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Gauge Theory and Integrability, II

Kevin Costello, Edward Witten, Masahito Yamazaki

TL;DR

This work provides a gauge-theory–driven derivation of RTT presentations for Yangians, quantum loop algebras, and elliptic quantum groups to all orders in ħ, starting from a 4D gauge theory on Σ×C and employing Wilson-line crossings to generate R-matrices. It systematically extends the RTT framework from the rational case for gl_N to so_N, sp_{2N}, sl_N, and exceptional algebras (g2, f4, e6, e7), introducing necessary extra relations (quantum determinants) and addressing boundary and framing anomalies. The authors show that for all simple algebras except e8, the R-matrix is uniquely fixed by Yang-Baxter, unitarity, and RTT-type constraints, with gl_N containing an intrinsic ambiguity linked to the U(1) factor. In the trigonometric and elliptic regimes they map gauge-theory parameters to deformation data, relate to quantum loop algebras and elliptic quantum groups, and connect these results to 3D Chern-Simons theory, offering a unified, geometry-grounded route to quantum-group structures relevant for integrable systems.

Abstract

Starting with a four-dimensional gauge theory approach to rational, elliptic, and trigonometric solutions of the Yang-Baxter equation, we determine the corresponding quantum group deformations to all orders in $\hbar$ by deducing their RTT presentations. The arguments we give are a mix of familiar ones with reasoning that is more transparent from the four-dimensional gauge theory point of view. The arguments apply most directly for $\mathfrak{gl}_N$ and can be extended to all simple Lie algebras other than $\mathfrak{e}_8$ by taking into account the self-duality of some representations, the framing anomaly for Wilson operators, and the existence of quantum vertices at which several Wilson operators can end.

Gauge Theory and Integrability, II

TL;DR

This work provides a gauge-theory–driven derivation of RTT presentations for Yangians, quantum loop algebras, and elliptic quantum groups to all orders in ħ, starting from a 4D gauge theory on Σ×C and employing Wilson-line crossings to generate R-matrices. It systematically extends the RTT framework from the rational case for gl_N to so_N, sp_{2N}, sl_N, and exceptional algebras (g2, f4, e6, e7), introducing necessary extra relations (quantum determinants) and addressing boundary and framing anomalies. The authors show that for all simple algebras except e8, the R-matrix is uniquely fixed by Yang-Baxter, unitarity, and RTT-type constraints, with gl_N containing an intrinsic ambiguity linked to the U(1) factor. In the trigonometric and elliptic regimes they map gauge-theory parameters to deformation data, relate to quantum loop algebras and elliptic quantum groups, and connect these results to 3D Chern-Simons theory, offering a unified, geometry-grounded route to quantum-group structures relevant for integrable systems.

Abstract

Starting with a four-dimensional gauge theory approach to rational, elliptic, and trigonometric solutions of the Yang-Baxter equation, we determine the corresponding quantum group deformations to all orders in by deducing their RTT presentations. The arguments we give are a mix of familiar ones with reasoning that is more transparent from the four-dimensional gauge theory point of view. The arguments apply most directly for and can be extended to all simple Lie algebras other than by taking into account the self-duality of some representations, the framing anomaly for Wilson operators, and the existence of quantum vertices at which several Wilson operators can end.

Paper Structure

This paper contains 29 sections, 2 theorems, 123 equations, 12 figures.

Table of Contents

  1. Introduction
  2. RTT Relation for Y(gl(n))
  3. Basics
  4. RTT relation in terms of an expansion of T(z)
  5. Coproduct
  6. A Different Set of Generators
  7. RTT Relation for Y(so(N)) and Y(sp(2N))
  8. RTT Relation for Y(sl(N))
  9. RTT Presentation for Y(g(2)), Y(f(4)), Y(e(6)) and Y(e(7))
  10. Y(g(2))
  11. Y(f(4)) and Y(e(6))
  12. Y(e(7))
  13. Uniqueness of the Rational R-matrix
  14. Overview
  15. The failure of uniqueness of the R-matrix for gl(N)
  16. ...and 14 more sections

Key Result

Proposition 6.1

Fix a simple Lie algebra $\mathfrak{g}$ which is not $\mathfrak{e}_8$, and let $V$ denote its smallest non-trivial representation. Then, there is a unique $R$-matrix which is a series in $\hbar$ whose coefficients are rational functions of $z$, and which has the following properties.

Figures (12)

  • Figure 1: A vertical Wilson line, equipped with an incoming state $\langle i |$ and an outgoing state $|j \rangle$, gives rise to an operator acting on the states of a horizontal Wilson line.
  • Figure 2: Two parallel vertical Wilson lines crossing a horizontal one lead to a composition of the corresponding operators.
  • Figure 3: Two vertical Wilson lines are "bent" to cross each other, above or below a given horizontal Wilson line. (To avoid extraneous considerations involving a framing anomaly, one can extend the "vertical" Wilson lines in this and subsequent pictures so that they are indeed asymptotically vertical.)
  • Figure 4: The first picture in Fig. \ref{['figure_bent']} can be evaluated as shown here.
  • Figure 5: The second picture in Fig. \ref{['figure_bent']} has this interpretation.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Proposition 6.1
  • proof
  • Definition 7.1
  • Proposition 8.1
  • proof
  • Definition 9.1