AKSZ construction for shifted Poisson structures

TL;DR

This paper proves the AKSZ theorem for shifted Poisson structures, showing how mapping stacks inherit shifted Poisson structures and extending the concept to derived prestacks with applications in deformation theory and BV formalism.

Contribution

It establishes the AKSZ theorem for shifted Poisson structures on derived stacks and extends the definition to derived prestacks with deformation theory, including new applications.

Findings

Mapping stacks of shifted Poisson derived stacks have induced shifted Poisson structures.

Equivalence between shifted Poisson structures and shifted Lagrangian thickenings.

Extension of shifted Poisson structures to derived prestacks with deformation theory.

Abstract

We prove the AKSZ theorem for shifted Poisson structures: if $X$ is an $n$-shifted Poisson derived stack, and $Y$ a $d$-oriented derived stack, then the mapping stack \[\underline{\mathrm{Map}}(Y,X)\] is naturally endowed with an $(n-d)$-shifted Poisson structure. For this, we prove that the data of an $n$-shifted Poisson structure on a derived Artin stack is equivalent to the data of an $(n+1)$-shifted Lagrangian thickening of it. We also extend the definition of shifted Poisson structures to derived prestacks having a deformation theory and give two applications, one for mapping stacks with a non-proper source and one in BV formalism.