On vertex algebra representations of the Schrödinger–Virasoro Lie algebra

Abstract

The Schrödinger–Virasoro Lie algebra sv is an extension of the Virasoro Lie algebra by a nilpotent Lie algebra formed with a bosonic current of weight 3 2 and a bosonic current of weight 1. It is also a natural infinite-dimensional extension of the Schrödinger Lie algebra, which — leaving aside the invariance under time-translation — has been proved to be a symmetry algebra for many statistical physics models undergoing a dynamics with dynamical exponent z = 2 . We define in this article general Schrödinger–Virasoro primary fields by analogy with conformal field theory, characterized by a ‘spin’ index and a (non-relativistic) mass, and construct vertex algebra representations of sv out of a charged symplectic boson and a free boson and its associated vertex operators. We also compute two- and three-point functions of still conjectural massive fields that are defined by an analytic continuation with respect to a formal parameter.