A Derived Legendrian Category for Shifted Contact Stacks

Abstract

For every integer $s$ and every $n\ge 1$, we construct a symmetric monoidal $(\infty ,n)$-category ${\mathrm{Leg}}_n^s$ whose objects are $s$-shifted contact derived Artin stacks and whose $i$-morphisms are $i$-fold Legendrian correspondences. The construction is based on the fact that an $s$-shifted contact structure on a derived Artin stack $X$ is an $s$-shifted symplectic structure on $X$ relative to $B{\mathbb{G}}_m$, twisted by the weight-one line bundle, and that Legendrian structures are the corresponding relative Lagrangian structures. This lets us obtain the $(\infty ,n)$-categories ${\mathrm{Leg}}_n^s$ as twisted forms over $B{\mathbb{G}}_m$ of the $(\infty ,n)$-categories of iterated Lagrangian correspondences of Calaque, Haugseng and Scheimbauer, with the fiber product over $B{\mathbb{G}}_m$ as tensor product. For an $s$-shifted contact stack $X$, we then obtain the derived Legendrian category ${\mathcal{F}}_c(X)$. Its objects are Legendrian morphisms into $X$, and its mapping categories are categories of Legendrians in derived Legendrian intersections, which carry $(s-1)$-shifted contact structures. The AKSZ construction gives oriented extended topological field theories with values in ${\mathrm{Leg}}_n^s$, as well as shifted contact structures on the quotient mapping stacks $[\mathrm{Map}(Y,\tilde{X})/{\mathbb{G}}_m]$. Our definitions of contact and Legendrian structures agree with those of Berktav where both are defined, for every shift.

arXiv comment41 pages, comments are welcome! V3: paper extended to the (\infty, n)-categorical setup of CHS, more details added to proofs