scholarly journals Comments on the multi-spin solution to the Yang–Baxter equation and basic hypergeometric sum/integral identity

2019 ◽  
Vol 34 (18) ◽  
pp. 1950140 ◽  
Author(s):  
Ilmar Gahramanov ◽  
Shahriyar Jafarzade
Keyword(s):  
Statistical Mechanics ◽  
Integral Identity ◽  
Hypergeometric Functions ◽  
Three Dimensional ◽  
Spin Model ◽  
Baxter Equation ◽  
Lattice Spin ◽  
Integrable Lattice ◽  
Basic Hypergeometric Functions

We present a multi-spin solution to the Yang–Baxter equation (YBE). The solution corresponds to the integrable lattice spin model of statistical mechanics with positive Boltzmann weights and parametrized in terms of the basic hypergeometric functions. We obtain this solution from a nontrivial basic hypergeometric sum-integral identity which originates from the equality of supersymmetric indices for certain three-dimensional [Formula: see text] = 2 Seiberg dual theories.

scholarly journals Integrable lattice models and holography

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Meer Ashwinkumar
Keyword(s):  
Correlation Functions ◽  
Three Dimensional ◽  
Lattice Models ◽  
Simons Theory ◽  
Boundary Theory ◽  
Baxter Equation ◽  
Rational Solutions ◽  
Integrable Lattice ◽  
Local Operators ◽  
Integrable Lattice Models

Abstract We study four-dimensional Chern-Simons theory on D × ℂ (where D is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an “analytically-continued” toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational R-matrix purely from the boundary theory.

Statement B and the Yang-Baxter Equation

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Statistical Mechanics ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Transfer Matrix ◽  
Baxter Equation ◽  
Partition Functions ◽  
Alternate Proof ◽  
Crystal Basis ◽  
Exactly Solved Models ◽  
Multiple Dirichlet Series

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

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