Invariant theory and the W_{1+\infty} algebra with negative integral central charge

TL;DR

This paper proves a conjecture in physics that the W_{1+ abla} algebra at negative integral central charge has a minimal generating set, using invariant theory and free field realizations, clarifying its representation structure.

Contribution

It establishes the minimal strong generating set for W_{1+ abla} at negative integer central charge, confirming a physics conjecture through invariant theory methods.

Findings

Confirmed the conjectured minimal generating set of W_{1+ abla} for negative integer c.

Connected the algebra's representations to a subvariety of complex space.

Utilized free field realization and invariant theory to prove the structure.

Abstract

The vertex algebra W_{1+\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\geq 1, it was conjectured in the physics literature that W_{1+\infty,-n} should have a minimal strong generating set consisting of n^2+2n elements. Using a free field realization of W_{1+\infty,-n} due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of GL_n, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of W_{1+\infty,-n} are parametrized by a closed subvariety of C^{n^2+2n}.