Gromov-Witten theory of banded gerbes over schemes

Abstract

Let X be a smooth complex projective algebraic variety. Let G be a Gbanded gerbe with G a finite abelian group. We prove an exact formula expressing genus g orbifold Gromov-Witten invariants of G in terms of those of X. Contents 1. Introduction Conventions Acknowledgments 2. Orbifold Gromov-Witten theory 2.1. Twisted stable maps 2.2. Virtual Fundamental Class 2.3. Gromov-Witten invariants and Descendant Potential 3. Twisted curves 3.1. The Picard group of prestable curves 3.2. The Picard group of twisted curves 3.3. The Brauer group 4. Rephrasing the moduli problem 5. The structure morphism p 6. Push-forward formula 6.1. Comparison of Obstruction Theories 6.2. Proof of the pushforward formula 7. Orbifold Gromov-Witten theory of banded gerbes 7.1. Orbifold Gromov-Witten invariants 7.2. Decomposition of Gromov Witten theory Appendix A. A few useful facts about degree of morphisms References

On orbifold Gromov–Witten theory in codimension one

“…The proof of this Lemma in our setting is identical to the one in[12] 4. See e.g [5,6]. for discussions on root gerbes.…”

“…The orbifold quantum Riemann-Roch formula [16] applies to express these Hurwitz-Hodge integrals in terms of Gromov-Witten invariants of D r . By the main results of [5,6], Gromov-Witten invariants of the µ r -gerbe D r over D are expressed in terms of Gromov-Witten invariants of D.…”

On the geometry of orbifold Gromov-Witten invariants

“…Upon first look, Conjecture 3.1 seems strange and unmotivated. It is thus surprising that Conjecture 3.1 turns out to be correct in all examples studied so far: it is proven for specific classes of examples in [27], [8,9,10], and [37,38]. In [38] Conjecture 3.1 is proven for G-gerbes Y → B whose bands Y → B are trivializable principal bundles.…”

Costello's pushforward formula: errata and generalization

“…The papers [Vak06], [JKK03], [Cav08], [Wis11] reference but don't use Costello's Formula. The paper [Rap10] uses other results from Costello's paper and not his formula, while [AJT11] uses it only for motivation.…”