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We consider 4d supersymmetric (special) unitary Γ quiver gauge theories on compact manifolds which are fibrations over . We show that their partition functions are correlators of vertex operators and screening charges of the modular double version of elliptic algebras. We also consider a generating function of BPS surface defects supported on and show that it can be identified with a particular coherent state in the Fock module over the elliptic Heisenberg algebra. [ABSTRACT FROM AUTHOR]

We review the holomorphic block factorisation of partition functions of supersymmetric theories on compact manifolds in various dimensions. We then show how to interpret 3d and 5d partition functions as correlation functions with underlying symmetry given by a deformation of the Virasoro algebra. [ABSTRACT FROM AUTHOR]

We present an elementary review of some aspects of Chern–Simons theory with complex gauge group . We discuss some of the challenges in defining the theory as a full-fledged TQFT, as well as some successes inspired by the 3d–3d correspondence. The 3d–3d correspondence relates partition functions (and other aspects) of complex Chern–Simons theory on a 3-manifold M to supersymmetric partition functions (and other observables) in an associated 3d theory . Many of these observables may be computed by supersymmetric localization. We present several prominent applications to 3-manifold topology and number theory in light of the 3d–3d correspondence. [ABSTRACT FROM AUTHOR]

We review the calculation and properties of the supersymmetric index for four-dimensional theories, illustrating its physical significance in several examples. [ABSTRACT FROM AUTHOR]

We review exact results in supersymmetric gauge theories defined on S4 and its deformation. We first summarize the construction of rigid SUSY theories on curved backgrounds based on off-shell supergravity, then explain how to apply the localization principle to supersymmetric path integrals. Closed formulae for partition function as well as expectation values of non-local BPS observables are presented. [ABSTRACT FROM AUTHOR]

In this review article we describe the localization of three dimensional supersymmetric theories on compact manifolds, including the squashed sphere, , the lens space, , and . We describe how to write supersymmetric actions on these spaces, and then compute the partition functions and other supersymmetric observables by employing the localization argument. We briefly survey some applications of these computations. [ABSTRACT FROM AUTHOR]

We give a pedagogical review of the computation of Gromov–Witten invariants via localization in 2D gauged linear sigma models. We explain the relationship between the two-sphere partition function of the theory and the Kähler potential on the conformal manifold. We show how the Kähler potential can be assembled from classical, perturbative, and non-perturbative contributions, and explain how the non-perturbative contributions are related to the Gromov–Witten invariants of the corresponding Calabi–Yau manifold. We then explain how localization enables efficient calculation of the two-sphere partition function and, ultimately, the Gromov–Witten invariants themselves. [ABSTRACT FROM AUTHOR]

This is an introductory review to localization techniques in supersymmetric two-dimensional gauge theories. In particular, we describe how to construct Lagrangians of theories on curved spaces, and how to compute their partition functions and certain correlators on the sphere, the hemisphere and other curved backgrounds. We also describe how to evaluate the partition function of theories on the torus, known as the elliptic genus. Finally we summarize some of the applications, in particular to probe mirror symmetry and other non-perturbative dualities. [ABSTRACT FROM AUTHOR]

In this paper we study the large N solution to matrix models describing the partition functions of 3d supersymmetric gauge theories on . The model we focus on has a single gauge group and fundamental fields, whose number scales with N, also known as the Veneziano limit. The novel point in our model is our choice of masses for the fundamental fields. Instead of vanishing or fixed masses, we consider a linear distribution. We show that the model can still be solved by standard large N techniques and explore the different phases of the model. We also comment about other natural mass distributions. [ABSTRACT FROM AUTHOR]

The present paper is part of our ongoing work on supersymmetric σ-models, their relation with the Potts model at q  =  0 and spanning forests, and the rigorous analytic continuation of the partition function as an entire function of N  −  2M, a feature first predicted by Parisi and Sourlas in the 1970s. Here we accomplish two main steps. First, we analyze in detail the role of the Ising variables that arise when the constraint in the model is solved, and we point out two situations in which the Ising and forest variables decouple. Second, we establish the analytic continuation for the model in some special cases: when the underlying graph is a forest, and for the Nienhuis action on a cubic graph. We also make progress in understanding the series-parallel graphs. [ABSTRACT FROM AUTHOR]