M5-branes wrapped on the 3-cycle in S7/Z is characterized by the homology class k H (S7/Z ) = Z , (4.4) 3 k k which is interpreted as the fractional M2-brane charge [17]. Clearly, this charge does not distinguish the parity of k. It might be the case that the even/odd k difference appears in the bulk theory as some sort of quantum effects in M-theory, which cannot be seen in the supergravity approximation. If this is true, it would be nice to understand this effect better. In the regime where 1 N5 k N , (4.5) ≪ ≪ the bulk theory is described by the type IIA string theory on AdS CP3. On the CFT 4 × side, this regime is related to the ’t Hooft limit of ABJM theory (1.1), and the classical type IIAsupergravity description becomes goodwhen the’tHooftcoupling t = N islarge. k When comparing the free energy F = logZ of ABJM theory and the classical action N,k − of the bulk supergravity theory, we need to perform an analytic continuation of Z as N,k a function of k and N. In particular, when determining the eigenvalue distribution for the matrix integral (2.2) in the ’t Hooft limit, the analytic continuation in k is implicitly assumed. Our result suggests that the analyticity in k is not obvious a priori, even in the large N regime. In some cases of Chern-Simons-matter theories, the analytic continuation in k requires the deformation of integration contour. However, the integral representation of Z in (3.4) is well-defined for k R without changing the integration contour of λ. 2,k ∈ From this integral representation (3.4), one can see that Z decreases monotonically as 2,k a function of k 4, and the expression (3.4) for k R serves as an interpolating function ∈ of our result (3.7) for integer k. It would be nice to see if similar analytic continuation is possible for N > 2 without deforming the integration contour. Acknowledgements I would like to thank Kazuo Hosomichi for a nice lecture on the localization method in supersymmetric gauge theories at Chubu Summer School 2011. This work is supported in part by JSPS Grant-in-Aid for Young Scientists (B) 23740178. 4We would like to thank the referee of Prog. Theor. Phys. for pointing this out. 9 A Computation of A ℓ,k InthisAppendix, byperformingtheintegrationoftwovariables, werewritethe2ℓ-variable integral A given in (2.9) into the integral of 2(ℓ 1) variables. Using this expression, ℓ,k − we find A = 1. We also find the expression of A as a single variable integral. 1,k k 2,k A.1 Writing A as the integral of 2(ℓ 1) variables ℓ,k − For readers convenience, we repeat the integral A in (2.9) ℓ,k ℓ 1 A = dℓσdℓσeiπk(σ2−σ2) . (A.1) ℓ,k e coshπ(σ σ )coshπ(σ σ ) i i i i+1 Z i=1 − − Y This integral can be simplifieed by the following change of variables e e (σ , ,σ ,σ , ,σ ) (λ , ,λ ,λ , ,λ ,σ ) (A.2) 1 ℓ 1 ℓ 1 ℓ 1 ℓ−1 ℓ ··· ··· → ··· ··· where e e e e e λ = σ σ (i = 1, ,ℓ) , λ = σ σ (i = 1, ,ℓ 1) . (A.3) i i i i i i+1 − ··· − ··· − In terms of these new variables, the integral becomes e e e ℓ ℓ−1 1 1 1 A = dℓλdℓ−1λdσ ℓ,k Z ℓYi=1 coshπλi Yi=1 coshπλi · coshπ( ℓi=1λi − iℓ=−11λi) e ℓ−1 i ℓ e P P exp 2πki λe λ +2πki λ σ . e (A.4) j i i ℓ × ! i=1 j=1 i=1 XX X e e Since the variable σ appears only in the exponent, the σ integral is just a δ-function ℓ ℓ ℓ ℓ 1 e dσ exp 2πki λ σ = δ e λ . (A.5) ℓ i ℓ i k ! ! Z i=1 i=1 X X After integrating out λ byesetting λ = ℓe−1λ by the above δ-function, we get ℓ ℓ − i=1 i ℓ−1 P 1 1 1 A = dℓ−1λdℓ−1λ ℓ,k k coshπλ coshπλ · coshπ( ℓ−1λ )coshπ( ℓ−1λ ) Z i=1 i i i=1 i i=1 i Y e ℓ−1 i P P exp 2πki eλ λ . e (A.6) j i × ! i=1 j=1 XX e A.2 A and A 1,k 2,k Let us look closely at the expression (A.6) for ℓ = 1,2. For ℓ = 1, there is no integral and the result is simply 1 A = . (A.7) 1,k k 10