Minimal models for minimal BCOV theories

Abstract

Minimal BCOV theory is a classical field theory which describes a subclass of deformations of the category of perfect complexes on a Calabi-Yau variety. We compute minimal models for $L_{\infty }$-algebras describing minimal BCOV theory and its variants on flat space and find that they give certain $L_{\infty }$-extensions of the infinite-dimensional simple Lie superalgebra $\mathrm{SHO}(d|d)$. We apply this computation to compare an ${\mathfrak{sl}}_2$ action on an odd two-dimensional central extension of $\mathrm{SHO}(3|3)$ first discovered by Kac to an action of ${\mathfrak{sl}}_2$ on a variant of minimal BCOV theory previously found by the authors.

18 pages, preliminary version, comments welcome!