Gauge theory and boundary integrability. Part II. Elliptic and trigonometric cases

Abstract

We consider the mixed topological-holomorphic Chern-Simons theory introduced by Costello, Yamazaki & Witten on a Z 2 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 Z 2 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.

Q-operators are 't Hooft lines

“…There is only one Weyl orbit of minuscule coweight in this case. The Levi factor is l = so(10) ⊕ so(2), and the subspaces n ± are the two spin representations of so (10). We choose our coweight so that n ± = S ± .…”

“…Let x i be a basis for for the vector representation of so (10), and ψ α , ψ α bases for the spin representations S ± . A basis for the 27 dimensional representation of E 6 is given by v, in the trivial representation, x i , and ψ α .…”

“…Four-dimensional Chern-Simons theory [1][2][3] is a unified approach to studying integrability which has been able to explain many examples of integrable systems: spin chains [1,2], integrable field theories [4][5][6][7][8], spin chains with boundary [9,10], and integrable string theories [11].…”

4-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 [75][76][77][78][79] and their integrable deformations [80][81][82][83] can also be obtained from the 4-dimensional Chern-Simons theory, as shown in the works [4,[30][31][32]34,45], following a strategy similar to the one reviewed in these lectures 19 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.…”

“…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 .…”