We study 3d N = 2 Chern-Simons (CS) quiver theories on S 3 and Σ g × S 1 . Using localization results, we examine their partition functions in the large rank limit and requiring the resulting matrix models to be local, find a large class of quiver theories that include quivers in one-to-one correspondence with the ADE Dynkin diagrams. We compute explicitly the partition function on S 3 for D quivers and that on Σ g × S 1 for AD quivers, which lead to certain predictions for their holographic duals. We also provide a new and simple proof of the "index theorem", extending its applicability to a larger class of theories than considered before in the literature. * d.jain@saha.ac.in † augniva.ray@saha.ac.in ∆ (a,b) + ∆ (b,a) = 1 .(1.2)Note that for N ≥ 3 case, this condition is automatic since the supersymmetry enhancement fixes the R-charges to be 1 2 and ADE classification was presented in [14]. We will then explicitly solve the large N matrix model of the N = 2 D quiver theories 1 , whose dual geometry involves certain 7-dimensional Sasaki-Einstein manifolds Y 7 . Computation of their volumes directly does not necessarily give the volume for the Calabi-Yau (CY) metric necessary for

We compute the partition function of $$ \mathcal{N} $$ N = 2 supersymmetric mixed dimensional QED on a squashed hemisphere using localization. Mixed dimensional QED is an abelian gauge theory coupled to charged matter fields at the boundary. The partition function is a function of the complex gauge coupling τ, the choice of R-symmetry and the squashing deformation. The superconformal R-symmetry is determined using the 3-dimensional F-maximization. The free energy as a function of squashing deformation allows computing correlation functions that contain the insertion of the energy-momentum tensor. We compute the 2-point correlation function of the energy-momentum tensor of 3-dimensional theory by differentiating the free energy with respect to the squashing parameter. We comment on the behaviour of the 2-point function as we change the complex coupling τ.

We find the exact quantum gravity partition function on the static patch of 3d de Sitter spacetime. We have worked in the Chern Simons formulation of 3d gravity. To obtain a non-perturbative result, we supersymmetrized the Chern Simons action and used the technique of supersymmetric localization. We have obtained an exact non-perturbative result for the spin-2 gravity case. We comment on the divergences present in the theory. We also comment on higher spin gravity theories and analyse the nature of divergences present in such theories.