Colour-independent partition functions in coloured vertex models

Abstract

We study lattice configurations related to S_n, the scalar product of an off-shell state and an on-shell state in rational A_n 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 A_n models, and to all n. This n-independence explains, in vertex-model terms, results from recent studies of S_2 [1, 2]. Namely, 1. S_2 which depends on two sets of Bethe roots, b_1 and b_2, and cannot (as far as we know) be expressed in single determinant form, degenerates in the limit b_1 -> infinity, and/or b_2 -> infinity, into a product of determinants, 2. Each of the latter determinants is an A_1 vertex-model partition function.

28 pages, LaTeX figures included in the text. Minor corrections and clarifications added. Version published in Nuclear Physics B

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• Multiple integral formulae for the scalar product of on-shell and off-shell Bethe vectors in SU(3)-invariant models
• Comments on World-Sheet Form Factors in AdS/CFT
• Scalar products and norm of Bethe vectors for integrable models based on $U_q({\widehat{\mathfrak{gl}}}_n)$
• A new generalisation of Macdonald polynomials
• Observables of coloured stochastic vertex models and their polymer limits
• Nested coordinate Bethe wavefunctions from the Bethe/gauge correspondence
• Stochastic symplectic ice
• Colored five-vertex models and Demazure atoms
• A class of partition functions associated with $E_{\tau ,\eta }(g{l}_3)$ by Izergin-Korepin analysis