Fermionic spectra in integrable systems

TL;DR

This paper reviews algebraic methods related to fermionic spectra in integrable systems, connecting quantum spin chains, conformal blocks, and algebraic structures like cluster algebras and difference equations.

Contribution

It provides a comprehensive overview of algebraic constructions and their interrelations in the context of fermionic spectra in integrable models.

Findings

Connections between fermionic formulas and conformal blocks

Relations between quantum cluster algebras and integrable systems

Insights into difference equations and noncommutative evolutions

Abstract

This is a brief review of several algebraic constructions related to generalized fermionic spectra, of the type which appear in integrable quantum spin chains and integrable quantum field theories. We discuss the connection between fermionic formulas for the graded dimensions of the spaces of conformal blocks of WZW theories, quantum cluster algebras, discrete integrable noncommutative evolutions and difference equations.