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

TL;DR

The paper investigates string propagation on stacks and gerbes, introducing noneffective orbifolds and the decomposition conjecture, which equates the CFTs of strings on gerbes to those on disjoint unions. It then applies these ideas to gauged linear sigma models, showing that Landau-Ginzburg points can realize branched double covers and, in some cases, noncommutative resolutions via Clifford-algebra–based sheaves, rather than being dictated by a superpotential. A key finding is that GLSM phases are connected by Kuznetsov's homological projective duality (HPD) rather than birational maps, challenging conventional lore about geometric phases. These results yield new geometric interpretations of GLSMs and broaden the interplay between string theory, algebraic geometry, and noncommutative geometry, with broad implications for GW theory and the geometric Langlands program.

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.'