Lagrangian correspondences and the generalized Viterbo restriction functor

TL;DR

This paper establishes the equivalence of two functors in wrapped Fukaya categories—Viterbo restriction and graph correspondence—extending the former to non-strongly-exact cases and connecting it with Legendrian homology.

Contribution

It proves the equivalence of Viterbo restriction and graph correspondence functors and extends the Viterbo functor to non-strongly-exact restrictions using Legendrian homology.

Findings

Viterbo restriction and graph correspondence functors agree on their common domain.

Extension of Viterbo restriction to non-strongly-exact cases via Legendrian homology.

The graph correspondence functor naturally extends to a larger category with Lagrangian immersions.

Abstract

We study two kinds of functors of wrapped Fukaya categories: 1) the Viterbo restriction functor for an inclusion of a Liouville sub-domain; 2) the Lagrangian correspondence functor associated to the graph of the completion of the inclusion of the sub-domain, named the graph correspondence functor. We prove that these two functors agree on the sub-category where the Viterbo restriction functor is defined. We also extend the Viterbo restriction functor to the case of non-strongly-exact restrictions of Lagrangian submanifolds, which yields a natural object-wise deformation of the wrapped Fukaya category, constructed using another theory - linearized Legendrian homology. On the other hand, the graph correspondence functor is naturally defined on the whole wrapped Fukaya category, a priori taking values in a suitable enlargement of the wrapped Fukaya category having certain Lagrangian immersions as objects. We prove that the graph correspondence functor agrees with the extension of the Viterbo restriction functor.