ABJ Theory in the Higher Spin Limit

We study the conjecture made by Chang, Minwalla, Sharma, and Yin on the duality between the N=6 Vasiliev higher spin theory on AdS_4 and the N=6 Chern-Simons-matter theory, so-called ABJ theory, with gauge group U(N) x U(N+M). Building on our earlier results on the ABJ partition function, we develop the systematic 1/M expansion, corresponding to the weak coupling expansion in the higher spin theory, and compare the leading 1/M correction, with our proposed prescription, to the one-loop free energy of the N=6 Vasiliev theory. We find an agreement between the two sides up to an ambiguity that appears in the bulk one-loop calculation.

To compare the 1/M expansion of the ABJ free energy with that of the HS free energy, an obstacle is the lack of an action for the Vasiliev theory from which to extract a weak coupling expansion 5 . In this paper, following Refs. [8,11], we circumvent this problem by computing the one-loop free energy, which can be computed without an action as long as we know the spectrum, and by comparing it with the ABJ free energy.
The organization of this paper is as follows: In section 2 we summarize our claim and the main results on the HS and ABJ free energy and the correspondence between the two sides. In section 3 we review the integral representation, sometimes referred to as "mirror description" of the ABJ partition function, using which we analyze the free energy in the HS limit and develop a systematic 1/M expansion. Some of the technical details in section 3 are provided in Appendices A and B. In section 4 we calculate the one-loop free energy of N = 6 Vasiliev HS theory. We close our paper with discussions in section 5.

The main results
We first summarize our claim and the main results on the correspondence between the N = 6 HS and ABJ free energies in the limit (1.1) with 1/M corrections.
Higher spin theories are dual to vector models. Our working assumption is that the vector where γ is a constant that cannot be fixed by the analysis of the current paper, t = M/|k| as defined in (1.1), and Z CS (M ) k and Z ABJM (N ) k = Z ABJ (N, N ) k are the partition function of Figure 1: The open-string interpretation of the field content of the ABJ theory in the type IIB UV description. N D3-branes are intersecting with an NS5-brane and with a (1, k) 5-brane, and wrap the horizontal direction which is periodically identified. M fractional D3-branes partially wrap the horizontal direction, ending on the 5-branes. (For more detail about the brane configuration, see [21,22].) The open strings stretching between D3-branes represent fields in the ABJ theory. In the case of the ABJ theory this is known as the Giveon-Kutasov-Seiberg duality under which the partition function Z ABJ (N, N + M ) k is invariant [21,42]. For the CS partition function Z CS (M ) k , this is nothing but the level-rank duality, whereas the invariance of the ABJM partition function Z ABJM (N ) k trivially holds. Note that the first of (2.3) can be written in terms of t as The identification of the Newton constant G HS in (2.2) can be inferred from the 1/M expansion of the U (N ) HS free energy F HS (G HS , θ 0 , N ) ≡ − log Z HS (G HS , θ 0 ; N ). Namely, with the identification (2.2), the 1/M expansion of the ABJ free energy, which is the content of section 3, implies the following G HS expansion of the HS free energy: 6 Note that the Newton constant G HS with the identification (2.2) is a duality invariant. It is also worth emphasizing that this identification agrees with the one suggested by the computation of three point functions of higher spin currents for non-supersymmetric theories which is an independent and a completely different analysis [43].
Besides the sensible identification of the Newton constant G HS , the free energy (2.5) has a few favorable features: (1) The leading 1/G HS term is linear in M , as opposed to M 2 as would be expected from the U (M ) vector degrees of freedom, and the dependence on the PV phase θ 0 is qualitatively similar to that of the N = 2 theory in [44] which exhibits the invariance The leading 1/M correction, the logarithmic term in (2.5), agrees with the one-loop free energy of the N = 6 HS theory whose contribution comes solely from U (N ) gauge fields, as calculated in section 4, up to the ambiguity of the constant γ.

The boundary side: ABJ theory
In this section, we study the HS limit of the partition function of the ABJ theory and develop a systematic way to derive its large M expansion. The expansion can be explicitly worked out any finite order in principle. In the next section, we will use the 1-loop part of the expansion for comparison with the bulk Vasiliev theory.

