The Chen-Ruan cohomology of moduli of curves of genus 2 with marked points

TL;DR

This paper computes the Chen-Ruan cohomology and stringy Chow ring of the moduli stacks of genus 2 pointed curves, providing explicit additive structures and algebra generators.

Contribution

It offers the first detailed description of the Chen-Ruan cohomology for genus 2 moduli stacks, including algebraic generators for the even cohomology.

Findings

Computed the additive structure of Chen-Ruan cohomology for genus 2 curves

Described algebra generators for the even Chen-Ruan cohomology ring

Established the algebraic correspondence with the stringy Chow ring

Abstract

In this work we describe the Chen-Ruan cohomology of the moduli stacks of smooth and stable genus 2 pointed curves, and its algebraic counterpart: the stringy Chow ring. In the first half of the paper we compute the additive structure of the Chen-Ruan cohomology ring for the moduli stack of stable pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen--Ruan cohomology ring as an algebra on the ordinary cohomology.