Non-birational twisted derived equivalences in abelian GLSMs

TL;DR

This paper explores abelian gauged linear sigma models that realize twisted derived equivalences between non-birational spaces, revealing new geometric and physical phenomena, including noncommutative spaces and the role of stacks.

Contribution

It introduces examples of non-birational Kahler phases in GLSMs, connecting them to Kuznetsov's homological projective duality and expanding understanding of noncommutative spaces in physics.

Findings

Gauged linear sigma models can realize non-birational Kahler phases.

Noncommutative spaces are physically realized in GLSMs.

Stacks are more prevalent in GLSMs than previously understood.

Abstract

In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with non-birational Kahler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kahler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov's `homological projective duality.' Along the way, we shall see how `noncommutative spaces' (in Kontsevich's sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.