We derive the partition function of 5d N = 1 gauge theories on the manifold S 3 b × Σ g with a partial topological twist along the Riemann surface, Σ g . This setup is a higher dimensional uplift of the two-dimensional A-twist, and the result can be expressed as a sum over solutions of Bethe-Ansatz-type equations, with the computation receiving nontrivial non-perturbative contributions. We study this partition function in the large N limit, where it is related to holographic RG flows between asymptotically locally AdS 6 and AdS 4 spacetimes, reproducing known holographic relations between the corresponding free energies on S 5 and S 3 and predicting new ones. We also consider cases where the 5d theory admits a UV completion as a 6d SCFT, such as the maximally supersymmetric N = 2 Yang-Mills theory, in which case the partition function computes the 4d index of general class S theories, which we verify in certain simplifying limits. Finally, we comment on the generalization to M 3 × Σ g with more general three-manifolds M 3 and focus in particular on M 3 = Σ g × S 1 , in which case the partition function relates to the entropy of black holes in AdS 6 . β 67 A.1 5d maximal theory and class S 68 A.2 Class S k theories 70 A.3 E-string theory 72 B Instanton partition function for N = 2 U(N ) and SU(N ) SYM 74This preserves an SU(2) × U(1) isometry, and the Killing vector appearing in the superalgebra is K = ∂ ψ . Then, in this background we have 44], and so the central charge in the N = (2, 2) algebra (2.15) iswhich defines a variable, σ, valued in the complexification of the Cartan subalgebra of the flavor symmetry, which is a holomorphic combination of the background real scalar, z, and component, A ψ , of the background gauge field along the direction K, as discussed in more detail in [11]. The 2d A-twist and Bethe Ansatz equationsAs observed above, the supersymmetric background on S 3 b × Σ g , when considered along the Σ g directions, takes the form of a 2d topological twist. Such setups, where a d-dimensional 5 We will return to consider more general lens spaces in section 5.

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 apply the matrix model of Kapustin, Willett and Yaakov to compute the free energy of N = 3 Chern-Simons matter theories with D n quivers in the large N limit. We conjecture a general expression for the free energy that is explicitly invariant under Seiberg duality and show that it can be interpreted as a sum over certain graphs known as signed graphs. Through the AdS/CFT correspondence, this leads to a prediction for the volume of certain tri-Sasaki Einstein manifolds. We also study the unfolding procedure, which relates these D n quivers to A 2n−5 quivers. Furthermore, we consider the addition of massive fundamental flavor fields, verifying that integrating these out decreases the free energy in accordance with the F-theorem.

We derive actions for projective N=2 superspace ("hyperspace") from those for harmonic hyperspace, including that for nonabelian Yang-Mills (a new result). The method uses Wick rotation of the sphere from complex conjugate coordinates to real, null ones, which can be treated as independent. The result can be considered "holographic" in that the dimension of the internal (R-symmetry) space is reduced from 2 to 1, by solving equations of motion or gauge conditions for dependence on the other coordinate. The auxiliary nature of the redundant dimension makes the hypergraph rules and evaluation almost identical.