[Paper Review] Integrability As Duality: The Gauge/YBE Correspondence

Masahito Yamazaki|arXiv (Cornell University)|Aug 13, 2018

Black Holes and Theoretical Physics101 references3 citations

TL;DR

This paper establishes a duality framework linking integrable models via the Yang-Baxter equation (YBE) to supersymmetric quiver gauge theories, showing that solutions to the YBE arise from gauge theory dualities—specifically Seiberg-like dualities—thereby explaining the existence of integrable models through quantum field theory principles. The key contribution is the Gauge/YBE correspondence, which maps R-matrices to gauge theory partition functions and spectral parameters to R-charges.

ABSTRACT

The Gauge/YBE correspondence states a surprising connection between solutions to the Yang-Baxter equation with spectral parameters and partition functions of supersymmetric quiver gauge theories. This correspondence has lead to systematic discoveries of new integrable models based on quantum-field-theory methods. We provide pedagogical introduction to the subject and summarizes many recent developments. This is a write-up of the lecture at the String-Math 2018 conference.

Motivation & Objective

• To explain why integrable models exist despite the over-constrained nature of the Yang-Baxter equation (YBE) with spectral parameters.
• To establish a systematic correspondence between solutions of the YBE and partition functions of 3D/4D supersymmetric quiver gauge theories.
• To demonstrate that integrability arises not from isolated models but from dualities in a theory space of quiver gauge theories.
• To clarify the physical origin of spectral parameters in integrable models by identifying them with R-charges in supersymmetric gauge theories.
• To provide a unified framework for discovering new integrable models using quantum field theory techniques, particularly Seiberg duality and supersymmetric localization.

Proposed method

• Utilizes supersymmetric localization to compute exact partition functions of 3D/4D quiver gauge theories, which encode statistical mechanical models.
• Constructs R-matrices from gauge theory building blocks (e.g., chiral multiplets, gauge multiplets) via the theory T[R] associated with each R-matrix.
• Models the gluing of R-matrices via gauging of global symmetries in the quiver diagram, corresponding to the composition of R-matrices in the YBE.
• Identifies spectral parameters in the YBE with R-charges of chiral multiplets in the gauge theory, linking algebraic structures to physical quantum numbers.
• Applies Seiberg-like duality in gauge theories to derive the Yang-Baxter equation as a duality relation between different quiver configurations.
• Uses zig-zag paths in the quiver diagram to track R-charges and derive the rapidity difference property of R-matrices.

Experimental results

Research questions

• RQ1 Why do integrable models exist despite the over-constrained nature of the Yang-Baxter equation with spectral parameters?
• RQ2 What is the physical origin of the spectral parameter in vertex and IRF models?
• RQ3 How can new integrable models be systematically constructed using quantum field theory methods?
• RQ4 What is the role of gauge theory dualities in generating solutions to the Yang-Baxter equation?
• RQ5 How are R-charges in supersymmetric gauge theories related to the spectral parameters in integrable models?

Key findings

• The Gauge/YBE correspondence provides a physical explanation for the existence of integrable models: they arise from Seiberg-like dualities in supersymmetric quiver gauge theories.
• Spectral parameters in the YBE are identified as R-charges of chiral multiplets in the gauge theory, with the rapidity difference property emerging from the R-charge structure.
• The star-triangle relation in the SU(2) case is derived from Seiberg duality in a 3D N=2 quiver gauge theory with fundamental and adjoint matter.
• New integrable models are systematically constructed via the super-master solution, which generalizes the Bazhanov-Sergeev master solution using gauge theory techniques.
• The correspondence maps the Yang-Baxter equation to a duality between different gauge theory configurations, with the YBE being equivalent to Yang-Baxter duality.
• Quantum parameters such as q and p in the YBE are interpreted as complex structure moduli of the underlying Riemann surface, making them continuous parameters from the outset.