Gravitational descendants in symplectic field theory

TL;DR

This paper extends the concept of gravitational descendants in symplectic field theory to general contact manifolds and explores their geometric and algebraic properties, linking them to integrable systems and Poisson-commuting functions.

Contribution

It generalizes the definition of gravitational descendants in SFT beyond circle bundles to all contact manifolds and provides geometric interpretations and explicit computations.

Findings

Gravitational descendants relate to branching conditions in holomorphic curves.

Sequences of Poisson-commuting functions are computed for unit cotangent bundles.

The algebraic formalism connects to quantum and classical integrable systems.

Abstract

It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of gravitational descendants in SFT from circle bundles in the Morse-Bott case to general contact manifolds. After we have shown that for the basic examples of holomorphic curves in SFT, that is, branched covers of cylinders over closed Reeb orbits, the gravitational descendants have a geometric interpretation in terms of branching conditions, we compute the corresponding sequences of Poisson-commuting functions when the contact manifold is the unit cotangent bundle of a Riemannian manifold.