The ABJ partition function
The partition function of the U (N 1 ) k × U (N 2 ) −k ABJ theory on S 3 has been written in the matrix model form [25,26] using the localization technique [27]. The explicit expression of the partition function is where ∆ sh and ∆ ch are the one-loop determinant of the vector multiplets and the matter multiplets in the bi-fundamental representation, respectively: Furthermore, k ∈ Z =0 is the Chern-Simons level, while N is the normalization factor [28] Because of the relation we can assume N 1 ≤ N 2 and k > 0 without loss of generality, as we will do henceforth. We set We write Z ABJ (N 1 , N 2 ) also as Z ABJ (N ; M ).
There are various ways to analyze the ABJ partition function (3.1), including the Fermi gas approach [29,45,46] extensively used in the literature. However, for the purpose of studying its HS limit, the most convenient starting point is the "mirror description" of the ABJ partition function found in [35], generalizing the mirror description of the ABJM partition function [47,48]. The "mirror description" of the ABJ partition function is as follows: where is the partition function for the U (M ) k CS theory and we defined the quantity 7 (3.9) 7 Note that Ψ defined in (3.9) is different from the one in [35] by the inclusion of the factor (−1) In the above, we defined and (a) n = (a; q) n ≡ n−1 j=0 (1 − aq j ) is the q-Pochhammer symbol. The contour of integration in (3.9) is C = [−i∞ + η, +i∞ + η] with the constant η chosen to lie in the following range: In [35], various consistency checks of the expression (3.7) were performed: (i) agreement of the perturbative expansion with the original matrix integral (3.1), (ii) vanishing of the partition function for k < M , in accord with the prediction [21] that there must be no SCFT in this range, and (iii) invariance under the Giveon-Kutasov-Seiberg duality (2.3). Later, the expression (3.7) was derived in [36] directly from the matrix integral (3.1) using the Cauchy-Vandermonde formula.

The large M expansion
We would like to develop a formulation to evaluate the ABJ partition function in the HS limit (1.1). The expression (3.7) is especially suitable for that purpose, since the number of integrals N is fixed in the HS limit. To begin with, let us rewrite (3.9) in the following way [46]: where we did the following change of variables and also defined with R(x) = log(2 cosh(πx)) (M = 2p : even), log(2 sinh(πx)) (M = 2p − 1 : odd). contour for x j in (3.12) corresponds to choosing η correctly in the range (3.11), and that x = 0 is the critical point of the function f (x) for both even and odd M . Therefore, the strategy is to expand f (x) around x = 0 and carry out the integration by expansion around that point, taking into account the HS limit (1.1). It is easy to show that f (x, k, t) is an even function in x.
As we have shown in Appendix A, using the Euler-Maclaurin formula, f (x, k, t) can be formally rewritten as 16) in the sense that the formal power expansion of (3.16) around x = 0 reproduces the formal power expansion of (3.14). Namely, the right hand side gives the asymptotic expansion of f (x, k, t). Let us write the expansion of (3. 16) in x as Here, the quantities f 2n (k, t) are defined as the expansion coefficients and their explicit expression is given by (3.16) as where B n are the Bernoulli numbers. Note that f 2n (k, t) is defined so that its 1/k expansion (which is equivalent to the 1/M expansion) starts with an O(k 0 ) term. The m = 0 term in f 0 is understood as where I(x) was defined in (2.6).
If we write down the first few terms of the expansion (3.17), we have (3.20) The first term gives a constant contribution irrelevant for the x integration, while the x 2 term suggests that we define a new variable ξ by so that the expansion (3.20) now reads gives a starting point for the large k (large M ) expansion of the integral (3.12).
In terms of ξ, the integral (3.12) can be rewritten as The integral (3.23) is a standard Hermitian matrix integral and can be straightforwardly evaluated, regarding the ξ 2 term as giving the propagator and all higher power terms as interactions. Here we do not present the detail of the computation but simply write down the resulting large M expansion: Note that the full ABJ free energy F ABJ = − log Z ABJ contains more terms coming from (3.7). The computational detail of (3.25) can be found in Appendix B. Because we used an asymptotic expansion in evaluating the integral, the large M expansion (3.25) is also an asymptotic expansion to be completed by non-perturbative corrections. 9 In this section we compute the one-loop free energy of the bulk HS theory dual to the ABJ theory in the higher spin limit (1.1). 8 It was conjectured in [20] that the ABJ theory in the higher spin limit corresponds to the N = 6 parity-violating U (N ) Vasiliev theory on AdS 4 .
The Vasiliev theory has three parameters: 1. The Newton constant G HS which is proportional to M −1 at large M , as mentioned in the Introduction and section 2. 3. The PV phase θ 0 which violates parity and higher spin symmetry. As stated in the Introduction, θ 0 is identified with the 't Hooft coupling t by θ 0 = πt/2 [20,49].
The partition function of the Vasiliev theory takes the following form in perturbation theory: The free energy F

