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Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems

Ben Webster

Abstract

In this paper, we consider how the approach of Bezrukavnikov and Kaledin to understanding the categories of coherent sheaves on symplectic resolutions can be applied to the Coulomb branches introduced by Braverman, Finkelberg and Nakajima. In particular, we construct tilting generators on resolved Coulomb branches, and give explicit quiver presentations of categories of coherent sheaves on these varieties, with the wall-crossing functors described by natural bimodules.

Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems

Abstract

In this paper, we consider how the approach of Bezrukavnikov and Kaledin to understanding the categories of coherent sheaves on symplectic resolutions can be applied to the Coulomb branches introduced by Braverman, Finkelberg and Nakajima. In particular, we construct tilting generators on resolved Coulomb branches, and give explicit quiver presentations of categories of coherent sheaves on these varieties, with the wall-crossing functors described by natural bimodules.

Paper Structure

This paper contains 23 sections, 47 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivationfromlinedefects
  3. Summaryofapproach
  4. QuantumCoulombbranches
  5. Background
  6. Theextendedcategory
  7. Representations
  8. Thehomogeneouspresentation
  9. Consequencesforrepresentationtheory
  10. Relationtogeometry
  11. Frobeniusconstantquantization
  12. Homogeneouscoordinaterings
  13. Infinitesimalsplittings
  14. Liftingtocharacteristic0
  15. Derivedlocalization
  16. ...and 8 more sections

Key Result

Theorem A

AnyBFNCoulombbranchwithaBFNresolutionhasanexplicit combinatoriallypresentednoncommutativeresolutionofsingularities $A$.Thecategory$D^b(A\operatorname{-mod})$isequivalenttothederivedcategoryofcoherentsheavesonanyBFNresolutionthroughanexplicittiltinggenerator.

Figures (2)

  • Figure 1: Apointofintersectionforweightandroothyperplanes
  • Figure 2: Theeffectof$p$throotconventionsonmatterhyperplanes

Theorems & Definitions (96)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.6
  • Theorem 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 86 more