Large $N$ twisted partition functions in 3d-3d correspondence and Holography

We study the large $N$ limit of twisted partition functions on $\mathcal{M}_{g,p}$, the $S^1$ bundle of degree $p$ over a Riemann surface of genus $g$, for 3D $\mathcal{N}=2$ superconformal field theories arising as low-energy limit of wrapped $N$ M5-branes on hyperbolic 3-manifold $M$. We study contributions from two Bethe vacua which correspond to two canonical irreducible $SL(N, \mathbb{C})$ flat connections on $M$ via 3D-3D correspondence. Using mathematical results on perturbtaive Chern-Simons invariants around the flat connections, we find universal expressions for the large $N$ twisted partition functions contributed from the two Bethe vacua in term of the hyperbolic volume of $M$. The two large $N$ partition functions perfectly match the on-shell actions for two Bolt-type solutions in the holographic dual $AdS_4$ gravity respectively.


INTRODUCTION AND SUMMARY
As a consistent theory of quantum gravity, string/M-theory is expected to provide microscopic understandings of quantum aspects of black holes (BHs). In a celebrated work [1], this hope was realized for extremal black holes in Minkowski spacetime, by reproducing the Bekenstein-Hawking entropy through counting D-brane bound states. Recently, this success was extended to black holes in asymptotically anti-de-Sitter spacetime. In particular, the entropy of magnetically charged supersymmetric black holes in AdS 4 supergravity can be explained using the holographic principle [2][3][4][5][6][7][8][9][10][11]. The AdS/CFT correspondence [12] says quantum gravity in asymptotically AdS 4 spacetime should be dual to a conformal field theory (CFT) on the 3-dimensional (3D) boundary. The field theories of interest have N ≥ 2 supersymmetry, and the black hole entropy on the gravity side turns out to be related to the so-called topologically twisted indices [13][14][15][16] in the dual field theory. They are the partition functions (ptns) on supersymmetric curved backgrounds M g,p=0 := Σ g × S 1 , with an appropriately chosen background magnetic flux coupled to R-symmetry current. The magnetic flux is turned on along the Riemann surface Σ g of genus g. The twisted indices can be then computed using the supersymmetric localization technique. Recently, the realm of localizable 3D manifolds has been further extended in [17], and we now have formulae for twisted partition functions Z g,p , i.e. the ptn on degree-p S 1bundles M g,p (see (4)) over Σ g .
In this letter, we study holographic duality for a large class of 3D N = 2 SCFTs T N [M ], defined in (2) (see also (3)), arising from wrapped M5-branes on closed hyperbolic 3manifolds M . The 3D theory is characterized by N , number of M5-branes, and the choice of a 3-manifold M . For each M , there is an associated AdS 4 /CFT 3 correspondence. The holographic dual of the wrapped M5-brane 3D SCFT was studied in [18,19]. Since there are infinitely many such 3-manifolds [20], the wrapped M5-branes system provides a huge set of AdS 4 /CFT 3 examples. We probe the holography using the twisted ptns. For p = 0, the ptn becomes a twisted index which counts ground states of M5-branes on Σ g × M . The counting is holographically dual to the microstates counting for a supersymmetric BH solution interpolating the asymptotic AdS 4 and its near-horizon limit AdS 2 × Σ g [11].
The 3D-3D correspondence [21][22][23] provides a novel way of analyzing the 3D N = 2 SCFTs. Schematically, the correspondence says Refer to [24][25][26][27][28][29][30][31] for more details on 3D-3D dictionary. Via the correspondence, some supersymmetric quantities of 3D T N [M ] theory can be evaluated without relying on a field theoretic description of the 3D SCFT. For example, as summarized in Table I, the twisted partition function Z g,p can be written in terms of basic perturbative invariants of the complex Chern-Simons (CS) theory. These invariants are mathematically well-defined and have been extensively studied in math literature.
