Perversely categorified Lagrangian correspondences

Abstract

In this article, we construct a $2$-category of Lagrangians in a fixed shifted symplectic derived stack S. The objects and morphisms are all given by Lagrangians living on various fiber products. A special case of this gives a $2$-category of $n$-shifted symplectic derived stacks $Sym{p}^n$. This is a $2$-category version of Weinstein's symplectic category in the setting of derived symplectic geometry. We introduce another $2$-category $Sym{p}^{or}$ of $0$-shifted symplectic derived stacks where the objects and morphisms in $Sym{p}^0$ are enhanced with orientation data. Using this, we define a partially linearized $2$-category $LSymp$. Joyce and his collaborators defined a certain perverse sheaf on any oriented $(-1)$-shifted symplectic derived stack. In $LSymp$, the $2$-morphisms in $Sym{p}^{or}$ are replaced by the hypercohomology of the perverse sheaf assigned to the $(-1)$-shifted symplectic derived Lagrangian intersections. To define the compositions in $LSymp$ we use a conjecture by Joyce, that Lagrangians in $(-1)$-shifted symplectic stacks define canonical elements in the hypercohomology of the perverse sheaf over the Lagrangian. We refine and expand his conjecture and use it to construct $LSymp$ and a $2$-functor from $Sym{p}^{or}$ to $LSymp$. We prove Joyce's conjecture in the most general local model. Finally, we define a $2$-category of $d$-oriented derived stacks and fillings. Taking mapping stacks into a $n$-shifted symplectic stack defines a $2$-functor from this category to $Sym{p}^{n-d}$.

v3: improved exposition; couple of results added

References in corpus (7)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Donaldson-Thomas invariants via microlocal geometry
• Gauge theory and mirror symmetry
• Orientability for gauge theories on Calabi-Yau manifolds
• Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles
• The stack of higher internal categories and stacks of iterated spans
• Generic Base Change, Artin's Comparison Theorem, and the Decomposition Theorem for Complex Artin stacks

Cited by in corpus (6)

• Dimensional reduction in cohomological Donaldson-Thomas theory
• Shifted symplectic reduction of derived critical loci
• A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes
• A motivic integral identity for $(-1)$-shifted symplectic stacks
• Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty }$-algebras
• Virtual classes via vanishing cycles