The star-square relation and the generalized star-triangle relation from 3d supersymmetric dualities I

We study duality transformations the star-square relation and the generalized star-triangle relation for Ising-like lattice spin models obtained via gauge/YBE correspondence. Solutions to those relations obtained by the use of integral identities coming from the duality of three-dimensional supersymmetric gauge theories allow us to construct spin lattice models with four-spin (the star-square relation) and three-spin (the generalized star-triangle relation) interactions.Comment: 17 pages, 2 figure

Liouville integrable binomial Hamiltonian system

In this study we work on a novel Hamiltonian system which is Liouville integrable. In the integrable Hamiltonian model, conserved currents can be represented as Binomial polynomials in which each order corresponds to the integral of motion of the system. From a mathematical point of view, the equations of motion can be written as integrable second-order nonlinear partial differential equations in 1 + 1 dimensions.Comment: 6 page

Lens Partition Functions and Integrability Properties

We study lens partitions functions for the three-dimensional $N=2$ supersymmetric gauge theories on $S_b^3/Zr$. We consider an equality as a new hyperbolic hypergeometric solution to the star-star relation via the gauge/YBE correspondence. The correspondence allows the construction of integrable lattice spin models of statistical mechanics by the use of integral identities. Additionally, we obtain new hyperbolic hypergeometric integral identities of gauge theories.Comment: minor correction

Hyperbolic and trigonometric hypergeometric solutions to the star-star equation

We construct the hyperbolic and trigonometric solutions to the star-star relation via the gauge/YBE correspondence by using the three-dimensional lens partition function and superconformal index for a certain N=2 supersymmetric gauge dual theories. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic and basic hypergeometric functions.Comment: 14 pages, v2: minor corrections and comments, v3: minor correction

Lens partition function, pentagon identity and star-triangle relation

We study the three-dimensional lens partition function for $\mathcal N=2$ supersymmetric gauge dual theories on $S^3/\mathbb{Z}_r$ by using the gauge/YBE correspondence. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic hypergeometric functions. We obtain such an integral identity which can be written as the star-triangle relation for Ising type integrable models and as the integral pentagon identity. The latter represents the basic 2-3 Pachner move for triangulated 3-manifolds. A special case of our integral identity can be used for proving orthogonality and completeness relation of the Clebsch-Gordan coefficients for the self-dual continuous series of $U_q(osp(1|2))$.Comment: 22 pages, v2: minor corrections and comments, v3: minor correction

On Bailey pairs for N=2\mathcal N=2 supersymmetric gauge theories on Sb3/ZrS_b^3/\mathbb{Z}_r

We study Bailey pairs construction for hyperbolic hypergeometric integral identities acquired via the duality of lens partitions functions for the three-dimensional $\mathcal N=2$ supersymmetric gauge theories on $S_b^3/\mathbb{Z}_r$. The novel Bailey pairs are constructed for the star-triangle relation, the star-star relation and the pentagon identity. The first two of them are integrability conditions for the Ising-type integrable lattice models. The last one corresponds to the representation of the basic $2-3$ Pachner move for triangulated 3-manifolds.Comment: 26 pages, v2: minor corrections, v3: minor corrections and comments adde

On Bailey pairs for N = 2 supersymmetric gauge theories on S 3 b /Zr

We study Bailey pairs construction for hyperbolic hypergeometric integral identities acquired via the duality of lens partitions functions for the three-dimensional N = 2 supersymmetric gauge theories on S3b/Zr . The novel Bailey pairs are constructed for the star-triangle relation, the star-star relation, and the pentagon identity. The first two of them are integrability conditions for the Ising-type integrable lattice models. The last one corresponds to the representation of the basic 2 − 3 Pachner move for triangulated 3-manifolds

Decorating the gauge/YBE correspondence

In this paper, we aim to study the three-dimensional $\mathcal N=2$ supersymmetric dual gauge theories on $S_b^3/\mathbb{Z}_r$ in the context of the gauge/YBE correspondence. We consider hyperbolic hypergeometric integral identities acquired via the equality of supersymmetric lens partition functions as solutions to the decoration transformation and the flipping relation in statistical mechanics. The solutions of those transformations aim at investigating various decorated lattice models possessing the Boltzmann weights of integrable Ising-like models obtained via the gauge/YBE correspondence. We also constructed The Bailey pairs for the decoration transformation and the flipping relation.Comment: 25 pages, 6 figure

Lens partition function, pentagon identity, and star-triangle relation

We study the three-dimensional lens partition function for N=2 supersymmetric gauge dual theories on S3/Zr by using the gauge/Yang-Baxter equation correspondence. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic hypergeometric functions. We obtain such an integral identity which can be written as the star-triangle relation for Ising type integrable models and as the integral pentagon identity. The latter represents the basic 2-3 Pachner move for triangulated 3-manifolds. A special case of our integral identity can be used for proving orthogonality and completeness relation of the Clebsch-Gordan coefficients for the self-dual continuous series of Uq(osp(1|2))

On Bailey pairs for N N \mathcal{N} = 2 supersymmetric gauge theories on S b 3 / ℤ r Sb3/Zr {S}_b^3/{\mathbb{Z}}_r

Abstract We study Bailey pairs construction for hyperbolic hypergeometric integral identities acquired via the duality of lens partitions functions for the three-dimensional N $$\mathcal{N}$$ = 2 supersymmetric gauge theories on S b 3 / ℤ r $${S}_b^3/{\mathbb{Z}}_r$$ . The novel Bailey pairs are constructed for the star-triangle relation, the star-star relation, and the pentagon identity. The first two of them are integrability conditions for the Ising-type integrable lattice models. The last one corresponds to the representation of the basic 2 − 3 Pachner move for triangulated 3-manifolds