GLSM's, gerbes, and Kuznetsov's homological projective duality

TL;DR

This paper explores string propagation on stacks, particularly gerbes, and applies the decomposition conjecture to gauged linear sigma models, revealing new geometric interpretations of Landau-Ginzburg points and connections to Kuznetsov's homological projective duality.

Contribution

It provides a novel application of the decomposition conjecture to GLSMs, linking phases to homological projective duality and offering geometric insights into nonperturbative effects.

Findings

Landau-Ginzburg points correspond to noncommutative resolutions of branched double covers.

Phases of GLSMs are related by homological projective duality, not just birational transformations.

String propagation on gerbes can be understood via the decomposition conjecture.

Abstract

In this short note we give an overview of recent work on string propagation on stacks and applications to gauged linear sigma models. We begin by outlining noneffective orbifolds (orbifolds in which a subgroup acts trivially) and related phenomena in two-dimensional gauge theories, which realize string propagation on gerbes. We then discuss the `decomposition conjecture,' equating conformal field theories of strings on gerbes and strings on disjoint unions of spaces. Finally, we apply these ideas to gauged linear sigma models for complete intersections of quadrics, and use the decomposition conjecture to show that the Landau-Ginzburg points of those models have a geometric interpretation in terms of a (sometimes noncommutative resolution of a) branched double cover, realized via nonperturbative effects, rather than as the vanishing locus of a superpotential. These examples violate old unproven lore on GLSM's (e.g., that geometric phases must be related by birational transformations), and we conclude by observing that in these examples (and conjecturing more generally in GLSM's), the phases are instead related by Kuznetsov's `homological projective duality.'