A Derived Legendrian Category for Shifted Contact Stacks

TL;DR

This paper constructs a derived Legendrian category for shifted contact stacks in derived algebraic geometry, introducing new categorical structures and applications to moduli theory.

Contribution

It develops the derived Legendrian category for shifted contact stacks and embeds it into an $( olinebreak ext{infty,}2)$-category of spans, with applications to Legendrian surgery.

Findings

Defines the derived Legendrian category $cal_c(X)$ for shifted contact stacks.

Embeds $cal_c(X)$ into an $( ext{infty,}2)$-category of spans via the AKSZ construction.

Introduces derived Legendrian surgery using topological cobordisms as Lagrangian correspondences.

Abstract

We construct the derived Legendrian category $\mathcal{F}_{c}(X)$ for an $n$-shifted contact derived Artin stack $X$ and the $(\infty,2)$-category $Leg_n$ of Legendrian correspondences in the context of derived algebraic geometry, with several applications to moduli theory. In brief, the objects of the category $\mathcal{F}_{c}(X)$ are Legendrian morphisms; the morphism spaces and composition operations are defined using equivariant descent. We also establish that $\mathcal{F}_{c}(X)$ embeds into an $(\infty, 2)$-category of spans defined by the AKSZ construction. We further evaluate topological cobordisms as Lagrangian correspondences to define derived Legendrian surgery.