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We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d sudefine these objects and study their dependence on equivariant parameters for non-compact toric Kahler quotients. We generalize the finite-difference equations (shift equations) obeyed by equivariant volumes to these partition functions. The partition functions are annihilated by differential/difference operators that represent equivariant quantum cohomology/K-theory relations of the target and the appearance of compact divisors in these relations plays a crucial role in the analysis of the non-equivariant limit. We show that the expansion in equivariant parameters contains information about genus-zero Gromov-Witten invariants of the target.

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 vertical bar 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.

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 vertical bar 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.

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 vertical bar 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.

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 vertical bar 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.

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 vertical bar 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.

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.

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.

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.

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.

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.

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical beta -deformed Hermitean matrix model and the q-deformed matrix models associated to 3d N = 2 supersymmetric gauge theories on D(2)x(q)S(1) and Sb3 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.

We revisit the vacuum structure of the N = 1 Intriligator-Seiberg-Shenker model on R-3 x S-1. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Unsal, 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., Title in WoS: From exact results to gauge dynamics on Double-struck capital R-3 x S-1

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical beta -deformed Hermitean matrix model and the q-deformed matrix models associated to 3d N = 2 supersymmetric gauge theories on D(2)x(q)S(1) and Sb3 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.

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical beta -deformed Hermitean matrix model and the q-deformed matrix models associated to 3d N = 2 supersymmetric gauge theories on D(2)x(q)S(1) and Sb3 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.

We revisit the vacuum structure of the N = 1 Intriligator-Seiberg-Shenker model on R-3 x S-1. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Unsal, 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., Title in WoS: From exact results to gauge dynamics on Double-struck capital R-3 x S-1

We revisit the vacuum structure of the N = 1 Intriligator-Seiberg-Shenker model on R-3 x S-1. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Unsal, 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., Title in WoS: From exact results to gauge dynamics on Double-struck capital R-3 x S-1

We revisit the vacuum structure of the N = 1 Intriligator-Seiberg-Shenker model on R-3 x S-1. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Unsal, 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., Title in WoS: From exact results to gauge dynamics on Double-struck capital R-3 x S-1

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical beta -deformed Hermitean matrix model and the q-deformed matrix models associated to 3d N = 2 supersymmetric gauge theories on D(2)x(q)S(1) and Sb3 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.

We revisit the vacuum structure of the N = 1 Intriligator-Seiberg-Shenker model on R-3 x S-1. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Unsal, 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., Title in WoS: From exact results to gauge dynamics on Double-struck capital R-3 x S-1

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical beta -deformed Hermitean matrix model and the q-deformed matrix models associated to 3d N = 2 supersymmetric gauge theories on D(2)x(q)S(1) and Sb3 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.

We revisit the vacuum structure of the N = 1 Intriligator-Seiberg-Shenker model on R-3 x S-1. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Unsal, 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., Title in WoS: From exact results to gauge dynamics on Double-struck capital R-3 x S-1

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 Sb3 with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level kappa(2) and Fayet- Illiopoulos parameter kappa(1). For these values of kappa(1) and kappa(2) the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent D(2)x S-1 theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.

We obtain the brane setup describing 3d N = 2 dualities for USp(2N(c)) and U(Nc) SQCD with monopole superpotentials. This classification follows from a complete analysis of affine and twisted affine compactifications from 4d. The analysis leads to a new duality for the unitary case that has previously been overlooked in the literature. We check this by matching the three-sphere partition function of the two sides of this new duality and find a perfect agreement. Furthermore, we use the partition function to predict new 3d N = 2 dualities for SQCD with monopole superpotentials and tensorial matter.

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 Sb3 with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level kappa(2) and Fayet- Illiopoulos parameter kappa(1). For these values of kappa(1) and kappa(2) the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent D(2)x S-1 theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.

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 Sb3 with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level kappa(2) and Fayet- Illiopoulos parameter kappa(1). For these values of kappa(1) and kappa(2) the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent D(2)x S-1 theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.

We obtain the brane setup describing 3d N = 2 dualities for USp(2N(c)) and U(Nc) SQCD with monopole superpotentials. This classification follows from a complete analysis of affine and twisted affine compactifications from 4d. The analysis leads to a new duality for the unitary case that has previously been overlooked in the literature. We check this by matching the three-sphere partition function of the two sides of this new duality and find a perfect agreement. Furthermore, we use the partition function to predict new 3d N = 2 dualities for SQCD with monopole superpotentials and tensorial matter.

We obtain the brane setup describing 3d N = 2 dualities for USp(2N(c)) and U(Nc) SQCD with monopole superpotentials. This classification follows from a complete analysis of affine and twisted affine compactifications from 4d. The analysis leads to a new duality for the unitary case that has previously been overlooked in the literature. We check this by matching the three-sphere partition function of the two sides of this new duality and find a perfect agreement. Furthermore, we use the partition function to predict new 3d N = 2 dualities for SQCD with monopole superpotentials and tensorial matter.

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 Sb3 with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level kappa(2) and Fayet- Illiopoulos parameter kappa(1). For these values of kappa(1) and kappa(2) the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent D(2)x S-1 theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.

We obtain the brane setup describing 3d N = 2 dualities for USp(2N(c)) and U(Nc) SQCD with monopole superpotentials. This classification follows from a complete analysis of affine and twisted affine compactifications from 4d. The analysis leads to a new duality for the unitary case that has previously been overlooked in the literature. We check this by matching the three-sphere partition function of the two sides of this new duality and find a perfect agreement. Furthermore, we use the partition function to predict new 3d N = 2 dualities for SQCD with monopole superpotentials and tensorial matter.

We obtain the brane setup describing 3d N = 2 dualities for USp(2N(c)) and U(Nc) SQCD with monopole superpotentials. This classification follows from a complete analysis of affine and twisted affine compactifications from 4d. The analysis leads to a new duality for the unitary case that has previously been overlooked in the literature. We check this by matching the three-sphere partition function of the two sides of this new duality and find a perfect agreement. Furthermore, we use the partition function to predict new 3d N = 2 dualities for SQCD with monopole superpotentials and tensorial matter.

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 Sb3 with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level kappa(2) and Fayet- Illiopoulos parameter kappa(1). For these values of kappa(1) and kappa(2) the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent D(2)x S-1 theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.