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We discuss some physical consequences of the resurgent structure of Painlevé equations and their related conformal block expansions. The resurgent structure of Painlevé equations is particularly transparent when expressed in terms of physical conformal block expansions of the associated tau functions. Resurgence produces an intricate network of inter-relations; some between expansions around different critical points, others between expansions around different instanton sectors of the expansions about the same critical point, and others between different non-perturbative sectors of associated spectral problems, via the Bethe-gauge and Painlevé-gauge correspondences. Resurgence relations exist both for convergent and divergent expansions, and can be interpreted in terms of the physics of phase transitions. These general features are illustrated with three physical examples: correlators of the 2d Ising model, the partition function of the Gross–Witten–Wadia matrix model, and the full counting statistics of one dimensional fermions, associated with Painlevé VI, Painlevé III and Painlevé V, respectively. [ABSTRACT FROM AUTHOR]

We review some exact results for the matrix models appearing in the localization of Chern–Simons-matter theories, focusing on the structure of non-perturbative effects and on the M-theory expansion of ABJM theory. We also summarize some of the results obtained for other Chern–Simons-matter theories, as well as recent applications to topological strings. [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]

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 five-dimensional AGT correspondence implies the connection between the q-deformed Virasoro block and the 5d Nekrasov partition function. In this paper, we determine a q-deformation of the four-point block in the Coulomb gas representation from the 5d Nekrasov function, and obtain an expression of the q-deformed vertex operator. If we use only one kind of the q-vertex operators, one of the insertion points of them must be modified in order to hold the 2d/5d correspondence. [ABSTRACT FROM AUTHOR]