Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Supersymmetric gauge theory and the Yangian

Kevin Costello

TL;DR

This work builds a deep link between a holomorphically twisted, deformed 4D N=1 gauge theory and the Yangian Hopf algebra. The central result shows that the operator algebra of Wilson lines in the twisted theory is governed by the Yangian, with circle observables corresponding to finite-dimensional Yangian representations and their tensor products matching the transfer-matrix structure of integrable spin chains. Through the lens of factorization algebras and Koszul duality, the $E_2$-algebra of local observables is dual to the Yangian, and the $z$-direction OPE is encoded by the universal R-matrix, enabling a categorical and Hochschild-homological understanding of surface operators. The paper also demonstrates a precise equivalence between Wilson-operator correlators on a doubly periodic lattice and spin-chain partition functions, and it situates these results within an extended TFT framework, including higher-categorical interpretations of surface operators. Collectively, the work provides a perturbative, rigorous path to exact results in a twisted N=1 theory and reveals a broad, structural bridge between holomorphic field theories and quantum integrable systems.

Abstract

This paper develops a new connection between supersymmetric gauge theories and the Yangian. I show that a twisted, deformed version of the pure N=1 supersymmetric gauge theory is controlled by the Yangian, in the same way that Chern-Simons theory is controlled by the quantum group. This result is used to give an exact calculation, in perturbation theory, of the expectation value of a certain net of n+m Wilson operators in the deformed N=1 gauge theory. This expectation value coincides with the partition function of a spin-chain integrable lattice model on an n-by-m doubly-periodic lattice.

Supersymmetric gauge theory and the Yangian

TL;DR

This work builds a deep link between a holomorphically twisted, deformed 4D N=1 gauge theory and the Yangian Hopf algebra. The central result shows that the operator algebra of Wilson lines in the twisted theory is governed by the Yangian, with circle observables corresponding to finite-dimensional Yangian representations and their tensor products matching the transfer-matrix structure of integrable spin chains. Through the lens of factorization algebras and Koszul duality, the -algebra of local observables is dual to the Yangian, and the -direction OPE is encoded by the universal R-matrix, enabling a categorical and Hochschild-homological understanding of surface operators. The paper also demonstrates a precise equivalence between Wilson-operator correlators on a doubly periodic lattice and spin-chain partition functions, and it situates these results within an extended TFT framework, including higher-categorical interpretations of surface operators. Collectively, the work provides a perturbative, rigorous path to exact results in a twisted N=1 theory and reveals a broad, structural bridge between holomorphic field theories and quantum integrable systems.

Abstract

This paper develops a new connection between supersymmetric gauge theories and the Yangian. I show that a twisted, deformed version of the pure N=1 supersymmetric gauge theory is controlled by the Yangian, in the same way that Chern-Simons theory is controlled by the quantum group. This result is used to give an exact calculation, in perturbation theory, of the expectation value of a certain net of n+m Wilson operators in the deformed N=1 gauge theory. This expectation value coincides with the partition function of a spin-chain integrable lattice model on an n-by-m doubly-periodic lattice.

Paper Structure

This paper contains 38 sections, 83 theorems, 507 equations.

Table of Contents

  1. Representations of the Yangian and Wilson operators
  2. Wilson loops and lattice models
  3. The Hilbert space and the transfer matrix.
  4. Integrability in two dimensions
  5. Factorization algebras
  6. The $R$-matrix and the operator product in the $z$-direction
  7. Higher-categorical interpretation of surface operators
  8. Sketch of the proof of the relationship between lattice models and Wilson operators
  9. Extended TFTs and the Hilbert space
  10. Generalizations
  11. Mathematical results
  12. Acknowledgments
  13. Physical $N=1$ super-symmetric gauge theory
  14. The $N=1$ supertranslation Lie algebra
  15. The $N=1$ theory in the BV formalism
  16. ...and 23 more sections

Key Result

Theorem 1

The twisted, deformed $N=1$ gauge theory admits a unique perturbative quantization (compatible with certain symmetries) on any complex surface $X$ equipped with a holomorphic volume form and a closed holomorphic $1$-form.

Theorems & Definitions (202)

  • Remark
  • Theorem
  • Theorem
  • Corollary
  • proof
  • Theorem
  • Remark
  • Lemma
  • Theorem
  • Theorem
  • ...and 192 more