The one-loop contribution
The N = 6 Vasiliev theory is constructed from the so-called n = 6 extended supersymmetric Vasiliev theory by imposing a set of SO(6) invariant boundary conditions [20,51]. The parityeven n = 6 Vasiliev theory can have 64 supercharges, but the boundary conditions and the parity violation reduce the number of supersymmetries to N = 6 with 24 supercharges. The spectrum of the N = 6 Vasiliev theory is given by [20,51] • 32 fields for each integer, s = 0, 1, · · · , and half-integer spin, s = 1 2 , 3 2 , 5 2 , . . .
• All integer and half-integer spin fields with s ≥ 2 obey the so-called ∆ + = s+1 boundary condition at the AdS 4 boundary. 8 We thank Rajesh Gopakumar for stimulating discussions which motivated us to carry out the calculation in this section.
• 31 of the spin-1 fields have the ∆ + = 2 boundary condition, whereas the remaining one satisfies the ∆ − = 1 condition. The ∆ − spin-1 field is most important in the following and corresponds to a gauge field.
We summarize the spectrum in Table 1. There is, however, a very important caveat: The boundary conditions, as stated here, are only true in the strict large M limit. In fact, ∆ ± is the dimension of CFT operators dual to higher spin fields and may thus receive 1/M corrections [49,52]. As we will see, the 1/M correction to the ∆ − spin-1 field is particularly important and contributes to the one-loop free energy, whereas all the rest of 1/M corrections, even if present, have no contributions to one-loop. In Table 1    We can now write down the bulk one-loop partition function where Z s,∆ ± is the partition function for a field with spin s and the boundary condition ∆ ± and can be expressed in terms of functional determinants of symmetric transverse traceless (STT) tensors in AdS 4 [8, 10, 11, 50]: 9 with the understanding that Roughly speaking, the spin-(s − 1) determinants in (4.3) are the contributions from the gauge fixing ghosts. These determinants can be explicitly computed by applying the techniques developed in [53,54]. To proceed, we first simplify (4.2) by using the result of Giombi and Klebanov for the type-A Vasiliev theory [8], So, the bosonic contribution to the one-loop free energy could come only from the spin-0 and spin-1 fields. This simplifies the calculation.
For the convenience of the subsequent calculations we introduce which has been computed by Camporesi and Higuchi [53,54] and is given in terms of the spectral zeta function where the spectral zeta function ζ (∆,s) (z) is defined by The parameter Λ in (4.8) is a UV cutoff. The logarithmic divergence arises in even dimensions and is related to the conformal anomaly. As we will show below, the logarithmic divergence actually cancels out in the N = 6 Vasiliev theory (in a certain regularization scheme). Hence the net contribution to the one-loop partition function comes solely from ζ (∆,s) . In particular, the O(log M ) correction observed in the ABJ theory comes entirely from the ∆ − spin-1 field and the consequence of the "induced gauge symmetry" [9].

