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Minimal models for minimal BCOV theories

Surya Raghavendran, Philsang Yoo

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 $\operatorname{SHO}(d|d)$. We apply this computation to compare an $\mathfrak{sl}_2$ action on an odd two-dimensional central extension of $\operatorname{SHO}(3|3)$ first discovered by Kac to an action of $\mathfrak{sl}_2$ on a variant of minimal BCOV theory previously found by the authors.

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 -algebras describing minimal BCOV theory and its variants on flat space and find that they give certain -extensions of the infinite-dimensional simple Lie superalgebra . We apply this computation to compare an action on an odd two-dimensional central extension of first discovered by Kac to an action of on a variant of minimal BCOV theory previously found by the authors.

Paper Structure

This paper contains 8 sections, 9 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. BCOV theory and its variants
  3. Minimal BCOV theory
  4. Introducing potentials
  5. Minimal models and extensions of $\mathop{\mathrm{SHO}}\nolimits(d|d)$
  6. Minimal BCOV theory and $\mathop{\mathrm{SHO}}\nolimits(d|d)$
  7. Extensions from potentials
  8. Two $\mathfrak{sl}_2$ actions

Key Result

Proposition 3.3

The minimal model for minimal BCOV theory $\Pi\mathcal{E}_{\mathrm{mBCOV}} (\mathop{\mathrm{\mathbb{C}}}\nolimits^d)$ is a one-dimensional odd central extension of $\mathop{\mathrm{SHO}}\nolimits (d | d)$.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3: Minimal BCOV theory for Calabi--Yau 3-folds
  • proof
  • Remark 2.5
  • Definition 3.1
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 11 more