Combining the 3D-3D dictionary with mathematical results, we obtain the large N behavior of the twisted partition functions contributed from two distinguished Bethe vacua in the T N [M ] theory. The Bethe vacua of our interest correspond to two irreducible SL(N, C) flat connections on M in 3D-3D dual complex CS theory. These two flat connections also have a natural interpretation in terms of hyperbolic geometry, see eq (18). They also give global minimum and maximum of the absolute value of the fibering operator F appearing in the ptn computation (12), see (20). We confirm that the large N twisted ptns from the two Bethe-vacua nicely match the onshell action for the two Bolt-type solutions [10] in the gravity dual respectively. The comparison is summarized in Table II, which is the main point of this letter.  For simplicity, we assume M is a closed (compact) hyperbolic 3-manifold without boundary. To preserve supersymmetry, we perform a partial topological twisting along the internal 3-manifold using SO(3) vector subgroup of SO(5) Rsymmetry of the 6D theory. The twisting preserves a quarter of supersymmetries and the resulting 3D theory becomes a 3D N = 2 SCFT with 4 supercharges. The 6D theory describes the low-energy effective world-volume theory of N coincident M5-branes in M-theory, and the 3D theory can be considered as an effective world-volume theory of N coincident M5-branes wrapped on the compact 3-manifold M , i.e.
Here T * M is the cotangent bundle of M , which is a local Calabi-Yau.
Let us comment on a subtle point in the setup. As emphasized in [32], in taking the twisted compactification we need to choose a connected subset of the vacuum moduli-space of the theory defined on R 3 , in order to have a genuine 3D SCFT. For a hyperbolic M , there is a natural choice (which is actually a single point) which is expected to become a discrete set of vacua when the theory is put on R 2 × S 1 . This discrete set of vacua corresponds to a subset of irreducible SL(N, C) flat connections on M . A field theoretic construction of the effective 3D gauge theory is proposed in [32], extending the beautiful construction in [23,33] for cusped 3-manifolds with at least one torus boundary component, by incorporating gauge theoretical operations corresponding to Dehn filling (removing torus boundaries) operations on 3-manifold.
We now turn to the case where T N [M ] is put on a large class of nontrivial backgrounds M g,p [17]: The metric can be written as where z,z are local coordinates on the Riemann surface and ψ ∼ ψ + 2π parameterizes the S 1 -fiber of length β. a is a 1-form on Σ g whose curvature F a := da is normalized as To preserve some supersymmetry, we turn on the following background gauge field coupled to U (1) R-symmetry.
with proper quantization conditions for (ν R , n R ) [10]. Here π * a is a 1-form on M g,p given as the pull-back of a using the projection map π : M g,p → Σ g . For later comparison with the bolt-type solutions in the supergravity, we follow [10] and choose Throughout the letter, we restrict our attention to the choice in (8) and some formulae below may not work for other cases. For example, the large N computation in Table II give incorrect answer for the usual round S 3 case which is M g=0,p=1 .
For small N the effective 3d theory T N [M ] might witness emergent symmetries in addition to R-symmetry, as pointed out in [32,34]. When N is large enough, on the other hand, there is no accidental symmetry and the U (1) R-symmetry in the IR should be simply inherited from the compact SO(2) subgroup of SO(5) R-symmetry in the 6D theory. It implies that the U (1) R-charge, R, should be properly quantized The Dirac quantization conditions for the U (1) R-symmetry flux on Σ g are From (9), we see that the Dirac quantizations are always satisfied for even p. In summary, for large enough N we can put the 3D T N [M ] theory on any M g∈Z ≥0 ,p∈2Z with supersymmetry preserving background gauge field, given in (7) and (8), coupled to the R-symmetry in the IR.

HOLOGRAPHIC DUAL OF TN [M ]
The gravity dual description is given by the uplift of a certain magnetically charged AdS 4 solution in the maximally supersymmetric D = 7 gauged supergravity [18,19]. Schematically, the D = 11 solution is a product of AdS 4 , hyperbolic 3-manifold M , and a squashed 4-sphereS 4 . Consistency of the truncation from D = 11 down to minimal N = 2, D = 4 gauged supergravity is established in [35] and it is guaranteed that we may replace the AdS 4 part with any nontrivial D = 4 solution and we still have an exact D = 11 solution.
