Dualities and Discretizations of Integrable Quantum Field Theories from 4d Chern-Simons Theory

TL;DR

The paper develops a comprehensive framework linking continuum 2d integrable field theories and integrable lattice models through 4d Chern-Simons theory with coupled 2d surface defects that discretize into 1d Wilson lines. It shows that discretization generically maps diverse defects to Wilson lines, creating a robust lattice description that recovers known and new integrable structures, and it analyzes anomaly inflow as a consistency condition for quantum integrability. A web of dualities among defects is uncovered, enabling infinite families of non-Abelian bosonizations between massless Thirring-type theories and coupled WZW models, with precise 1d-2d correspondences and a string-theoretic embedding via D-brane polarization to realize the thermodynamic limit. The work further connects the algebraic underpinnings of defects, such as Zhu’s algebra and twisted differential operators, to lattice discretizations, and outlines a broad program for extending integrable discretizations to curved β-γ systems and VA defects. Overall, the approach unifies discrete and continuous integrable systems within a single 4d CS framework and suggests deep links to string theory and representation theory, with potential for systematic quantization of integrable QFTs.

Abstract

We elucidate the relationship between 2d integrable field theories and 2d integrable lattice models, in the framework of the 4d Chern-Simons theory. The 2d integrable field theory is realized by coupling the 4d theory to multiple 2d surface order defects, each of which is then discretized into 1d defects. We find that the resulting defects can be dualized into Wilson lines, so that the lattice of discretized defects realizes integrable lattice models. Our discretization procedure works systematically for a broad class of integrable models (including trigonometric and elliptic models), and uncovers a rich web of new dualities among integrable field theories. We also study the anomaly-inflow mechanism for the integrable models, which is required for the quantum integrability of field theories. By analyzing the anomalies of chiral defects, we derive a new set of bosonization dualities between generalizations of massless Thirring models and coupled Wess-Zumino-Witten (WZW) models. We study an embedding of our setup into string theory, where the thermodynamic limit of the lattice models is realized by polarizations of D-branes.

Figures (24)

• Figure 1: From from the perspectives of the 4d CS theory, the lattice discretization of the integrable field theory is understood as a discretization of 2d surface defects into 1d defects. When we reduce along the spectral curve $C$ we obtain the more traditional discretization of integrable field theories into integrable lattice models.
• Figure 2: Order surface defects located at various points on $C$. Black dots represent chiral defects while red dots represent anti-chiral defects.
• Figure 3: Tree-level gluon exchange between crossed Wilson lines
• Figure 4: The Yang-Baxter equation
• Figure 5: The lightcone lattice where the discrete modes of the currents are supported along the null segments.