On Gromov-Witten theory of root gerbes

TL;DR

This paper calculates genus 0 Gromov-Witten invariants of root gerbes over smooth schemes, confirming the decomposition conjecture relating gerbe and base theories, and extends verification to all genera for toric cases.

Contribution

It provides a complete genus 0 Gromov-Witten theory calculation for root gerbes and verifies the decomposition conjecture in genus 0 and all genera for toric gerbes.

Findings

Confirmed genus 0 decomposition conjecture for root gerbes.

Verified the conjecture in all genera for toric gerbes.

Provided explicit calculations of Gromov-Witten invariants for root gerbes.

Abstract

This research announcement discusses our results on Gromov-Witten theory of root gerbes. A complete calculation of genus 0 Gromov-Witten theory of $\mu_{r}$-root gerbes over a smooth base scheme is obtained by a direct analysis of virtual fundamental classes. Our result verifies the genus 0 part of the so-called decomposition conjecture which compares Gromov-Witten theory of \'etale gerbes with that of the bases. We also verify this conjecture in all genera for toric gerbes over toric Deligne-Mumford stacks.