Closed string mirrors of symplectic cluster manifolds

TL;DR

This paper constructs the closed string mirror of symplectic cluster manifolds by computing the relative symplectic cohomology sheaf and demonstrating its structure as a pushforward of a structure sheaf of a rigid analytic space, using axiomatic properties.

Contribution

It introduces a new axiomatic approach to compute relative symplectic cohomology sheaves for nodal Lagrangian fibrations, leading to the construction of the closed string mirror in symplectic geometry.

Findings

Computed the relative symplectic cohomology sheaf for nodal Lagrangian fibrations.

Identified the sheaf as the pushforward of a structure sheaf of a rigid analytic space.

Established axiomatic properties enabling local model computations without ad hoc analysis.

Abstract

For the base $B$ of a Maslov $0$ Lagrangian torus fibration with singularities consider the sheaf assigning to each $P\subset B$ the relative symplectic cohomology in degree $0$ of its pre-image. We compute this sheaf for nodal Lagrangian torus fibrations on four dimensional symplectic cluster manifolds. We show that it is the pushforward of the structure sheaf of a certain rigid analytic space under a non-archimedean torus fibration. The rigid analytic space is constructed in a canonical way from the relative SH sheaf and is referred as the \emph{closed string mirror}. The construction relies on computing relative SH for local models by applying general axiomatic properties rather than ad hoc analysis of holomorphic curves. These axiomatic properties include previously established ones such as the Mayer-Vietoris property and locality for complete embeddings; and new ones such as the Hartogs property and the holomorphic volume form preservation property of wall crossing in relative $SH$. We indicate some higher dimensional settings where the same techniques apply.