Gauge Theory And Integrability, III

Abstract

We study two-dimensional integrable field theories from the viewpoint of the four-dimensional Chern-Simons-type gauge theory introduced recently. The integrable field theories are realized as effective theories for the four-dimensional theory coupled with two-dimensional surface defects, and we can systematically compute their Lagrangians and the Lax operators satisfying the zero-curvature condition. Our construction includes many known integrable field theories, such as Gross-Neveu models, principal chiral models with Wess-Zumino terms and symmetric-space coset sigma models. Moreover we obtain various generalization these models in a number of different directions, such as trigonometric/elliptic deformations, multi-defect generalizations and models associated with higher-genus spectral curves, many of which seem to be new.

Figures (19)

• Figure 1: One expects that the thermodynamic limit of an integrable lattice model gives rise an integrable field theory. In the language of four-dimensional Chern-Simons theory, this is the limit where an infinite parallel Wilson lines, in both vertical and horizontal Wilson lines, fill out the two-dimensional plane, thereby becoming a surface defect. While we do not directly take advantage of this mental picture, this is of help in understanding the relation between the present paper and the part I, II of the series Costello:2017dsoCostello:2018gyb. Note that vertical and horizontal Wilson lines in this Figure are placed at the same point $z$ and $z'$ of the spectral curve. We can more generally inhomogeneous lattice models and Wilson lines at more than two points of the spectral curve, which will lead to the setup with multiple surface defects.
• Figure 2: A tree-level diagram contributing to the effective action. Since the propagator has a power of $\hbar$, and since each of the two vertices contributes a factor of $\hbar^{-1}$, this Feynman diagram contributes with the power of $\hbar^{1-2}=\hbar^{-1}$, and hence should be included in the computation of the effective two-dimensional action.
• Figure 3: The Feynman diagram computation of Figure \ref{['fig:gluon_exchange']} is the same as in the tree-level Feynman diagram computation as in this figure, which gives the first tree-level contribution to the $R$-matrix, namely the computes the classical $r$-matrix.
• Figure 4: The tree-level diagram in Figure \ref{['fig:gluon_exchange']} is the only non-zero Feynman diagram. For example, the tree-level diagram in this Figure is zero, since the only vertex in the Chern-Simons theory involves all the three components $A_w, A_{\overline{w}}, A_{\overline{z}}$ of the gauge field, while in our gauge the propagator does not involve the $A_{\overline{z}}$ component.
• Figure 5: The tree-level diagram for the computation of the Lax operator. We insert $A_w(z)$ at the position of the cross. We have a similar diagram for the $A_{\overline{w}}(z)$. The two diagrams are the only non-zero tree-level diagrams.

Theorems & Definitions (4)

• Conjecture
• Conjecture 14.1
• Definition 15.1
• Definition 15.2