Sigma Models and Phase Transitions for Complete Intersections

TL;DR

This paper explores the relationship between different phases of gauged linear sigma models associated with complete intersections, establishing a genus-zero correspondence that generalizes known dualities and extends the Landau-Ginzburg/Calabi-Yau correspondence.

Contribution

It develops foundational properties of GLSMs in the negative phase and proves a genus-zero comparison theorem, extending dualities to complete intersections.

Findings

Established a genus-zero comparison theorem for GLSMs.

Proved a correspondence between phases generalizing Landau-Ginzburg/Calabi-Yau duality.

Connected Gromov-Witten theory and Landau-Ginzburg models through phase analysis.

Abstract

We study a one-parameter family of gauged linear sigma models (GLSMs) naturally associated to a complete intersection in weighted projective space. In the positive phase of the family we recover Gromov-Witten theory of the complete intersection, while in the negative phase we obtain a Landau--Ginzburg-type theory. Focusing on the negative phase, we develop foundational properties which allow us to state and prove a genus-zero comparison theorem that generalizes the multiple log-canonical correspondence and should be viewed as analogous to quantum Serre duality in the positive phase. Using this comparison result, along with the crepant transformation conjecture and quantum Serre duality, we prove a genus-zero correspondence between the GLSMs which arise at the two phases, thereby generalizing the Landau-Ginzburg/Calabi-Yau correspondence to complete intersections.