Integrable lattice models and holography

Meer Ashwinkumar

doi:10.1007/jhep02(2021)227

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.

Highlights

Given the importance of the 2d chiral Wess-Zumino-Witten (WZW) model dual to 3d Chern-Simons theory as a straightforward example of a holographic dual, and for describing edge modes of the nonabelian fractional quantum Hall effect, it is of interest to investigate the existence of a holographic dual of 4d Chern-Simons theory
We study four-dimensional Chern-Simons theory on D ×C, 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

Summary

Current algebra via canonical quantization

Note that we have found a central extension term at the classical level This is analogous to the case of the standard 2d WZW model with kinetic term in “chiral” form, i.e., using lightcone coordinates, wherein such a central extension term appears in the classical Poisson bracket involving currents when taking one of the lightcone directions to be time, as shown in [12]. Employing the orthogonality of these modes, the resulting algebra takes the form This has the form of a two-toroidal Lie algebra (or a centrally-extended double loop algebra), which, in particular, arises as the current algebra of the four-dimensional WZW model studied in [13,14,15,16]. Based on the results of [19, 20], this integrable sigma model is likely to be described by some type of affine Gaudin model

Wilson lines and boundary local operators

Non-perpendicular Wilson lines

Crossed Wilson lines at order 2

Parallel Wilson lines

Three Wilson lines