Large N Free Energy of 3d N=4 SCFTs and AdS/CFT

We provide a non-trivial check of the AdS_4/CFT_3 correspondence recently proposed in arXiv:1106.4253 by verifying the GKPW relation in the large N limit. The CFT free energy is obtained from the previous works (arXiv:1105.2551, arXiv:1105.4390) on the S^3 partition function for 3-dimensional N=4 SCFT T[SU(N)]. This is matched with the computation of the type IIB action on the corresponding gravity background. We unexpectedly find that the leading behavior of the free energy at large N is 1/2 N^2 ln N. We also extend our results to richer theories and argue that 1/2 N^2 ln N is the maximal free energy at large N in this class of gauge theories.


Introduction
In this paper we study a class of 3d N = 4 SCFT T ρ ρ [SU (N )] introduced in [4], where ρ,ρ are partitions of N . This theory is a 1/2 BPS domain wall theory inside the 4d N = 4 SU (N ) Yang-Mills theory, and plays crucial roles in the generalizations [5,6] of the AGT correspondence, as well as the connection with the 3d SL(2) Chern-Simons theory [7,8,9]. The T ρ ρ [SU (N )] theories also appear as the basic building blocks for the 3d mirror of the 4d N = 2 Gaiotto theories [10] compactified on S 1 [11,3].
The type IIB supergravity dual for T ρ ρ [SU (N )] theories has recently been constructed in [1]. In this paper we provide further quantitative consistency checks of this AdS4/CFT3 correspondence by verifying the GKPW relation [12,13] in the leading large N limit.
On the CFT side, we take the large N limit of the S 3 partition functions of [2,3], evaluated at the conformal point. On the gravity side, we evaluate the gravity action in the gravity background of [1]. We find that in both cases the leading contribution of the free energy in the large N limit scales as More detailed statements will be given momentarily in section 2.2. As we will see, on the CFT side N 2 ln N comes from the asymptotic behavior of the Barnes G-function. On the gravity side, a factor of N 2 comes from the local scaling of the supergravity Lagrangian, and an extra ln N comes from the size of the geometry.
The organization of this paper is as follows. We first summarize the notations and the main results (section 2). We then give the derivations the results in gauge theory (section 3) and gravity (section 4). We also include two appendices.
2 Summary of the Results

Review of T ρ ρ [SU (N )] Theories
Let us first briefly summarize the basics of T ρ ρ [SU (N )] theories needed for the understanding of this paper (see [4] for details).
As stated in the introduction, the T ρ ρ [SU (N )] theory is specified by two partitions ρ and ρ of N : (1) , .., l (1) , l (2) , l (2) , .., l (2) , ... , l (1) ,l (1) , ..,l (1) ,N (2) 5 l (2) ,l (2) , ..,l (2) , ... ,N To construct the 3d theory, it is useful to use the brane configurations of [14]. Namely, we consider a D3-D5-NS5-brane configuration with N D3-branes suspended between NS5branes on the left and D5-branes on the right, where l (a) D3-branes (l (a) D3-branes) end on the i-th D5-brane (NS5-brane). We can identify the 3d theory after suitable exchanges of D5 and NS5-branes. The result is a 3d N = 4 quiver gauge theory. This theory has a non-trivial irreducible IR fixed point only when [4] ρ T >ρ ⇔ρ T > ρ , (2.3) where ρ >ρ for ρ = [n 1 , n 2 , . . .] andρ = [m 1 , m 2 , . . .] is defined by for all k. When the inequality is saturated for some value of i, the quiver breaks into pieces and the IR fixed point consists of products of irreducible theories. The global symmetry of Tρ ρ [SU (N )] is given by G ρ × Gρ, where G ρ is a subgroup of SU (N ) commuting with the embedding ρ G ρ is a symmetry of the Lagrangian, and acts non-trivially on the Higgs branch, whereas Gρ is a quantum mechanical symmetry acting on the Coulomb branch 1 . We can weakly gauge these symmetries to introduce a set of real mass parameters and FI parameters, which we collectively denote by m andm, respectively. The two global symmetries are related by 3d mirror symmetry [15] exchanging Higgs and Coulomb branches, together with real mass and FI parameters. This is simply the S-duality of the D3-D5-NS5 system, and in particular,

