Combinatorics of Generalized Bethe Equations

Author： Kozlowski Karol   Sklyanin Evgeny  

Publisher： Springer Publishing Company

ISSN： 0377-9017

Source： Letters in Mathematical Physics, Vol.103, Iss.10, 2013-10, pp. : 1047-1077

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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 $${mathbb{Z}^M}$$ Z M , and on the other hand, they count integer points in certain M-dimensional polytopes.

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