arXiv subjects

Andrew P. Kels

Publications and source records attributed to Andrew P. Kels.

At least 19 recordsLinked to original sources

arXiv2025

Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations

The quasi-classical expansion of a multicomponent spin solution of the star-star relation with hyperbolic Boltzmann weights is investigated. The equations obtained in a quasi-classical limit provide n-1-component extensions of certain scalar 5-point equations (corresponding to n=2) that were previously investigated by the author in the context of integrability and consistency of equations on face-centered cubics.

math-ph↗

arXiv2024

Algebraic entropy for hex systems

Hex systems were recently introduced [A. P. Kels. Integrable systems on hexagonal lattices and consistency on polytopes with quadrilateral and hexagonal faces. 2022. arXiv:2205.02720 [math-ph]] as systems of equations defined on two-dimensional honeycomb lattices. We give a definition of algebraic entropy for such systems and use it to check the integrability of specific examples.

nlin.SI↗

arXiv2022

Two-component Yang-Baxter maps and star-triangle relations

It is shown how Yang-Baxter maps may be directly obtained from classical counterparts of the star-triangle relations and quantum Yang-Baxter equations. This is based on reinterpreting the latter equation and its solutions which are given in terms of special functions, as a set-theoretical form of the Yang-Baxter equation whose solutions are given by quadrirational Yang-Baxter maps. The Yang-Baxter maps obtained through this approach are found to satisfy two different types of Yang-Baxter equations, one that is the usual equation involving a single map, and another equation that involves a pair of maps, which is a case of what is also known as an entwining Yang-Baxter equation. Apart from the elliptic case, each of these Yang-Baxter maps are quadrirational, but only maps that solve the former type of Yang-Baxter equation are reversible. The Yang-Baxter maps are expressed in terms of two-component variables, and two-component parameters, and have a natural QRT-like composition of separate maps for each component. Through this approach, sixteen different Yang-Baxter maps are derived from known solutions of the classical star-triangle relations.

math-ph↗

arXiv2022

Integrable systems on hexagonal lattices and consistency on polytopes with quadrilateral and hexagonal faces

The new concept of a system of hex equations is introduced as an overdetermined system of six five-point face-centered quad equations defined on six vertices of a hexagon. For a consistent system of hex equations, two variables on neighbouring vertices of the hexagon can be solved for uniquely in terms of the other four. A consistent system of hex equations has a well-defined unique evolution in the hexagonal lattice under suitable initial value problems defined on a single connected staircase of points. Multidimensional consistency for systems of hex equations is proposed in terms of their consistency on certain polytopes which have both hexagonal and quadrilateral faces, and specific examples are presented for the hexagonal prism, the elongated dodecahedron, the truncated octahedron, and the 6-6-duoprism. Consistent systems of hex equations on such polytopes may be constructed from face-centered quad equations which satisfy consistency-around-a-face-centered-cube, in combination with regular quad equations that satisfy consistency-around-a-cube.

math-ph↗

arXiv2022

Discrete integrable equations on face-centered cubics: consistency and Lax pairs of corner equations

A new set of discrete integrable equations, called face-centered quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang-Baxter equation. These equations satisfy a new formulation of multidimensional consistency, known as consistency-around-a-face-centered-cube (CAFCC), which requires consistency of an overdetermined system of fourteen five-point equations on the face-centered cubic unit cell. In this paper a new formulation of CAFCC is introduced where so-called type-C equations are centered at faces of the face-centered cubic unit cell, whereas previously they were only centered at corners. This allows type-C equations to be regarded as independent multidimensionally consistent integrable systems on higher-dimensional lattices and is used to establish their Lax pairs.

nlin.SI↗

arXiv2021

Entanglement of two disjoint intervals in conformal field theory and the 2D Coulomb gas on a lattice

In the conformal field theories given by the Ising and Dirac models, when the system is in the ground state, the moments of the reduced density matrix of two disjoint intervals and of its partial transpose have been written as partition functions on higher genus Riemann surfaces with $\mathbb{Z}_n$ symmetry. We show that these partition functions can be expressed as the grand canonical partition functions of the two-dimensional two component classical Coulomb gas on certain circular lattices at specific values of the coupling constant.

