Gauge theory and boundary integrability. Part II. Elliptic and trigonometric cases
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Springer
Received: March 28, 2020 Accepted: May 15, 2020 Published: June 12, 2020
Roland Bittleston and David Skinner Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
E-mail: [email protected], [email protected] Abstract: We consider the mixed topological-holomorphic Chern-Simons theory introduced by Costello, Yamazaki & Witten on a Z2 orbifold. We use this to construct semiclassical solutions of the boundary Yang-Baxter equation in the elliptic and trigonometric cases. A novel feature of the trigonometric case is that the Z2 action lifts to the gauge bundle in a z-dependent way. We construct several examples of K-matrices, and check that they agree with cases appearing in the literature. Keywords: Chern-Simons Theories, Lattice Integrable Models, Wilson, ’t Hooft and Polyakov loops ArXiv ePrint: 1912.13441
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)080
JHEP06(2020)080
Gauge theory and boundary integrability. Part II. Elliptic and trigonometric cases
Contents 1 Introduction 1.1 CWY theory 1.2 The boundary Yang-Baxter equation
1 2 4 8 8 9 10
3 Trigonometric 3.1 Trigonometric solutions of the Yang-Baxter equation 3.2 Trigonometric solutions of the boundary Yang-Baxter equation 3.3 Examples of trigonometric solutions 3.4 Generalization to z-dependent automorphisms 3.5 Examples of K-matrices associated to z-dependent automorphisms
13 13 15 16 20 21
4 Comments on uniqueness
23
A Asymptotic behaviour of elliptic K-matrices in the ~ → 0 limit
24
B Asymptotic behaviour of trigonometric K-matrices in the ~ → 0 limit
27
C Determination of allowed automorphisms in the elliptic case
28
D Determination of allowed automorphisms in the trigonometric case
30
E An aside on loop algebras
32
1
Introduction
In [1] rational solutions of the boundary Yang-Baxter equation were generated as the vacuum expectation values of Wilson lines in a mixed topological-holomorphic analogue of Chern-Simons theory on a Z2 -orbifold. This extended the link between gauge theory and quantum integrability developed by Costello, Witten, and Yamazaki in the papers [2–5] to integrable models with boundary. The mixed topological-holomorphic analogue of ChernSimons theory underpinning this link was first proposed by Nekrasov in his Ph.D. thesis [6], see also the related papers [7, 8]. In this paper we expand this construction to elliptic and trigonometric solutions of the boundary Yang-Baxter equation. We begin by reviewing the CWY approach to 2d quantum integrable lattice models. We shall be brief, and refer the reader to [2, 4] for more detailed discussions of the theory
–1–
JHEP06(2020)080
2 Elliptic solutions 2.1 Elliptic solutions of the Yang-Baxter equation 2.2 Elliptic solutions of the boundary Yang-Baxter equation 2.3 Construction of elliptic solutions
1.1
CWY theory
where A(w) = Ax (w)dx + Ay (w)dy + Az¯(w)d¯ z is a partial connection on a G-bundle over M . Note that
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