Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Seiberg-like duality for resolutions of determinantal varieties

Nathan Priddis, Mark Shoemaker, Yaoxiong Wen

Abstract

We study the genus-zero Gromov-Witten theory of two natural resolutions of determinantal varieties, termed the PAX and PAXY models. We realize each resolution as lying in a quiver bundle, and show that the respective quiver bundles are related by a quiver mutation. We prove that generating functions of genus-zero Gromov-Witten invariants for the two resolutions are related by a specific cluster change of variables. Along the way, we obtain a quantum Thom-Porteous formula for determinantal varieties and prove a Seiberg-like duality statement for certain quiver bundles.

Seiberg-like duality for resolutions of determinantal varieties

Abstract

We study the genus-zero Gromov-Witten theory of two natural resolutions of determinantal varieties, termed the PAX and PAXY models. We realize each resolution as lying in a quiver bundle, and show that the respective quiver bundles are related by a quiver mutation. We prove that generating functions of genus-zero Gromov-Witten invariants for the two resolutions are related by a specific cluster change of variables. Along the way, we obtain a quantum Thom-Porteous formula for determinantal varieties and prove a Seiberg-like duality statement for certain quiver bundles.
Paper Structure (14 sections, 23 theorems, 130 equations)

This paper contains 14 sections, 23 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. The connection to quiver bundles
  3. Gromov--Witten theory of determinantal varieties
  4. Seiberg-like duality for the PAX and PAXY resolution
  5. Seiberg-like duality for Grassmannian bundles
  6. Quivers and mutations
  7. The basic relative quiver mutation
  8. The quasimap $I$-functions
  9. The proof of basic Seiberg-like duality
  10. PAX/PAXY correspondence
  11. Determinantal varieties and resolutions
  12. PAX/PAXY models as a relative quiver mutation
  13. A quantum Thom--Porteous formula
  14. The proof of Theorem \ref{['Thm.PAX/PAXY']}

Key Result

Theorem 1.1

The PAX and PAXY models can each be constructed as critical loci of specified superpotentials on fiber bundles over $B$ with fibers given by quiver varieties. The respective quiver bundles are related by a mutation.

Theorems & Definitions (49)

  • Theorem 1.1: Theorem \ref{['thm.pax/paxy_mutation']}
  • Theorem 1.2: Theorem \ref{['Thm.PAX/PAXY']}
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • ...and 39 more