On Complex Supermanifolds with Trivial Canonical Bundle

TL;DR

This paper characterizes when a complex supermanifold has a trivial canonical bundle using algebraic methods, and applies this to Calabi-Yau supermanifolds with explicit formulas involving superholonomy and connections.

Contribution

It provides a new algebraic criterion for trivial canonical bundles on supermanifolds and extends the understanding of Calabi-Yau supermanifolds with explicit formulas.

Findings

Algebraic characterization of trivial canonical bundle

Explicit formula for Calabi-Yau supermanifolds

Use of superholonomy and complex integral forms

Abstract

We give an algebraic characterisation for the triviality of the canonical bundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky superalgebra structure. As an application, we study the Calabi-Yau case, in which an explicit formula in terms of the Levi-Civita connection is achieved. Our methods include the use of complex integral forms and the recently developed theory of superholonomy.