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This paper introduces a concrete relation between genus zero closed Gromov–Witten invariants of Calabi–Yau threefolds and genus zero open Gromov–Witten invariants of a Lagrangian A-brane in the same threefold. Symplectic cutting is a natural operation that decomposes a symplectic manifold (X , ω) with a Hamiltonian U(1) action into two pieces glued along an invariant divisor. In this paper we study a quantum uplift of the cut construction defined in terms of equivariant gauged linear sigma models. The nexus between closed and open Gromov–Witten invariants is a quantum Lebesgue measure associated to a choice of cut, that we introduce and study. Integration of this measure recovers the equivariant quantum volume of the whole CY3, thereby encoding closed Gromov–Witten invariants. Conversely, the monodromies of the quantum measure around cycles in Kähler moduli space encode open Gromov–Witten invariants of a Lagrangian A-brane associated to the cut. Both in the closed and the open string sector we find a remarkable interplay between worldsheet instantons and semiclassical volumes regularized by equivariance. This leads to equivariant generating functions of GW invariants that extend smoothly across the entire moduli space, and which provide a unifying description of standard GW potentials. The latter are recovered in the non-equivariant limit in each of the different phases of the geometry. [ABSTRACT FROM AUTHOR]

We propose formulas for the large N expansion of the generating function of connected correlators of the β -deformed Gaussian and Wishart–Laguerre matrix models. We show that our proposal satisfies the known transformation properties under the exchange of β with 1 / β and, using Virasoro constraints, we derive a recursion formula for the coefficients of the expansion. In the undeformed limit β = 1 , these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic β , we define the higher genus Catalan polynomials C g , ν (β) whose coefficients are integer numbers. [ABSTRACT FROM AUTHOR]

We consider the matrix model of U(N) refined Chern–Simons theory on S 3 for the unknot. We derive a q-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as q-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent q-difference operator for a similar refinement of ABJ theory and show that the corresponding q-Virasoro constraints are equal to those of refined Chern–Simons for a gauge super-group U(N|M). Our equations and solutions are manifestly symmetric under Langlands duality q ↔ t - 1 which correctly reproduces 3d Seiberg duality when q is a specific root of unity. [ABSTRACT FROM AUTHOR]

We revisit the Virasoro constraints and explore the relation to the Hirota bilinear equations. We furthermore investigate and provide the solution to non-homogeneous Virasoro constraints, namely those coming from matrix models whose domain of integration has boundaries. In particular, we provide the example of Hermitean matrices with positive eigenvalues in which case one can find a solution by induction on the rank of the matrix model. [ABSTRACT FROM AUTHOR]

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical β-deformed Hermitean matrix model and the q-deformed matrix models associated to 3d N = 2 supersymmetric gauge theories on D2×qS1 and S b 3 by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the W -algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results. [ABSTRACT FROM AUTHOR]

Using the ideas from the BPS/CFT correspondence, we give an explicit recur- sive formula for computing supersymmetric Wilson loop averages in 3d N = 2 Yang-Mills-Chern-Simons U(N) theory on the squashed sphere S b 3 with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level κ2 and Fayet- Illiopoulos parameter κ1. For these values of κ1 and κ2 the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent D2× S1 theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed. [ABSTRACT FROM AUTHOR]

Within the framework of four dimensional conformal supergravity we consider $$ \mathcal{N}=1,\;2,\;3,\;4 $$ supersymmetric theories generally twisted along the abelian subgroups of the R-symmetry and possibly other global symmetry groups. Upon compactification on constant curvature Riemann surfaces with arbitrary genus we provide an extensive classification of the resulting two dimensional theories according to the amount of supersymmetry that is preserved. Exploiting the c-extremization prescription introduced in we develop a general procedure to obtain the central charge for 2d $$ \mathcal{N}=\left(0,2\right) $$ theories and the expression of the corresponding R-current in terms of the original 4d one and its mixing with the other abelian global currents. [ABSTRACT FROM AUTHOR]

We revisit the vacuum structure of the N = 1 Intriligator-Seiberg-Shenker model on ℝ3× S1. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Ünsal, we argue that previously overlooked non-perturbative effects generate a Higgs-type potential on the classical Coulomb branch of the low-energy effective 3d N = 2 theory. In particular, on part of the Coulomb branch we encounter the first instance of a dynamically-generated quintic monopole superpotential. [ABSTRACT FROM AUTHOR]

We study various aspects of the 4d/3d reduction of N = 1 dualities involving USp(2Nc) gauge theories with 2Nf fundamentals and one antisymmetric. We discuss the non-trivial role played by the monopole superpotentials in the reduction and obtain new 3d dualities for models with both symplectic and unitary gauge groups. For Nf = 4 we observe interesting webs of dualities and symmetry enhancements, recovering and extending some results already appeared in the literature. [ABSTRACT FROM AUTHOR]