Gromov-Witten theory of root Gerbes I: Structure of genus 0 moduli spaces

Abstract

Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r > 1 one defines the stack r L/X of r-th roots of L. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus 0 twisted stable maps to r L/X. Our main results are explicit constructions of moduli stacks of genus 0 twisted stable maps to r L/X starting from moduli stack of genus 0 stable maps to X. As a consequence, we prove an exact formula expressing genus 0 Gromov-Witten invariants of r L/X in terms of those of X.Date: November 1, 2018. 1 2 ELENA ANDREINI, YUNFENG JIANG, AND HSIAN-HUA TSENG References 36The Gromov-Witten theoretic version of the decomposition conjecture, which can be formulated for arbitrary G-gerbes more general than G-banded gerbes, states that Gromov-Witten theory of the G-gerbe is equivalent to certain twist of the Gromov-Witten theory of someétale cover of the base. A detailed discussion of the conjecture in full generality can be found in [38]. For G-banded gerbes this conjecture states that the Gromov-Witten theory of a G-banded gerbe over X is equivalent to (certain twists of) the Gromov-Witten theory of the disjoint union of |Conj(G)| copies 1 of X after a change of variables. Computations of Gromov-Witten invariants ofétale gerbes is thus intimated connected to the decomposition conjecture.The simplest examples of G-gerbes are trivial gerbes. The trivial G-gerbe over a Deligne-Mumford stack X is the product X × BG. In [7] the computation of Gromov-Witten invariants of X × BG is handled as a special case of a general product formula for orbifold Gromov-Witten invariants of product Deligne-Mumford stacks X × Y. As a consequence the decomposition conjecture is proven for trivial G-gerbes.An interesting class of non-trivial gerbes is provided by root gerbes associated to line bundles. This is the first of two papers in which we study Gromov-Witten theory of root gerbes of line bundles over smooth projective varieties, with the decomposition conjecture in mind. The present paper is devoted to study the genus 0 Gromov-Witten theory of root gerbes.Let X be a smooth complex projective variety and L → X a line bundle. Given an integer r > 0, let G := r L/X → X 1 Here Conj(G) is the set of conjugacy classes of G.

Notes on generalized global symmetries in QFT

“…This decomposition makes predictions for Gromov‐Witten invariants, which have been checked rigorously (see e.g . ), and also plays a role in understanding phases of certain GLSMs (see for a more complete list of references and reviews). It would be interesting to understand if there are analogues of decomposition for any notion of gauge theories in some dimension.…”

Categorical Equivalence and the Renormalization Group

“…This has since been proven in work of H.-H. Tseng, Y. Jiang, and collaborators in e.g. [28][29][30][31][32][33]. ii) Phases of gauged linear sigma models (GLSMs).…”

“…Tseng, Y. Jiang, and collaborators in e.g. []. ii)Phases of gauged linear sigma models (GLSMs). Phases of certain GLSMS, which were previously obscure, now have a solid understanding utilizing decomposition.…”

Mirror theorems for root stacks and relative pairs

“…First, recall that IX r = ⊔ r−1 j=0 X. Let ι j ∶ X → IX r be the isomorphism of X with the component of age j r. By [6], the small J-function of X and the one of X r has a simple relation.…”