Perversely categorified Lagrangian correspondences

In this article, we construct a $2$-category of Lagrangians in a fixed shifted symplectic derived stack $\mathrm{S}$. 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 $\mathsf{Symp}^n$. This is a $2$-category version of Weinstein’s symplectic category in the setting of derived symplectic geometry. We introduce another $-$category $\mathsf{Symp}$ or of $0$-shifted symplectic derived stacks where the objects and morphisms in $\mathsf{Symp}^0$ are enhanced with orientation data. Using this, we define a partially linearized $2$-category $\mathsf{LSymp}$. Joyce and his collaborators defined a certain perverse sheaf on any oriented $(-1)$-shifted symplectic derived stack. In $\mathsf{LSymp}$, the $2$-morphisms in $\mathsf{Symp}$ or are replaced by the hypercohomology of the perverse sheaf assigned to the $(-1)$-shifted symplectic derived Lagrangian intersections. To define the compositions in $\mathsf{LSymp}$ 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 $\mathsf{LSymp}$ and a $2$-functor from $\mathsf{Symp}$ or to $\mathsf{LSymp}$. 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 an $n$-shifted symplectic stack defines a $2$-functor from this category to $\mathsf{Symp}^{n-d}$.

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We would like to thank Dominic Joyce for sharing some of his insights on this topic and many helpful discussions. We would also like to thank Tony Pantev, Damien Calaque and Bertrand To¨en for helpful conversations.

The first author was supported by EPSRC grant EP/J016950/1. The second author acknowledges the support of Oxford University and the European Commission under the Marie Curie Programme for the IEF grant which enabled this research to take place.

The contents of this article reflect the author’s views and not the views of the European Commission.

Published 7 July 2017