Integrable lattice models and holography

Abstract

We study four-dimensional Chern-Simons theory on D × ℂ (where D is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an “analytically-continued” toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational R-matrix purely from the boundary theory.

Four-dimensional Chern–Simons theory and integrable field theories

“…Other important examples of integrable σ-models include models on coset spaces G/H such as symmetric spaces (which contain for instance spheres and Anti-de Sitter spaces in any dimension). These integrable coset models [81][82][83][84][85] and their integrable deformations [86][87][88][89] can also be obtained from the four-dimensional Chern-Simons theory, as shown in the works [4, 32-34, 36, 48], following a strategy similar to the one reviewed in these lectures 21 for non-coset models but supplemented with an averaging process over the action of a finite order automorphism of the underlying Lie group (we will not enter into more details about this approach here). We additionally note that some of these references start with an extension of the theory considered in these lectures where the Lie algebra g underlying the construction is replaced by a Lie superalgebra, resulting in σ-models which also possess fermionic degrees of freedom.…”

“…It was proposed in [8] that this four-dimensional theory can provide a geometric origin to quantum affine algebras, which are algebraic structures underlying the theory of integrable systems. In the reference [9] and in subsequent works [10][11][12][13] of Costello, Yamazaki and Witten, it was shown that integrable lattice models (spin chains) can be naturally obtained using this four-dimensional theory (see also [14][15][16][17][18][19][20][21][22][23] for further developments) 2 . The extension of this approach to also generate integrable two-dimensional field theories was put forward by Costello and Yamazaki in [4] and has been the subject of many subsequent works .…”

4d Chern-Simons theory as a 3d Toda theory, and a 3d-2d correspondence

“…Consider the 4d CS theory on a four-manifold with a single boundary along the topological plane. If one of the complex gauge fields (namely Az) is set to vanish at the boundary, a 3d analogue of the chiral Wess-Zumino-Witten (WZW) model can be obtained [13].…”

“…where δ (ξ − ξ ) = ∂ ξ δ(ξ − ξ ). This takes the form of the "analytically-continued" toroidal Lie algebra seen in [13].…”

4-dimensional Chern-Simons theory and integrable field theories

“…The 4-dimensional semi-holomorphic Chern-Simons theory was initially proposed by K. Costello in [8]. In this reference and in subsequent works [9][10][11][12] of K. Costello, M. Yamazaki and E. Witten, it was shown that integrable lattice models (spin chains) can be naturally obtained using this 4dimensional theory (see also [13][14][15][16][17][18][19][20][21][22] for further developments). The extension of this approach to also generate integrable 2-dimensional field theories was put forward by K. Costello and M. Yamazaki in [4] and has been the subject of many subsequent works .…”