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Quantum Geometric Langlands Categories from N = 4 Super Yang-Mills Theory

Chris Elliott, Philsang Yoo

Abstract

We describe the family of supersymmetric twists of $\mathcal N = 4$ super Yang--Mills theory using derived algebraic geometry, starting from holomorphic Chern--Simons theory on $ \mathcal N = 4$ super twistor space. By considering an ansatz for categorical geometric quantization of the family of further twists of a fixed holomorphic twist, we give a quantum field-theoretic synthesis of the categories of twisted D-modules occuring in the quantum geometric Langlands correspondence.

Quantum Geometric Langlands Categories from N = 4 Super Yang-Mills Theory

Abstract

We describe the family of supersymmetric twists of super Yang--Mills theory using derived algebraic geometry, starting from holomorphic Chern--Simons theory on super twistor space. By considering an ansatz for categorical geometric quantization of the family of further twists of a fixed holomorphic twist, we give a quantum field-theoretic synthesis of the categories of twisted D-modules occuring in the quantum geometric Langlands correspondence.

Paper Structure

This paper contains 23 sections, 23 theorems, 54 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Quantum Geometric Langlands
  3. Review of Previous Work
  4. Goals of this Paper
  5. Conventions
  6. N=4 Super Yang--Mills Theory
  7. Twisting Classical Field Theory
  8. N=4 Super Yang--Mills Theory and Twistors
  9. Description of the Supersymmetry Action
  10. Classification of Twists
  11. Twisting N=4 Theory
  12. Dimensional Reduction
  13. Holomorphic Twists
  14. Further Twists
  15. Rank (2,0) and (0,2) Twists
  16. ...and 8 more sections

Key Result

Theorem 1.6

The holomorphic twist of 4d $\mathcal{N}=4$ super Yang--Mills theory is defined on a complex algebraic surface $X$, and has the classical moduli space of solutions where $\mathop{\mathrm{Higgs}}\nolimits_G(X)$ is the stack of $G$-Higgs bundles on $X$; that is, of pairs $(P,\phi)$, where $P$ is an algebraic $G$-bundle and $\phi$ is a section of the sheaf $K_X \otimes \mathfrak{g}^*_P$ such that $[

Figures (1)

  • Figure 1: A schematic illustrating the relationship between the twists we have discussed in this paper. For each twist we give the rank of the relevant supercharge, the notation we have used to indicate the supercharge, the result in which a BV description for the twist is given, and then the differential on $\Omega^{\bullet,\bullet }(\mathbb{C}^2; \mathfrak{g}[\varepsilon][1])$ occurring in this description. The operator $\lambda \partial_\varepsilon$ is indicated by $\lambda$-dR. Deformations, or "further" twists, are indicated with dotted lines.

Theorems & Definitions (85)

  • Remark 1.1
  • Conjecture 1.2: Quantum Geometric Langlands Correspondence
  • Remark 1.3
  • Conjecture 1.4: Categorical Geometric Langlands Correspondence ArinkinGaitsgory
  • Remark 1.5
  • Theorem 1.6: EY1
  • Definition 1.7
  • Theorem 1.8: EY1
  • Remark 1.9
  • Remark 1.11
  • ...and 75 more