The computation of holographic free energy can be also first done in D = 4 setup, and substitute the Newton constant with [36] Here, the hyperbolic volume is defined as The hyperbolic metric is normalized as R µν = −2g µν . The Mostow's rigidity theorem [37] guarantees the uniqueness of the hyperbolic metric and thus the volume is actually a topological invariant.
As gravity duals of the boundary theory put on M g,p , we utilize the supersymmetric AdS-Taub-NUT and bolt solutions constructed in [38]. Since these solutions have non-vanishing Maxwell field, which in D = 11 uplift appears as a twisting of the R-symmetry angle inS 4 , one might worry about a conflict with the quantization condition g, p. But it turns out, since the R-symmetry angle is part ofS 4 and we have a standard periodicity of 2π, the regularity condition for D = 4 NUT/Bolt is enough. This is in line with the field theory side discussion, in particular (9). A comment is in order here, in comparison with the uplifts involving Sasaki-Einstein 7-manifolds. In that case, the periodicity of the R-symmetry angle from the regularity of Bolt solution should be compatible with the periodicity condition due to collapsing cycles in the Kähler-Einstein base manifold of the Sasaki-Einstein space. The readers are referred to [10] for more details, where the authors considered an explicit example of Sasaki-Einstein manifolds such as V 5,2 = SO(5)/SO(3).

TWISTED PARTITION FUNCTIONS OF TN [M ] IN 3D-3D CORRESPONDENCE
The twisted partition function Z g,p on the M g,p for general 3D N = 2 SCFTs is given as the following finite sum [17,29] Here α labels the so-called Bethe vacua [39] of the 3D theory. It is obtained by extremizing the effective 2d twisted superpotential in the compactification on R 2 × S 1 . The number of vacua is equal to the Witten index [40,41] of the 3D SCFT.
H and F are called handle-gluing and fibering operators respectively. The explicit forms of H and F for any given ultraviolet (UV) Lagrangian are available in [17]. Let us emphasize that, the formula (12) applied to the case of S 3 partition function which corresponds to (g, p) = (0, 1) is apparently different from the more familiar Coulmob branch integral expression [42]. But their equivalence is illustrated for a number of examples in [17].   (2). The twisted ptns for these theories can be analyzed using the 3D-3D dictionaries summarized in Table I. Twisted ptns in 3D-3D correspondence were studied in [29,30]. In the table, the The Chern-Simons functional is Note that the counterpart of F and H are simply tree level and one-loop contributions in perturbation theory! More explicitly, the perturbative coefficients are given as Tor R [A α , M ] is the Ray-Singer torsion of an associated vector bundle in a representation R ∈ Hom[SL(N, C) → GL(V R )] twisted by a flat connection A α . Here V R is the vector space for representation R and GL(V R ) is the general linear group on the V R . The analytic torsion is defined as follows [45,47,48] Tor Here ∆ n (R, A α ) is a Laplacian acting on V R -valued n-form twisted by a flat connection A α . det ′ ∆ denotes the zeta function regularized determinant of the Laplacian ∆. For the oneloop part, the denominator comes from gauge field fluctuations δA while the numerator comes from the ghosts associated to a gauge fixing [49]. The 3D-3D dictionary in Table I can be derived combining several known results in literatures. The Bethe-vacua (vacua on R 2 × S 1 ) of 3D T N [M ] theory are in one-to-one correspondence to a subset of irreducible flat-connections on M [21,32]. According to a dictionary of 3D-3D relation, the asymptotic expansion Z α CS pert in (14) is equal to the perturbative expansion of holomorphic block B α (q) [21,27] associated to the Bethe-vacuum α in the limit q → 1, , as an asymptotic expansion in → 0.
