Representations of the Virasoro algebra from lattice models

TL;DR

This paper explores how the Virasoro algebra emerges in the scaling limit of lattice models like XXZ and RSOS, using a novel approach involving lattice stress-energy tensors and Bethe-ansatz solutions.

Contribution

It formulates a new conjecture for the lattice stress-energy tensor and verifies algebraic relations in the scaling limit, connecting lattice models with conformal field theory.

Findings

Virasoro algebra features observed in lattice models

Numerical results match theoretical predictions

Connections established between lattice integrability and conformal invariance

Abstract

We investigate in details how the Virasoro algebra appears in the scaling limit of the simplest lattice models of XXZ or RSOS type. Our approach is straightforward but to our knowledge had never been tried so far. We simply formulate a conjecture for the lattice stress-energy tensor motivated by the exact derivation of lattice global Ward identities. We then check that the proper algebraic relations are obeyed in the scaling limit. The latter is under reasonable control thanks to the Bethe-ansatz solution. The results, which are mostly numerical for technical reasons, are remarkably precise. They are also corroborated by exact pieces of information from various sources, in particular Temperley-Lieb algebra representation theory. Most features of the Virasoro algebra (like central term, null vectors, metric properties...) can thus be observed using the lattice models. This seems of general interest for lattice field theory, and also more specifically for finding relations between conformal invariance and lattice integrability, since basis for the irreducible representations of the Virasoro algebra should now follow (at least in principle) from Bethe-ansatz computations.