Large N Free Energy
We will verify the GKPW relation in the large N limit: where Z CFT is a CFT partition function on S 3 , F CFT := − ln Z CFT is the free energy, and S gravity is the action for the type IIB supergravity holographic dual to the CFT. Our findings are summarized as follows.
• The simplest prototypical example is the T [SU (N )] theory, which is a T ρ ρ [SU (N ))] theory with In this case we find The Cartan of this symmetry is the shift of the dual photon, and is present in the Lagrangian.
• More generally we consider the casep = 1, i.e., l (2) , l (2) , .., l (2) , ... , l ,l, ..,l . (2.9) We take the scaling limit where we take N large, while keeping κ a , λ (a) , γ a ,γ finite. We require The first condition is necessary for ρ to be partition, and the second ensures that the N (i) 5 becomes large, hence justifying the validity of the supergravity solution. We also have from (2.2) the constraint p a=1 γ a λ (a) =γl = 1 . (2.12) In this more general case we find (CFT analysis will be provided forl = 1, and gravity analysis for generall): (2.13) In particular when all κ a = 0, i.e. when all l (a) are finite, the leading large N behavior coincides with that in (2.8).
Note the number inside the bracket in (2.13) is a non-negative number smaller than 1 due to (2.11). Motivated by this result we conjecture (2.14) for all ρ,ρ satisfying (2.3). It would be interesting to see if some of the above inequalities could be explained in terms of the F-theorem [16,17] and the RG flows between the fixed points. The rest of this paper will be devoted to the derivation of (2.8) and (2.13).

CFT Analysis
In this section we analyze the CFT free energy F CFT .

The S 3 Partition Function
Let us begin with the T [SU (N )] theory (2.7). The partition function of this theory was computed by localization [18] to be [2,3] (see also [19]): 2 Here m ρ ,mρ are N -vectors, and each of their components is associated with a box of the Young diagram (also denoted by ρ,ρ) corresponding to the partitions ρ,ρ. For later purposes let us describe them by dividing the boxes of ρ into p blocks, where the a-th block is a rectangle with rows of length N where a is the label for the block and i (α) is the label for the column (row) inside the a-th block. The same applies toρ. In this notation, we have (m ρ ) (a,i,α) = ı(w l (a) ) α + m a,i , (mρ) (a,i,α) = ı(wl (a) ) α +m a,i , (3.18) where w N is a Weyl vector of the SU (N ) Lie algebra defined by Also, ∆ ρ (m) and ∆ρ(m) are defined by where [p, q] represents a box inside ρ,ρ at row p and column q. Note that the (m ρ ) [p,q] are simply a relabeling of the (m ρ ) (a,i,α) introduced previously. Several remarks are now in order. First, the partition function (3.17) is manifestly invariant under the simultaneous exchange of ρ,ρ and m,m. This is a manifestation of the 3d mirror symmetry.
Second, (3.17) vanishes unless ρ T ≥ρ [3]. This is consistent with the condition (2.3) for the existence of a non-trivial IR SCFT. This condition has a counterpart in the gravity dual [1].
Third, the expression (3.15) is either real or pure imaginary, however there is an ambiguity of the phase of the S 3 partition function and we will hereafter concentrate on the absolute value of the S 3 partition function.

