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

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 $xy=z^m{w}^n$. 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.

39 pages

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