Non-birational twisted derived equivalences in abelian GLSMs

TL;DR

The paper demonstrates that abelian GLSMs can realize twisted derived equivalences between non-birational geometries and proposes Kuznetsov's homological projective duality as the unifying mechanism linking distinct Kähler phases. By analyzing detailed quadrics-based examples, it shows that Landau-Ginzburg points often correspond to noncommutative resolutions rather than geometric branched covers, with matrix factorizations and Clifford-algebra structures encoding the nc data. Monodromy computations confirm CY-like behavior at LG points in Calabi–Yau cases, while non-Calabi–Yau instances reveal a rich tapestry of gerbes, orbifolds, and HPD-dual nc spaces. The work generalizes these insights to higher dimensions and more general complete intersections, suggesting a broad framework in which GLSM phases are governed by HPD and nc geometry, with many new CFTs awaiting exploration. Overall, the paper strengthens the bridge between GLSM phase structure and advanced algebraic geometry, offering a concrete physical realization of noncommutative spaces via D-brane categories and matrix factorizations.

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.