Colour-independent partition functions in coloured vertex models

Abstract

We study lattice configurations related to Sn, the scalar product of an off-shell state and an on-shell state in rational An integrable vertex models, n∈{1,2}. The lattice lines are colourless and oriented. The state variables are n conserved colours that flow along the line orientations, but do not necessarily cover every bond in the lattice.Choosing boundary conditions such that the positions where the colours flow into the lattice are fixed, and where they flow out are summed over, we show that the partition functions of these configurations, with these boundary conditions, are n-independent. Our results extend to trigonometric An models, and to all n.This n-independence explains, in vertex-model terms, results from recent studies of S2 (Caetano and Vieira, 2012, [1], Wheeler, arXiv:1204.2089, [2]). Namely, 1.S2, which depends on two sets of Bethe roots, {b1} and {b2}, and cannot (as far as we know) be expressed in single determinant form, degenerates in the limit {b1}→∞, and/or {b2}→∞, into a product of determinants, 2. Each of the latter determinants is an A1 vertex-model partition function.