Special polynomials related to the supersymmetric eight-vertex model: A summary

Abstract

We introduce and study symmetric polynomials, which as very special cases include polynomials related to the supersymmetric eight-vertex model, and other elliptic lattice models with $\Delta =\pm 1/2$. There is also a close relation to affine Lie algebra characters. After a natural change of variables, our polynomials satisfy a non-stationary Schrödinger equation with elliptic potential, which is related to the Knizhnik-Zamolodchikov-Bernard equation and to the canonical quantization of Painlevé VI. Moreover, specializations of our polynomials can be identified with tau functions of Painlevé VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of Bäcklund transformations. In the present work, our results on these topics are summarized with a minimum of technical details.

29 pages. This is a summary of our preprints arXiv:1305.0666, arXiv:1312.5879 and arXiv:1405.5318

References in corpus (8)

• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• The eight-vertex model and Painleve VI
• Spin chains with dynamical lattice supersymmetry
• Ground-state properties of a supersymmetric fermion chain
• The master Painlevé VI heat equation
• Special polynomials related to the supersymmetric eight-vertex model. III. Painlevé VI equation
• A staggered fermion chain with supersymmetry on open intervals
• "Quantum" linearization of Painlevé equations as a component of their $L,A$ pairs

Cited by in corpus (9)

• Elliptic solid-on-solid model's partition function as a single determinant
• On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions
• On the transfer matrix of the supersymmetric eight-vertex model. I. Periodic boundary conditions
• Elliptic pfaffians and solvable lattice models
• "Quantizations" of isomonodromic Hamiltonian Garnier system with two degrees of freedom
• Series Solutions of the Non-Stationary Heun Equation
• A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain
• Sum rules for the supersymmetric eight-vertex model
• On the elliptic ${\mathfrak{gl}}_2$ solid-on-solid model: functional relations and determinants