T [SU (N )]
Let us study the large N behavior of our partition functions.
For clarity, let us begin with the T [SU (N )] theories, whose partition function is given in (3.15). When the parameters m andm are generic and kept finite in the limit, 4 we have w∈S N ∼ O(N !), whose logarithm contributes O(N ln N ) to the F CFT . The remaining contributions come from the two sinh Vandermonde determinants, each of which involves This is not surprising since after all our theories are standard gauge theories. However, the scaling behavior could change if we consider non-generic values of m and m. This is exactly happens to our CFT case, where we need to take the limit m,m → 0 of (3.15): We choose to take the limit in two steps. First, let us take them → 0 limit of (3.17) withρ = [1, . . . , 1]. This is conveniently done by settingm = w N and by taking → 0, where w N is defined in (3.19). Using the Weyl denominator formula, we have (3.23) In the limit 2 → 0, this cancels the factor ∆( We next need to take the limit m → 0. This is easy for our case, ρ = [1, . . . , 1]; which gives which gives (2.8).

T ρ ρ [SU (N )]
Let us consider the more general case given in (2.9). As long asρ = [1, . . . , 1] the argument of the previous subsection works up until (3.24). In (3.24) we already have a factor of G 2 (N + 1). Just as in the T [SU (N )] case, this contributes (3.27) to the free energy. Next, let us send the FI parameters to zero in (3.24). The denominator ∆ ρ (m) goes to zero in the limit, but the same is true for the numerators, yielding the finite answer. We obtain powers of 2π in this process from the limit of ∆ ρ (m), however this only gives a subleading contribution of order N 2 .
There are still contributions from the numerator i<j [(m ρ ) i − (m ρ ) j ], which we have not yet taken into account. In the notation of the previous section the limit of this contribution When the two boxes are in the same block, this contributes a factor where the factor N (a) 5 2 accounts for the degeneracy from the column labels i. This contributes, under the scaling (2.10), to the free energy. When the two boxes are in the different blocks a, b with l (a) ≥ l (b) , κ a ≥ κ b , the contribution to the free energy is The expression inside the bracket gives Thus the contribution amounts to Collecting all the contributions (3.27), (3.28) and (3.29), we have From (2.12) we can show that this coincides with (2.13). In all of the examples above, the leading contribution to the partition function comes from the Barnes G-functions. It is curious to note that the same function appears in the formula for the volumes of Lie group SU (N ) [20], and hence in the measure for the SU (N ) gauge theory. This is probably not a coincidence, since in the correspondence in [3] the S 3 partition function of T ρ ρ [SU (N )] theory is identified with an overlap of wavefunctions of a 1d quantum mechanics, which is obtained from a dimensional reduction of the 2d Yang-Mills theory. The measure of 2d Yang-Mills contains a volume factor for the gauge group U (N ). The same N 2 ln N type behavior appears in a number of different contexts, such as Gaussian matrix models, c = 1, topological string on the conifold or more recently in the weak coupling expansion of the ABJM theory [21].

Gravity Analysis
In this section we analyze the type IIB supergravity action S gravity in the holographic dual.

Summary of the Gravity Solution
First we summarize the holographic duals of the Tρ ρ [SU (N )] theories constructed in [1] (see also [22] for related work), which is based on earlier solutions found in [23,24].
The geometry of the type IIB backgrounds is an AdS 4 × S 2 × S 2 fibration over a twodimensional Riemann surface Σ. We will parameterize Σ by an infinite strip, although it will turn out that Σ has finite volume and is really compact. Next we introduce complex coordinates on Σ as z,z. We will also make use of the real coordinates defined by writing z = x + ıy. After fixing Σ, the solution is then determined by two real harmonic functions, h 1 and h 2 , on Σ.
The metric can be written as where the warp factors are given by and we defined the auxiliary functions This geometry is supported by non-vanishing "matter" fields, which include the dilaton field in addition to non-vanishing 3-form and 5-form fluxes which are given in appendix B.
We now turn to the specific solutions corresponding to Tρ ρ [SU (N )]. The classical supergravity solutions describing the near horizon limit of D3-branes suspended between p stacks of D5-branes andp stacks of NS5-branes is given by the two harmonic functions: with −∞ < x < ∞ and 0 ≤ y ≤ π/2. Here δ 1 < δ 2 < ... < δ p are the positions of D5-brane singularities on the upper boundary of the strip (y = π/2), whereasδ 1 >δ 2 > ... >δp are the positions of NS5-brane singularities on the lower boundary (y = 0) (see fig. 2). The points at x = ±∞ are regular interior points of the ten-dimensional geometry. The coefficients of the logarithms determine the number of 5-branes located at the singularities. The number of D5-branes located at δ a is denoted by N