hep-th↗

arXiv2021

Algebraic entropy for face-centered quad equations

In this paper we define the algebraic entropy test for face-centered quad equations, which are equations defined on vertices of a quadrilateral plus an additional interior vertex. This notion of algebraic entropy is applied to a recently introduced class of these equations that satisfy a new form of multidimensional consistency called consistency-around-a-face-centered-cube (CAFCC), whereby the system of equations is consistent on a face-centered cubic unit cell. It is found that for certain arrangements of equations (or pairs of equations) in the square lattice, all known CAFCC equations pass the algebraic entropy test possessing either quadratic or linear growth.

nlin.SI↗

arXiv2021

Lax matrices for lattice equations which satisfy consistency-around-a-face-centered-cube

There is a recently discovered formulation of the multidimensional consistency integrability condition for lattice equations, called consistency-around-a-face-centered-cube(CAFCC), which is applicable to equations defined on a vertex and its four nearest neighbours on the square lattice. This paper introduces a method of deriving Lax matrices for the equations which satisfy CAFCC. This method gives novel Lax matrices for such equations, which include previously known equations of discrete Toda-, or Laplace-type, as well as newer equations which have only appeared in the context of CAFCC.

math-ph↗

arXiv2021

Interaction-round-a-face and consistency-around-a-face-centered-cube

There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations which satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former. This paper extends this correspondence to interaction-round-a-face (IRF) models, resulting in a new formulation of the consistency-around-a-cube (CAC) integrability condition applicable to five-point equations in the square lattice. Multidimensional consistency for these equations is formulated as consistency-around-a-face-centered-cube (CAFCC), which namely involves satisfying an overdetermined system of fourteen five-point lattice equations for eight unknown variables on the face-centered cubic unit cell. From the quasi-classical limit of IRF models, which are constructed from the continuous spin solutions of the star-triangle relations associated to the Adler-Bobenko-Suris (ABS) list, fifteen sets of equations are obtained which satisfy CAFCC.

math-ph↗

arXiv2019

Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations

In a previous paper, the author has established an extension of the Z-invariance property for integrable edge-interaction models of statistical mechanics, that satisfy the star-triangle relation (STR) form of the Yang-Baxter equation (YBE). In the present paper, an analogous extended Z-invariance property is shown to also hold for integrable vector models and interaction-round-a-face (IRF) models of statistical mechanics respectively. As for the previous case of the STR, the Z-invariance property is shown through the use of local cubic-type deformations of a 2-dimensional surface associated to the models, which allow an extension of the models onto a subset of next nearest neighbour vertices of $\mathbb{Z}^3$, while leaving the partition functions invariant. These deformations are permitted as a consequence of the respective YBE's satisfied by the models. The quasi-classical limit is also considered, and it is shown that an analogous Z-invariance property holds for the variational formulation of classical discrete Laplace equations which arise in this limit. From this limit, new integrable 3D-consistent multi-component quad equations are proposed, which are constructed from a degeneration of the equations of motion for IRF Boltzmann weights.

math-ph↗

arXiv2019

Lens Generalisation of $τ$-functions for the Elliptic Discrete Painlevé Equation

We propose a new bilinear Hirota equation for $τ$-functions associated with the $E_8$ root lattice, that provides a "lens" generalisation of the $τ$-functions for the elliptic discrete Painlevé equation. Our equations are characterized by a positive integer $r$ in addition to the usual elliptic parameters, and involve a mixture of continuous variables with additional discrete variables, the latter taking values on the $E_8$ root lattice. We construct explicit $W(E_7)$-invariant hypergeometric solutions of this bilinear Hirota equation, which are given in terms of elliptic hypergeometric sum/integrals.

nlin.SI↗

arXiv2018

Integrable quad equations derived from the quantum Yang-Baxter equation

This paper presents an explicit correspondence between two different types of integrable equations; the quantum Yang-Baxter equation in its star-triangle relation form, and the classical 3D-consistent quad equations in the Adler-Bobenko-Suris (ABS) classification. Each of the 3D-consistent ABS quad equations of $H$-type, are respectively derived from the quasi-classical expansion of a counterpart star-triangle relation. Through these derivations it is seen that the star-triangle relation provides a natural path integral quantization of an ABS equation. The interpretation of the different star-triangle relations is also given in terms of (hyperbolic/rational/classical) hypergeometric integrals, revealing the hypergeometric structure that links the two different types of integrable systems. Many new limiting relations that exist between the star-triangle relations/hypergeometric integrals are proven for each case.

