New non-commutative resolutions of determinantal Calabi-Yau threefolds from hybrid GLSM

TL;DR

This paper explores non-commutative resolutions of singular Calabi-Yau threefolds, constructing hybrid GLSMs, analyzing their topological string properties, and computing refined invariants, revealing new geometric and physical insights.

Contribution

It introduces new non-commutative resolutions of determinantal Calabi-Yau threefolds using hybrid GLSMs and computes associated topological string invariants.

Findings

Small resolutions have 2-torsional exceptional curves and are non-Kähler.

M-theory on these spaces exhibits a $ ext{Z}_2$ gauge symmetry.

Calculated $ ext{Z}_2$-refined Gopakumar-Vafa invariants for examples.

Abstract

We study topological strings on non-commutative resolutions of singular Calabi-Yau threefolds that are double covers of $\mathbb{P}^3$, ramified over determinantal octic surfaces. Using conifold transitions to complete intersections in toric ambient spaces, we prove that any small resolution has 2-torsional exceptional curves and is necessarily non-K\"ahler. The same transitions imply that M-theory develops a $\mathbb{Z}_2$ gauge symmetry on the singular space. We then construct gauged linear sigma models with hybrid phases that flow to the worldsheet theories of strings propagating on the determinantal double solids in the presence of a flat but topologically non-trivial B-field. Localizing the sphere partition function allows us to calculate the fundamental periods of the mirror Calabi-Yau manifolds, then we check agreement with the periods of the Borisov-Li mirrors. We find that the corresponding variations of Hodge structure either correspond to one of the 14 hypergeometric cases or to a double cover thereof. We then use mirror symmetry and integrate the holomorphic anomaly equations to calculate $\mathbb{Z}_2$-refined Gopakumar-Vafa invariants for several examples.