Abstract:We study the partition function of the ABJ theory, which is the N = 6 superconformal Chern-Simons matter theory with gauge group U(N ) × U(N + M ) and Chern-Simons levels (k, −k). We exactly compute the ABJ partition function on a three sphere for various k, M and N via the Fermi gas approach. By using these exact data, we show that the ABJ partition function is completely determined by the refined topological string on local P 1 × P 1 , including membrane instanton effects in the M-theory dual.

We study vacuum expectation values (VEVs) of circular half BPS Wilson loops in arbitrary representations in ABJM theory. We find that those in hook representations are reduced to elementary integrations thanks to the Fermi gas formalism, which are accessible from the numerical studies similar to the partition function in the previous studies. For non-hook representations, we show that the VEVs in the grand canonical formalism can be exactly expressed as determinants of those in the hook representations. Using these facts, we can study the instanton effects of the VEVs in various representations. Our results are consistent with the worldsheet instanton effects studied from the topological string and a prescription to include the membrane instanton effects by shifting the chemical potential, which has been successful for the partition function.

We study supersymmetric index of 4d SU (N ) N = 4 super Yang-Mills theory on S 1 × M 3 . We compute asymptotic behavior of the index in the limit of shrinking S 1 for arbitrary N by a refinement of supersymmetric Cardy formula. The asymptotic behavior for the superconformal index case (M 3 = S 3 ) at large N agrees with the Bekenstein-Hawking entropy of rotating electrically charged BPS black hole in AdS 5 via a Legendre transformation as recently shown in literature. We also find that the agreement formally persists for finite N if we slightly modify the AdS/CFT dictionary between Newton constant and N . This implies an existence of non-renormalization property of the quantum black hole entropy. We also study the cases with other gauge groups and additional matters, and the orbifold N = 4 super Yang-Mills theory. It turns out that the entropies of all the CFT examples in this paper are given by 2π Q 1 Q 2 + Q 1 Q 3 + Q 2 Q 3 − 2c(J 1 + J 2 ) with charges Q 1,2,3 , angular momenta J 1,2 and central charge c. The results for other M 3 make predictions to the gravity side. * mh974ATdamtp.cam.ac.uk 1 See also [10] for another embedding.

We study N = 2 supersymmetric gauge theories on squashed 3-sphere and S 1 × S 2 . Recent studies have shown that the partition functions in a class of N = 2 theories have factorized forms in terms of vortex and anti-vortex partition functions by explicitly evaluating matrix integrals obtained from Coulomb branch localization. We directly derive this structure by performing Higgs branch localization. It turns out that more general N = 2 theories have this factorization property. We also discuss the factorization of supersymmetric Wilson

Abstract:We study 4d N = 1 supersymmetric theories on a compact Euclidean manifold of the form S 1 × M 3 . Partition functions of gauge theories on this background can be computed using localization, and explicit formulas have been derived for different choices of the compact manifold M 3 . Taking the limit of shrinking S 1 , we present a general formula for the limit of the localization integrand, derived by simple effective theory considerations, generalizing the result of [1]. The limit is given in terms of an effective potential for the holonomies around the S 1 , whose minima determine the asymptotic behavior of the partition function. If the potential is minimized in the origin, where it vanishes, the partition function has a Cardy-like behavior fixed by Tr(R), while a nontrivial minimum gives a shift in the coefficient. In all the examples that we consider, the origin is a minimum if Tr(R) ≤ 0.

We show that the ABJM theory, which is an N = 6 superconformal U(N ) × U(N ) Chern-Simons gauge theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model that can be derived from the theory by using the localization technique. This opens up the possibility of probing the quantum aspects of M-theory and testing the AdS 4 /CF T 3 duality at the quantum level. Here we calculate the free energy, and confirm the N 3/2 scaling in the M-theory limit predicted from the gravity side. We also find that our results nicely interpolate the analytical formulae proposed previously in the M-theory and type IIA regimes. Furthermore, we show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion. The method can be easily generalized to the calculations of BPS operators and to other theories that reduce to matrix models. *

Abstract:We study the partition function of the three-dimensional N = 6 U (N ) k × U (N + M ) −k superconformal Chern-Simons matter theory known as the ABJ theory. We prove that the ABJ partition function on S 3 is exactly the same as a formula recently proposed by Awata, Hirano and Shigemori. While this formula was previously obtained by an analytic continuation from the L(2, 1) lens space matrix model, we directly derive this by using a generalization of the Cauchy determinant identity. We also give an interpretation for the formula from brane picture.

We study weak coupling perturbative series in 4D N=2 and 5D N=1 supersymmetric gauge theories with Lagrangians. We prove that the perturbative series of these theories in the zero-instanton sector are Borel summable for various observables. Our result for the 4D N=2 case supports an expectation from a recent proposal on a semiclassical realization of infrared renormalons in QCD-like theories, where the semiclassical solution does not exist in N=2 theories and the perturbative series are unambiguous, namely, Borel summable. We also prove that the perturbative series in an arbitrary number of instanton sectors are Borel summable for a wide class of theories. It turns out that exact results can be obtained by summing over the Borel resummations with every instanton number.