math-ph↗

arXiv2018

Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity

We study the root of unity limit of the lens elliptic gamma function solution of the star-triangle relation, for an integrable model with continuous and discrete spin variables. This limit involves taking an elliptic nome to a primitive $rN$-th root of unity, where $r$ is an existing integer parameter of the lens elliptic gamma function, and $N$ is an additional integer parameter. This is a singular limit of the star-triangle relation, and at subleading order of an asymptotic expansion, another star-triangle relation is obtained for a model with discrete spin variables in $\mathbb{Z}_{rN}$. Some special choices of solutions of equation of motion are shown to result in well-known discrete spin solutions of the star-triangle relation. The saddle point equations themselves are identified with three-leg forms of "3D-consistent" classical discrete integrable equations, known as $Q4$ and $Q3_{(δ=0)}$. We also comment on the implications for supersymmetric gauge theories, and in particular comment on a close parallel with the works of Nekrasov and Shatashvili.

math-ph↗

arXiv2018

Elliptic hypergeometric sum/integral transformations and supersymmetric lens index

We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the elliptic gamma function that depends on an additional integer variable, as well as a complex variable and two elliptic nomes. As an application of our results, we prove an equality between $S^1\times S^3/\mathbb{Z}_r$ supersymmetric indices, for a pair of four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories related by Seiberg duality, with gauge groups $SU(n+1)$ and $Sp(2n)$. This provides one of the most elaborate checks of the Seiberg duality known to date. As another application of the $A_n$ integral, we prove a star-star relation for a two-dimensional integrable lattice model of statistical mechanics, previously given by the second author.

math-ph↗

arXiv2017

Exactly solved models on planar graphs with vertices in $\mathbb{Z}^3$

It is shown how exactly solved edge interaction models on the square lattice, may be extended onto more general planar graphs, with edges connecting a subset of next nearest neighbour vertices of $\mathbb{Z}^3$. This is done by using local deformations of the square lattice, that arise through the use of the star-triangle relation. Similar to Baxter's Z-invariance property, these local deformations leave the partition function invariant up to some simple factors coming from the star-triangle relation. The deformations used here extend the usual formulation of Z-invariance, by requiring the introduction of oriented rapidity lines which form directed closed paths in the rapidity graph of the model. The quasi-classical limit is also considered, in which case the deformations imply a classical Z-invariance property, as well as a related local closure relation, for the action functional of a system of classical discrete Laplace equations.

math-ph↗

arXiv2016

The star-triangle relation, lens partition function, and hypergeometric sum/integrals

The aim of the present paper is to consider the hyperbolic limit of an elliptic hypergeometric sum/integral identity, and associated lattice model of statistical mechanics previously obtained by the second author. The hyperbolic sum/integral identity obtained from this limit, has two important physical applications in the context of the so-called gauge/YBE correspondence. For statistical mechanics, this identity is equivalent to a new solution of the star-triangle relation form of the Yang-Baxter equation, that directly generalises the Faddeev-Volkov models to the case of discrete and continuous spin variables. On the gauge theory side, this identity represents the duality of lens ($S_b^3/\mathbb{Z}_r$) partition functions, for certain three-dimensional $\mathcal N = 2$ supersymmetric gauge theories.

math-ph↗

arXiv2016

Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single "master solution", which is expressed through the elliptic gamma function and have continuous spins taking values of the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the star-triangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.

math-ph↗

arXiv2015

New solutions of the star-triangle relation with discrete and continuous spin variables

A new solution to the star-triangle relation is given, for an Ising type model that involves interacting spins, that contain integer and real valued components. Boltzmann weights of the model are given in terms of the lens elliptic-gamma function, and are based on Yamazaki's recently obtained solution of the star-star relation. The star-triangle given here, implies Seiberg duality for the $4\!-\!d$ $\mathcal{N}=1$ $S_1\times S_3/\mathbb{Z}_r$ index of the $SU(2)$ quiver gauge theory, and the corresponding two component spin case of the star-star relation of Yamazaki. A proof of the star-triangle relation is given, resulting in a new elliptic hypergeometric integral identity. The star-triangle relation in this paper contains the master solution of Bazhanov and Sergeev as a special case. Two other limiting cases are considered one of which gives a new star-triangle relation in terms of ratios of infinite $q$-products, while the other case gives a new way of deriving a star-triangle relation previously obtained by the author.

math-ph↗