Contact invariants of Q-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

TL;DR

This paper explores the relationship between contact invariants of Q-Gorenstein toric contact manifolds, Ehrhart polynomials, and Chen-Ruan cohomology, revealing deep connections between contact geometry, combinatorics, and orbifold cohomology.

Contribution

It establishes new links between cylindrical contact homology invariants and Ehrhart polynomials, Chen-Ruan cohomology, and toric geometry.

Findings

Contact invariants relate to Ehrhart polynomials of toric diagrams.

Cylindrical contact homology connects to Chen-Ruan cohomology of orbifold resolutions.

Results unify contact geometry with combinatorial and orbifold topological invariants.

Abstract

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to (i) the Ehrhart (quasi-)polynomial of its toric diagram; (ii) the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity; (iii) the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization.