On Extensions of $\widehat{\mathfrak{gl}(m|n)}$ Kac-Moody algebras and Calabi-Yau Singularities

TL;DR

This paper explores extensions of superalgebra-based vertex operator algebras linked to Calabi-Yau singularities, proposing a free-field realization and revealing deep connections between algebra, geometry, and gauge theory.

Contribution

It introduces a new class of vertex operator algebras associated with Calabi-Yau singularities and generalizes the Miura transformation for their free-field realizations.

Findings

Establishment of a correspondence between algebra truncations and holomorphic functions on singularities.

Development of a free-field realization generalizing the Miura transformation.

Identification of relations leading to bosonization-like equivalences.

Abstract

We discuss a class of vertex operator algebras $\mathcal{W}_{m|n\times \infty}$ generated by a super-matrix of fields for each integral spin $1,2,3,\dots$. The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to $xy=z^mw^n$. We propose a free-field realization of such truncations generalizing the Miura transformation for $\mathcal{W}_N$ algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory.