Partial domain wall partition functions

TL;DR

This paper introduces and analyzes the partial domain wall partition function in the six-vertex model, providing determinant expressions, proving their equivalence, and extending the structure to include 1-loop corrections relevant in N=4 SYM.

Contribution

It presents two determinant formulas for the partial domain wall partition function, proves their equivalence, and extends the determinant structure to 1-loop corrected cases.

Findings

Two determinant expressions for the partial domain wall partition function are equivalent.

Both determinants are shown to be discrete KP tau-functions.

Introducing 1-loop corrections preserves the determinant structure.

Abstract

We consider six-vertex model configurations on an n-by-N lattice, n =< N, that satisfy a variation on domain wall boundary conditions that we define and call "partial domain wall boundary conditions". We obtain two expressions for the corresponding "partial domain wall partition function", as an (N-by-N)-determinant and as an (n-by-n)-determinant. The latter was first obtained by I Kostov. We show that the two determinants are equal, as expected from the fact that they are partition functions of the same object, that each is a discrete KP tau-function, and, recalling that these determinants represent tree-level structure constants in N=4 SYM, we show that introducing 1-loop corrections, as proposed by N Gromov and P Vieira, preserves the determinant structure.