The Gravity Action
The type IIB action in Einstein frame is 7 where one imposes the self-duality conditionF (5) = * F (5) as a supplementary equation. The coupling κ 10 is related to the string scale α by 2κ 2 10 = (2π) 7 (α ) 4 . Due to the presence of the self-duality condition, the action (4.37) cannot be directly used to compute the on-shell value of the action. One way to deal with this is to relax the requirement of Lorentz invariance of the action. In this case an action principle could be obtained along the lines of [25]. As suggested in [26], perhaps the easiest way to implement this for the full type IIB supergravity action is to make a T-duality transformation of the type IIA action. A simpler method is to first dimensionally reduce the theory to 4-dimensions.
After carrying out the dimensional reduction, one can then truncate the theory to the 4dimensional graviton. To see this is consistent, one may check that the solutions of [23,24] can be extended by replacing the AdS 4 space with any space which obeys the same Einstein equations. Thus truncating to the 4-dimensional graviton is a consistent truncation. 8 The effective action for this mode is given by where the cosmological constant has been chosen so that the unit AdS 4 space is a solution.
The subscript (4) reminds us that g (4) is the 4-dimensional metric and R (4) is the associated Ricci scalar. The quantity vol 6 follows from the initial dimensional reduction and is the volume of the internal space dressed appropriately with the warp factor of AdS 4 The specific solution we are interested in is AdS 4 with Ricci scalar R (4) = −12. Thus the on-shell action becomes simply where we have used the regularized volume of AdS 4 , vol AdS 4 = (4/3)π 2 , which may be computed using holographic regularization [27,28,29,30] (see for example section 5 of [31]).
We now wish to take the large N limit of this configuration. It will turn out that locally the Lagrangian density will scale with a factor of N 2 at leading order in N . Secondly, as N goes to infinity, the positions δ of the 5-brane stacks are sent to infinity in opposite directions (see fig. 3). This leaves a large region of geometry between −δ and δ of size ln N , which will reproduce the ln N behavior of the partition function. Thus one can understand the leading behavior of the T [SU (N )] partition function as coming from the geometry located between the two stacks of 5-branes.
To make this more explicit and also compute the exact numerical coefficient, we now work out the large N expansion. First we re-scale the x coordinate so that z = δx + ıy and then expand the harmonic functions h 1 and h 2 around large N . At leading order we obtain From (4.43) we find that the only contribution to the action at this order comes from the central region −1 < x < 1. In this region W is given by W = − 1 2 e −2δ N 2 (α ) 2 sin(2y).
Computing the volume of the internal space, (4.39), and plugging into the expression for the effective action, (4.40), we find This reproduces exactly the leading order behavior of the CFT partition function (3.26). Finally we note that including higher order terms in the expansions of the harmonic functions will give additional contributions of order N 2 .

