Combinatorics of generalized Bethe equations

TL;DR

This paper explores a generalized form of Bethe equations with multiple zeros and poles, revealing new combinatorial structures and enumerations related to Fuss-Catalan numbers, with implications for mathematical physics.

Contribution

It introduces a generalized Bethe ansatz with multiple zeros and poles, and provides novel combinatorial interpretations of Fuss-Catalan numbers related to these equations.

Findings

Enumeration of solutions using Fuss-Catalan numbers

New combinatorial interpretations involving permutation groups

Counting integer points in specific polytopes

Abstract

A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over Z^M, and on the other hand, they count integer points in certain M-dimensional polytopes.