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

We derive non-Abelian localization formulae for supersymmetric Yang-Mills-Chern-Simons theory with matters on a Seifert manifold M , which is the threedimensional space of a circle bundle over a two-dimensional Riemann surface Σ, by using the cohomological approach introduced by Källén. We find that the partition function and the vev of the supersymmetric Wilson loop reduces to a finite dimensional integral and summation over classical flux configurations labeled by discrete integers. We also find the partition function reduces further to just a discrete sum over integers in some cases, and evaluate the supersymmetric index (Witten index) exactly on S 1 × Σ. The index completely agrees with the previous prediction from field theory and branes. We discuss a vacuum structure of the ABJM theory deduced from the

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. *

We study three dimensional N = 2 supersymmetric Chern-Simons-Matter theories on the direct product of a circle and a two dimensional hemisphere (S1 × D2) with specified boundary conditions by the method of localization. We construct boundary interactions to cancel the supersymmetric variation of the three dimensional superpotential term and the Chern-Simons term and show inflows of the bulk-boundary anomalies. It finds that the boundary conditions induce two dimensional N = (0, 2) type supersymmetry on the boundary torus. We also study the relation between the 3d-2d coupled partition function of our model and three dimensional holomorphic blocks.