Laurent polynomial mirrors for quiver flag zero loci

TL;DR

This paper develops a method to find Laurent polynomial mirrors for Fano quiver flag zero loci, extending mirror symmetry techniques to higher-dimensional varieties and providing conjectural mirrors for numerous cases.

Contribution

It generalizes Gelfand--Cetlin degenerations to Y-shaped quiver flag varieties and constructs conjectural Laurent polynomial mirrors for 99 four-dimensional Fano quiver flag zero loci.

Findings

Constructed conjectural mirrors for 99 Fano quiver flag zero loci.

Extended Gelfand--Cetlin degenerations to Y-shaped quiver flag varieties.

Validated mirrors by matching period sequences up to 20 terms.

Abstract

The classification of Fano varieties is an important open question, motivated in part by the MMP. Smooth Fano varieties have been classified up to dimension three: one interesting feature of this classification is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). There is a program to use mirror symmetry to classify Fano varieties in higher dimensions. Fano varieties are expected to correspond to certain Laurent polynomials under mirror symmetry; given such a Fano toric complete intersections, one can produce a Laurent polynomial via the Hori--Vafa mirror. In this paper, we give a method to find Laurent polynomial mirrors to Fano quiver flag zero loci in $Y$-shaped quiver flag varieties. To do this, we generalise the Gelfand--Cetlin degeneration of type A flag varieties to Fano $Y$-shaped quiver flag varieties, and give a new description of these degenerations as canonical toric quiver varieties. We find conjectural mirrors to 99 four dimensional Fano quiver flag zero loci, and check them up to 20 terms of the period sequence.