Minimal models for minimal BCOV theories

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 $L_{\mathrm{\infty }}$-algebras describing minimal BCOV theory and its variants on flat space and find that they give certain $L_{\mathrm{\infty }}$-extensions of the infinite-dimensional simple Lie superalgebra $\mathrm{SHO}(d\mathrm{\mid }d)$. We apply this computation to compare an ${\mathfrak{s}\mathfrak{l}}_2$ action on an odd two-dimensional central extension of $\mathrm{SHO}(3\mathrm{\mid }3)$ first discovered by Kac to an action of ${\mathfrak{s}\mathfrak{l}}_2$ on a variant of minimal BCOV theory previously found by the authors.