The bosonic contributions
We first consider the bosonic part F HS, B of the one-loop free energy. As commented on below (4.6), there are only contributions from the spin-0 and spin-1 fields. Moreover, as it will turn out, it is free of logarithmic divergences. For integer spins, the spectral zeta function ζ (∆,s) (0) has been calculated by Giombi and Klebanov [8]: (4.10) Noting that ∆ + − 3/2 = −(∆ − − 3/2), this expression implies, due to the invariance under Thus the logarithmic divergence in the bosonic part of the free energy cancel out between the contributions from different boundary conditions, namely, where . . . | log div means the logarithmically divergent part.
Turning to the finite piece, we first calculate the spin-1 free energy. Again borrowing the result from [8], we have where x − x 3 ψ(x + 1/2) (4.14) with ψ(z) being the digamma function. Here, as emphasized in the discussion of the spectrum, we need special care in dealing with the conformal dimensions ∆ ± . Generically, the dimensions ∆ ± may receive the finite M corrections where c ± are some constants. In fact, it has been shown [49,52]  corrections, an explicit calculation shows where the c + dependence yielded an O(1/M ) contribution, whereas the c − dependence O(M 0 ). We thus find that

The fermionic contributions
We next consider the fermionic part F HS, F of the one-loop free energy. Again, as it will turn out, it is free of logarithmic divergences. Moreover, it has no log M corrections.
We first show the absence of the logarithmic divergences: For s ∈ Z + 1/2, we can rewrite the spectral zeta function ζ (∆,s) (z) as a sum of two terms ζ (∆,s) (z) = 8(2s + 1) 3π (g 1 (ν, s; z) + g 2 (ν, s; z)) , (4.20) where (4.21) By explicit calculations, these two terms are given by This sum, as it stands, is divergent, and must be regularized. We adopt the regularization used in the analysis [11]. 10 This yields where we used (4.22) to find the second line. Thus the fermionic part of the one-loop free energy is also free of logarithmic divergences. We next evaluate the finite part. For s ∈ Z ≥0 + 1/2, an explicit computation yields where

The full one-loop free energy
Altogether, we find the full bulk one-loop free energy to be Note that the leading O(log M ) contribution comes entirely from the ∆ − spin-1 field, the U (N ) gauge fields, and, as in [9], is the consequence of the "induced gauge symmetry." The bulk one-loop free energy

Discussions
In the last two sections, we have calculated the free energies of the ABJ theory in the HS limit and the N = 6 Vasiliev theory at one-loop. We are now ready to discuss the correspondence between the two theories. However, it is not as straightforward as comparing the free energy of the ABJ theory (3.25) and that of the N = 6 HS theory (4.30) as they are, and it requires some considerations to make the correspondence more precise. As already mentioned in section 2, the ABJ theory, even in the HS limit (1.1), has more degrees of freedom than necessary to describe the N = 6 HS dual. For instance, the free energy of the ABJ theory in the limit (1.1) goes as M 2 , since the ABJ theory is a theory of Here we first recall our proposal made in section 2 and then elaborate on it. The proposed correspondence is given in (2.1): where the "vector model subsector" of the partition function is identified as which agrees with the one suggested in [43] for non-supersymmetric theories. It must be stressed that the quotient by the ABJM partition function Z ABJM (N ) k is crucial for us to be able to make this identification. To elaborate on this point, we note that the free energies for the partition functions appearing in (5.2) have the following expansions: Subtracting (5.7) from (5.6), we find the free energy for the "vector model subsector" to be As is evident, the logarithmic terms in (5.6) and (5.7) conspire to yield the combination M sin(πt)/(πt) = γ G −1 HS as well as the correct dependence on the U (N ) volume. Besides being duality invariant, this provides a nontrivial justification for the quotient by the ABJM partition function in (5.2). We also note that the appearance of the combination M sin(πt)/(πt) persists in higher orders of the 1/M expansion, as can be seen in (3.25).
All indicate that our proposal ( A Formal expansion of f (x, k, t) In this Appendix, we derive the formal expansion (3.16) of the quantity f (x, k, t) defined in (3.14).
First, let us do the following trivial rewriting of (3.14) as The quantity f 2n (k, t), which was defined in (3.17) and can be written as is computed from the expression (A.1) as follows. First, for even M , 11 (n ≥ 1).
(A.3) 11 Recall that the summation is always done in steps of one.
Here, we used the relation ∂ x = −i∂ m and the formula [56, eq. 1.518.2] For odd M , some care is needed in setting x = 0, because the singularity at x = 0 coming from the m = 0 term in the second sum of (A.1) cancels against the singularity coming from R(x). Using the formula [56, eq. 1.518.1] we obtain, for odd M , Because the summand in the first terms of (A.3), (A.6) is regular at m = 0 thanks to the rewriting (A.1), it can be safely evaluated using the Euler-Maclaurin formula. The version of the Euler-Maclaurin formula relevant here is the one that uses the midpoint trapezoidal rule and is given by (see e.g. [57]) where the remainder function is and ζ(s, q) is the Hurwitz zeta function. Generally, R w does not vanish in the w → ∞ limit and, therefore, sending w → ∞ and dropping R w in (A.7) gives a non-convergent asymptotic expansion. For n ≥ 1, the second terms of (A.3) and (A.6) involve the generalized harmonic number, Its asymptotic expansion for large q is [58] where "∼" means an asymptotic expansion and ζ(s) is the Riemann zeta function. By expanding this in r around r = 0 and collecting the O(r) terms, we obtain the asymptotic expansion which we can use for evaluating the n = 0 case of (A.3) and (A.6).
Applying the above formulas (A.7), (A.10) and (A.11) to (A.3) and (A.6) and massaging the resulting expression, we obtain the following asymptotic expansion: where, for even M , while, for odd M , Similar cancellations happen for the log terms for n = 0. Actually, as we will show below, f 0 = f 2n = 0. Therefore, (A.12) actually becomes where it is understood that, for n = l = 0, • Proof of f 2n = 0 Let us show that f 2n = 0 as mentioned above. For simplicity, let us consider the case with odd M and n ≥ 1. The relevant expression is (A.16). First, because B 0 = 1, B 1 = −1/2 and B 2n+1 = 0 for n ≥ 1, we can combine the two terms in the second line to get the following expression: When expanded in 1/M , the second term is equal to Now, recalling the relation between the Bernoulli polynomial B n (x) and the Bernoulli numbers Therefore, Because the summand vanishes for q = 1 and because B 2n+1 = 0 for n ≥ 1, we can set q = 2l, l ≥ 0. Then this cancels the first term in (A.23). So, we have shown f 2n = 0.
In a quite similar manner, using Bernoulli polynomial/number identities, we can show that f 0 = 0 for even M and f 0 = f 2n = 0 (n ≥ 1) for odd M .

