Gauge Theory and Boundary Integrability

Abstract

We study the mixed topological / holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold $(\Sigma \times \mathbb{C})/{\mathbb{Z}}_2$, obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order $\hslash$ calculation we derive a formula for the the asymptotic behaviour of $K$-matrices associated to rational, quasi-classical $R$-matrices. The ${\mathbb{Z}}_2$-action on $\Sigma \times \mathbb{C}$ 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 $RTT$ presentation of the boundary Yang-Baxter equation.

53 pages, 22 figures

References in corpus (16)

• TikZ-Feynman: Feynman diagrams with TikZ
• Introduction to Yangian symmetry in integrable field theory
• Integrable open spin chains from giant gravitons
• Reflecting magnons
• Quantum group symmetry in sine-Gordon and affine Toda field theories on the half-line
• Boundary remnant of Yangian symmetry and the structure of rational reflection matrices
• General boundary conditions for the sl(N) and sl(M|N) open spin chains
• Kinks in the Kondo problem
• Representations of reflection algebras
• Integrable Boundary Conditions for the One-Dimensional Hubbard Model
• Rational K-matrices and representations of twisted Yangians
• Chern-Simons Gauge Theory on Orbifolds: Open Strings from Three Dimensions
• Classification of reflection matrices related to (super) Yangians and application to open spin chain models
• Coideal algebras from twisted Manin triples
• Twisted Yangians for symmetric pairs of types B, C, D
• Yang-Baxter equation and reflection equations in integrable models

Cited by in corpus (9)

• A unifying 2d action for integrable $\sigma$-models from 4d Chern-Simons theory
• 4-dimensional Chern-Simons theory and integrable field theories
• Integrable deformations of coupled sigma-models
• 't Hooft lines of ADE-type and Topological Quivers
• Superspin Chains Solutions from 4D Chern-Simons Theory
• The solutions of the Yang-Baxter equation for the $(n+1)(2n+1)$-vertex models through a differential approach
• Integrable Lattice Models and Holography
• Gauge Theory and Boundary Integrability II: Elliptic and Trigonometric Case
• On the classification of rational K-matrices