For general 3D N = 2 theory, the asymptotic expansion coefficients S 0 and S 1 of holomorphic block are related to the operators F and H as given in Table I [50,51]. Here ω and e are respectively the spin-connection and vielbein of the unique hyperbolic metric on M . They are both locally so(3)-valued 1-forms and the complex combinations ω ± ie form an SL(2, C) flat-connections on M . ρ N is the N -dimensional irreducible representation of sl(2, C) = su(2) C , and obviously ρ N · (ω ± ie) become also irreducible SL(N, C) flat connections. A crucial property of these two flat-connections is that they take the minimum (maximum) value of Im[S 0 ] among all SL(N, C) flat connections. Namely, for any flat connection A α which is neither A geom nor A geom . Combined with the 3D-3D dictionary in Table I, it is implied for any Bethe-vacuum α which is neither (geom) nor (geom). Classical actions S α 0 for the two connections above can be computed as follows In the second line, T a (a = 1, 2, 3) are Pauli matrices and ρ N ·(T a ) are generators in the N -dimensional irreducible representation. From a simple group theoretical fact the expected N 3 -scaling of T N [M ] theory follows. In the third line of (21), we use the fact that the imaginary part of Chern-Simons functional of A = ω + ie is equal to the Einstein-Hilbert action with unit negative cosmology constant up to an overall numerical factor [52]. The action for the unique hyperboilc metric is twice of the hyperbolic volume of 3-manifold with a minus sign. The large N asymptotic behavior of the 1-loop coefficients, S geom 1 and S geom 1 , can be analyzed using a following mathematical theorem [53], Here ρ 2m+1 is the (2m + 1)-dimensional irreducible representation of sl(2, C) = su (2) Combining the 3D-3D dictionaries in Table I with the large N analysis in (21) and (25), we finally obtain following universal large N behavior of the twisted ptns (12) F geom g,p They nicely match the on-shell actions I Bolt± g,p of Bolt ± solution in [10]. The large N computations are summarized in Table II. theory. The 4d Newton constant G4 is given in (11).
The prescription to compute the twisted partition function Z g,p for T N [M ] through 3D-3D correspondence naturally shares several ingredients with the corresponding computation of a squashed 3-sphere partition function Z b (T N [M ]) studied in [36,54]. The squashed 3-sphere S 3 b of our interest is a supersymmetric curved background introduced in [55], defined as Setting b = 1 gives the usual round 3-sphere. According to the 3D-3D relation [22,23], the extreme squashing limit b ∈ R → 0 corresponds to a weakly coupled limit of the Chern-Simons theory. More concretely Z b is determined by the perturbative invariants S geom Combined with the large N behaviors of S geom n for n = 1, 2 in eqs. (21) and (25), we see that the asymptotic expansion is compatible with the gravity dual side of free-energy I gravity b [54], up to o( 0 ). Motivated from the comparison, it was further conjectured that [54] lim N →∞ This conjecture was checked numerically for a number of concrete examples. Now we compare the two large N analysis and see From the comparison, we obtain a general relation of the following form, between the twisted and the squashed S 3 partition functions in the large N limit which holds for every 3D T N [M ] theory. To arrive the conclusion, we use following universal relation between perturbative invariants The same universal relation (32) for p = 0 was observed in [4,11] for different class of 3D N = 2 SCFTs.

CONCLUSION
In this letter, we probe a large class of AdS 4 /CFT 3 associated to M5-branes wrapped on 3-manifolds by computing large N twisted ptns. For M2 and D2-branes and their Chern-Simons-matter theories, the large N computations have been performed already in [2][3][4][5][6][7][8][9][10][11]. A nicer feature of our analysis is that we map the large N analysis to a mathematical problem via 3D-3D correspondence which can be solved from known mathematical results, such as (19), (21) and (25). The results hold for any closed hyperbolic 3-manifold M and one does not need to perform the large N analysis for individual AdS 4 /CFT 3 model associated to each M . We can also give a simple explanation why there is a universal relation (see (32)) between the twisted ptns and S 3 -ptn in the large N limit. Both types of ptns are related to the same perturbative invariants of a complex CS theory through the 3D-3D correspondence in the large N limit. We hope that the improvement made in our analysis may provide a better way of understanding the subleading corrections to the large N twisted ptns which might be related to quantum corrections to the Bekenstein-Hawking entropy.