AKSZ Construction for Shifted Contact Structures

Abstract

This paper establishes the AKSZ theorem for shifted contact structures and its applications. In brief, to resolve certain obstructions, we first define the quotient mapping stack as the quotient of the symplectified mapping space by the constant multiplicative group action. We then prove that if $X$ is an $n$-shifted contact derived Artin stack and $Y$ is an $\mathcal{O}$-compact, $d$-oriented derived stack, the quotient mapping stack $$[\mathrm{Map}(Y, \widetilde{X})/\mathbb{G}_m]$$ admits an $(n-d)$-shifted contact structure. In addition, by formalizing the derived analogue of the graded contact AKSZ formalism, we also introduce the notion of weak shifted contact structures in derived algebraic geometry and prove that under global trivialization of the contact line bundle, the unmodified mapping stack inherits a weak contact structure. Extending this setup to spaces with boundary, we demonstrate that derived fillings naturally induce Legendrian morphisms between quotient mapping stacks, and that topological gluing of cobordisms evaluates to derived Legendrian intersections. Furthermore, we trace the transgression of the canonical shifted $1$-form to prove that our quotient mapping stacks satisfy the derived Classical Master Equation (CME). As applications of our quotient mapping stack formalism, we define the derived analogues of specific topological field theories, including the Jacobi, Courant-Jacobi, and Loop Space Sigma Models. Next, we establish the contact analogue of the PTVV theorem for moduli spaces of perfect complexes. Finally, by composing this geometric construction with the perverse linearization in a companion paper, we elevate these moduli spaces to generate Cohomological Contact Extended Topological Field Theories.