Topological recursion relations in the symplectic field theory of mapping tori and local symplectic field theory

TL;DR

This paper establishes topological recursion relations in symplectic field theory for mapping tori and local SFT, enabling the reconstruction of descendant invariants from primary data, revealing richer geometric information.

Contribution

It proves topological recursion relations for the Hamiltonian in SFT of symplectic mapping tori and local SFT, and introduces a reconstruction theorem for descendants from primaries.

Findings

Descendant Hamiltonian cannot be derived solely from Hamiltonian without descendants.

Only the first descendant Hamiltonian, counting curves tangent to the fiber, is needed for reconstruction.

Descendant SFT invariants encode more geometric information than primary invariants.

Abstract

Based on the localization result for descendants in rational SFT moduli spaces from our last joint paper, we prove topological recursion relations for the Hamiltonian in SFT of symplectic mapping tori and in local SFT. Combined with the dilaton equation in SFT, we use them to prove a reconstruction theorem for descendants from primaries. While it turns out that (in contrast to Gromov-Witten theory and non-equivariant cylindrical contact homology) the descendant Hamiltonian cannot be computed from the Hamiltonian without descendants alone, we get that the only additional piece of information needed is the first descendant Hamiltonian, which counts holomorphic curves tangent to the symplectic fibre. As already known from the explicit computations in local SFT, it follows that the descendant SFT invariants in general contain more geometric information than the primary SFT invariants.