AKSZ construction for shifted Poisson structures

TL;DR

This work establishes an AKSZ-type construction for shifted Poisson structures by proving that, for an $n$-shifted Poisson derived stack $X$ and a $d$-oriented derived stack $Y$, the mapping stack $\\underline{Map}(Y,X)$ inherits an $(n-d)$-shifted Poisson structure. Central to the approach is the equivalence between $n$-shifted Poisson structures on $X$ and $(n+1)$-shifted Lagrangian thickenings, extended to prestacks with deformation theory via formal localization. The paper also extends shifted Poisson geometry to general prestacks, establishes descent for Lagrangian thickenings, and provides two applications: Poisson structures on moduli of flat connections for non-proper sources and a BV-formalism influenced construction via a $(-1)$-shifted coisotropic structure. The AKSZ construction then leverages mapping stacks to transfer Lagrangian thickenings and produce $(n-d)$-shifted Poisson structures, offering a robust framework for new Poisson structures on moduli problems and potential quantizations in derived settings.

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.

Theorems & Definitions (62)

• Theorem A: \ref{['thm:aksz']}
• Theorem B: \ref{['thm:lagthick']}
• Theorem C
• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• proof
• Lemma 1.5
• proof
• Proposition 1.6