(4.46)
It is interesting to note that this is exactly the limiting geometry of Janus found in [32] for the case of an infinite jump in the coupling. 9 The radius L of the Janus space is related to N by L 2 = 2 √ 2α N e −δ . In the case we consider here, the Σ space comes with a natural cutoff at |x| = δ, while for Janus the space is unbounded.
(4.47) Due to the large N limit, throughout most of the region we have α R 1. However, due to the presence of D5-branes, as one approaches x = 1, α R is of order one and one expects higher curvature corrections to play a role. Since these corrections are localized only in the region near x = 1, we expect that they do not receive the ln N enhancement and therefore contribute only at order N 2 . A similar argument can be made when one examines the geometry near the D5-branes using (4.41) before taking the large N limit.
Due to the presence of N5-branes, the second issue for our calculation is to understand if the string coupling, g s , is small so that string loop corrections can be ignored. The dilaton in the central region, −1 < x < 1, is given by We observe that the dilaton is small in the region 0 < x < 1 but is big in the region −1 < x < 0. We first focus our attention on the region 0 < x < 1. In the large N limit, the string coupling is small except in the neighborhood of x = 0, where it is of order one. Thus we expect string loop corrections to be important, but again we argue that since they are localized near x = 0, they will give contributions at most of order N 2 . For the region −1 < x < 0, we find that the string coupling is generically large and one might expect string loop corrections to modify the leading N 2 ln N behavior. From this point of view, the exact match between gravity and CFT partition functions is surprising and we do not have a good a priori argument for why string loop corrections do not modify the N 2 ln N behavior. One possible explanation can be given in terms of a local S-duality transformation in this region. To be more precise, we divide the manifold into three regions −1 < x < − , − < x < and < x < 1 with 1. In the first region, we make an S-duality transformation, while in the third region the theory is already weakly coupled. The middle region then has to interpolate between two different S-duality frames and we do not know how to compute the action there. However, since the ln N enhancement requires the entire internal space and patching only needs to occur locally in the region near x = 0, one might hope that the middle region does not receive the ln N enhancement. Of course this argument is only heuristic and it would be interesting to either make it more precise or determine the exact mechanism for why the loop corrections are suppressed. Similar situations arise when one examines the geometry near the NS5-branes using (4.41) before taking the large N limit.

T ρ ρ [SU (N )]
We now consider more general partitions which take the form (2.9). In this case, there is a single NS5-brane stack and the charge relations, (4.36), can be easily inverted to express the phases δ a andδ in terms of the partitions ρ andρ: To analyze the large N behavior, we proceed analogously to the T [SU (N )] case and consider the limit whereδ → −∞ and the δ a → ∞. In this case, we approximate the harmonic functions by the following expressions  Using this in (4.39) we find where we define δ 0 ≡δ. Plugging into (4.40) and combining all of the numerical factors, we obtain We now consider the scaling behavior defined by (2.10). The idea is to introduce separations between the δ a which are of order ln N . In this case each region between a given δ a and δ a+1 will contribute to the action at order N 2 ln N . In terms of this scaling the action becomes which coincides with (2.13).

Subleading Terms
So far we have concentrated on the leading N 2 ln N contributions to the free energy and it is a natural question to ask about the subleading N 2 contributions. Comparing the CFT and gravity partition functions, we find that the subleading N 2 contributions do not match. 10 However, this is not surprising since the gravity solution contains 5-brane singularities around which supergravity approximation breaks down. Additionally, there are regions in the bulk of Σ where the string coupling becomes large. It would be interesting to interpret and if possible match the subleading contributions to the CFT partition function with higher curvature corrections, coming from both string and loop corrections, on the gravity side. For the T [SU (N )] theory, we note that near the D5-brane singularity, the Ricci scalar, (4.47) does not depend on N and so all powers of R will contribute at order N 2 . Similarly, one may check that other contractions of the Riemann tensor will also contribute at order N 2 . Thus even at order N 2 , the CFT partition function contains information about all orders of the higher curvature corrections.
Affairs through the Inter-University Attraction Poles Programme," Belgian Science Policy P6/11-P, as well as the European Science Foundation Holograv Network.

A Barnes G-function
Let us briefly summarize the properties of the Barnes G-function. Barnes G-function G 2 (z) satisfies G 2 (z + 1) = Γ(z)G 2 (z), G 2 (1) = 1 . where ω 45 and ω 67 are the volume forms of the unit-radius spheres S 2 1 and S 2 2 , while the gauge potentials b 1 and b 2 are given by In this expression one needs the dual harmonic functions, defined by where ω 0123 is the volume form of the unit-radius AdS 4 , F is a 1-form onρ with the property that f 4 4 F is closed, and * 2 denotes Poincaré duality with respect to theρ metric. The explicit expression for F is given by where C and D are defined by ∂C = A 1 ∂A 2 − A 2 ∂A 1 and D =Ā 1 A 2 + A 1Ā2 .