Gauge theory and boundary integrability

Abstract

We study the mixed topological / holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold (Σ × C)/Z 2 , obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order calculation we derive a formula for the the asymptotic behaviour of K-matrices associated to rational, quasi-classical R-matrices. The Z 2 -action on Σ × C fixes a line L, and line operators on L are shown to be labelled by representations of the twisted Yangian. The OPE of such a line operator with a Wilson line in the bulk is shown to give the coproduct of the twisted Yangian. We give the gauge theory realisation of the Sklyanin determinant and related conditions in the RT T presentation of the boundary Yang-Baxter equation. Introduction The Yang-Baxter EquationInteractions of an integrable spin chain are determined by an object known as an R-matrix. This is a linear mappair of complex vector spaces and z, z ′ complex spectral parameters on which the R-matrix depends meromorphically. The spin chain is integrable if the R-matrix obeys the Yang-Baxter equation (YBE)(1.1) around the identity 1 V ⊗V ′ . R-matrices admitting such an expansion are called quasiclassical, and r(z, z ′ ) is known as the classical r-matrix. This classical r-matrix takes values in g ⊗ g for g a finite dimensional, complex, simple Lie algebra, acting in representations associated to V ⊗ V ′ . Expanding the YBE to second order in shows that the classical r-matrix obeys the classical Yang-Baxter equation,Solutions of the classical Yang-Baxter equations were classified by Belavin & Drinfeld [4], under the (mild) assumption that the r-matrix is non-degenerate. The solutions can be separated into three families, distinguished by whether the classical r-matrix can be written in terms of rational, trigonometric, or elliptic functions. In this work we will concentrate on the rational case. Each family of solutions to is associated with an algebra, which in the rational case is the Yangian Y(g). Indeed, in [5] it was demonstrated that all rational solutions of the YBE of the form (1.1) determine a representation of the Yangian on V , and are themselves determined by such a representation. Accordingly, the line operators in Costello's theory are actually associated with representations of Y(g). These include ordinary Wilson lines in certain representations of g itself, but also more general line operators. As shown in [1-3], the more general line operators arise even in the OPE of two ordinary Wilson lines.Since the YBE is homogeneous, its solutions are defined only up to multiplication by a function f (z 1 , z 2 ) of the spectral parameters. This degeneracy can be removed by required that the R-matrix obeys a constraint known as the quantum determinant condition. In the rational case, for V the defining vector representation of g = sl n (C), the quantum determinant [6] condition is enough to guarantee existence of a unique quasi-classical Rmatrix in this representation for a given classical r-matrix. Similar results e...

Gauge Theory And Integrability, II

“…See[4,5,6,7,8,9,10,11,12] for recent discussion of four-dimensional Chern-Simons theory 2. As we will see, the model we find is related to the pure-spinor formulation, not the Green-Schwarz formulation.…”

A unifying 2D action for integrable $$\sigma $$-models from 4D Chern–Simons theory

“…The second approach, proposed recently in [CY], is based on a four-dimensional variant of Chern-Simons theory which was used in the earlier works [C1,C2,W,CWY1,CWY2,BS] to describe integrable lattice models. In fact, two types of integrable field theories were considered in [CY], associated with so-called order and disorder defects, respectively.…”

Integrable deformations of coupled σ-models

“…This theory was introduced in [35] and was related to integrable systems and in particular integrable lattice models in [36][37][38][39]. More recently, it was shown in [40] how to generate integrable two-dimensional field theories from this four-dimensional theory (see also [41][42][43] for further developments). The reference [40] treated two different classes of models, corresponding to so-called order and disorder defects.…”

“…In this section, we explain how the models considered in this article can be obtained using the approach proposed recently by Costello and Yamazaki to generate integrable 2d field theories from 4d semi-holomorphic Chern-Simons theory [40] (see [35][36][37][38][39][41][42][43] for additional references on this variant of Chern-Simons theory and its relation to integrable systems). Note that, in the terminology of [40], we restrict our attention here to 4d Chern-Simons theory with disorder defects.…”