Virasoro and KdV

TL;DR

This paper explores the representation theory of the Virasoro algebra, revealing how its structures decompose and connect to integrable systems like the KP hierarchy, with implications for differential operators in mathematical physics.

Contribution

It introduces new decomposition results for Virasoro algebra representations and links these to the multicomponent KP hierarchy and differential operators.

Findings

Decomposition of Virasoro representations as tensor products

Factorization properties of solutions to related differential equations

Connection established between Virasoro algebra and multicomponent KP hierarchy

Abstract

We investigate the structure of representations of the (positive half of the) Virasoro algebra and situations in which they decompose as a tensor product of Lie algebra representations. As an illustration, we apply these results to the differential operators defined by the Virasoro conjecture and obtain some factorization properties of the solutions as well as a link to the multicomponent KP hierarchy.