B Evaluation of the matrix integral (3.23)
In this appendix, we would like to systematically evaluate the integral (3.23), which we write down here again for convenience: Note that F defined here is different from the full ABJ free energy F ABJ = − log Z ABJ which contains more terms coming from (3.7).
Because f 2n = f 2n (k, t) = O(k 0 ), we can treat the ξ 2 term in the exponential of (B.1) as the propagator and all higher power terms as interactions, and evaluate the integral perturbatively in a 1/k expansion. The last term in the exponential can be written as j<m log tanh where we used the relation [56, eq. 1.518.3] To avoid clutter, let us use the shorthand notation First, note that the Gaussian integral of the quadratic term is given by where G 2 (N ) is the Barnes G-function. For a quantity O(ξ), let us define its expectation value by where we used the relation G 2 (z + 1) = Γ(z)G 2 (z).
The above is sufficient for computing F(N ; M ) k in principle, but the following observation makes the computation simpler. Note that ∆(ξ) 2 is nothing but the Fadeev-Popov determinant for going from the matrix model of an N × N Hermitian matrix X to the diagonal gauge where ξ j , j = 1, . . . , N are the eigenvalues of X. So, the expectation value of O defined in (B.6) can be written as the expectation value in a Hermitian matrix model as where X is an N × N Hermitean matrix. When going from the eigenvalue basis in terms of ξ j back to the Hermitean matrix model, we do the following replacements in O: In the above expressions, we set f 2 = 1 for simplicity, but the correct powers of f 2 can be recovered on dimensional grounds. When computing correlators such as (B.12), diagrams get out of hand quickly as the power grows. Rather than directly dealing with diagrams, it is easier to assume that a given correlator is an even/odd polynomial in N with certain degree, and determine the coefficients by computer for some small values of N .
So, in terms of the Hermitian matrix model, the "free energy" F(N ; M ) k can be computed as follows: where conn means the connected part; for example, (tr X 2 ) 2 conn = (tr X 2 ) 2 − tr X 2 2 . (B.14) Carrying out the diagram expansion in (B.13) to a few orders and using the large k expansion of f 2n (k, t) given in (3.18), we obtain the following large k expansion for F(N ; M ) k :