Contact homology of good toric contact manifolds

TL;DR

This paper establishes that good toric contact manifolds have well-defined cylindrical contact homology and provides a combinatorial method for computation, applying it to specific Sasaki-Einstein examples to discover new contact structures.

Contribution

It proves the existence of well-defined cylindrical contact homology for good toric contact manifolds and introduces a combinatorial approach for its calculation.

Findings

Computed cylindrical contact homology for specific Sasaki-Einstein examples.

Discovered a new infinite family of non-equivalent contact structures on S^2 x S^3.

Established the combinatorial method from the moment cone for contact homology calculation.

Abstract

In this paper we show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As an application we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett-Martelli-Sparks-Waldram on Sasaki-Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on $S^2 \times S^{3}$ in the unique homotopy class of almost contact structures with vanishing first Chern class.