Geometrical Loci and CFTs via the Virasoro Symmetry of the mKdV-SG hierarchy: an excursus

TL;DR

This paper explores the connection between algebraic KdV potentials, Virasoro symmetry, and conformal field theories, revealing geometric loci of complex points and their relation to Verma modules.

Contribution

It introduces a novel geometric and algebraic framework linking KdV potentials with Virasoro symmetry and conformal field theory representations.

Findings

Algebraic KdV potentials arise from Virasoro vector fields.

Geometric loci describe almost rational KdV fields.

Solutions relate to non-degenerate conformal Verma modules.

Abstract

We will describe the appearance of specific algebraic KdV potentials as a consequence of a requirement on a integro-differential expression. This expression belongs to a class generated by means of Virasoro vector fields acting on the KdV field. The ``almost'' rational KdV fields are described in terms of a geometrical locus of complex points. A class of solutions of this locus has recently appeared as a description of any conformal Verma module without degeneration.