Gauge theory and integrability. I
type: Item
instance of: scholarly article
title: Gauge theory and integrability. I
zbMATH Open document ID: 1405.81043
author: Kevin Costello
author: Masahito Yamazaki
author: Edward Witten
published in: ICCM Notices
publication date: 2018-10-23T00:00:00Z
review text: This very exhaustive discussion of integrable models in the context of gauge theories is based on evaluation of the Yang-Baxter equation, which states that for three pairwise intersecting Wilson lines the order of the intersections does not matter. The authors extend an earlier approach by one of the authors [the first author, Proc. Symp. Pure Math. 88, 3--23 (2014; Zbl 1321.81033)] and relating to a deduction of the Yang-Baxter equation in two dimensions from a higher dimensional theory [\textit{M. Atiyah}, ibid. 48, 285--299 (1988; Zbl 0667.57018)]. The paper starts with an extensive review on the Yang-Baxter equation and integrability in two and four dimensions. Wilson lines play a crucial role, which are introduced in Section 3. Furthermore, the so-called \(R\)-matrix, that is a solution to the Yang-Baxter equation is defined in Section 4. The framing anomaly describes a quantum effect on Wilson lines and plays an important role in networks of Wilson lines. Detailed two-loop calculations show that a further anomaly arises at higher quantum levels which is canceld by a non-trivial quantum deformation of the infinite-dimensional algebra. This deformation is outlined in an accompanying paper by the same authors [the first author et al., ICCM Not. 6, No. 1, 120--146 (2018; Zbl 1405.81044)]. Eventually, explicit solutions to the Yang-Baxter equation of the trigonometric and elliptic type.
reviewed by: Wolfgang G. Hollik
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 81T50
Mathematics Subject Classification ID: 81Q30
Mathematics Subject Classification ID: 81R12
zbMATH DE Number: 6962556
zbMATH Keywords: Yang-Baxter equation
zbMATH Keywords: Wilson lines
zbMATH Keywords: Lie groups
zbMATH Keywords: anomalies
MaRDI profile type: Publication
Wikidata QID: Q125271477
DOI: 10.4310/ICCM.2018.V6.N1.A6
description: scientific article; zbMATH DE number 6962556