Correlators on the wall and $\mathfrak{sl}_n$ spin chain

Abstract

We study algebras and correlation functions of local operators at half-BPS interfaces engineered by the stacks of D5 or NS5 branes in the 4d N = 4 super Yang-Mills. The operator algebra in this sector is isomorphic to a truncation of the Yangian Y(gl n ). The correlators, encoded in a trace on the Yangian, are controlled by the inhomogeneous sl n spin chain, where n is the number of fivebranes: they are given in terms of matrix elements of transfer matrices associated to Verma modules, or equivalently of products of Baxter's Q-operators. This can be viewed as a novel connection between the N = 4 super Yang-Mills and integrable spin chains. We also remark on analogous constructions involving half-BPS Wilson lines.

Interfaces and Quantum Algebras, I: Stable Envelopes

“…Acknowledgements. We are grateful to M. Aganagic We should also point out that some of our models can be mirror mapped to the N = 2 26 Transfer matrices labeled by the complex spins of sl 2 , but in the XXX spin chain with the usual spin 1/2 at each site, have also appeared in [186], in a seemingly quite different context, but in exactly the same SU (N ), N f = 2N gauge theory, this time with a codimension-one defect inserted. In that case, it is also true that inhomogeneities and the complex spins of representations of the auxiliary space are determined by the 2N masses.…”

Superspin chains from superstring theory

“…As an example, take (m|n) = (3|2) and ( 1 , 2 , 3 , 4 , 5 ) = (ε 1 , ε 2 , δ 1 , δ 2 , ε 3 ). This choice of ordered basis gives the Dynkin diagram (15). There are two odd simple roots, α 2 = ε 2 − δ 1 and α 4 = δ 2 −ε 3 , represented by the crossed nodes.…”

“…Many of the phenomena that are expected to constitute this web of dualities are yet to be uncovered, but their specializations to the case of gl(m|0) = gl(m) are known and have been studied in recent years. Besides the Bethe/gauge correspondence already described, the structures of rational gl(m) spin chains (and their trigonometric and elliptic generalizations) have appeared in quantization of the Seiberg-Witten geometries of four-dimensional N = 2 supersymmetric gauge theories [4][5][6], the action of surface and line defects on supersymmetric indices of four-dimensional supersymmetric gauge theories [7][8][9][10][11][12], quantization of the Coulomb branches of three-dimensional N = 4 supersymmetric gauge theories [13,14], and correlation functions of local operators on interfaces in four-dimensional N = 4 super Yang-Mills theory [15], to name a few.…”

Lax operator and superspin chains from 4D CS gauge theory

“…These parallel lines are aligned along a direction of R 2 and are simultaneously crossed by a perpendicular magnetic 't Hooft line at z ′ = z. The 't Hooft line defect plays an important role in this modeling as it was interpreted in terms of the transfer (monodromy) matrix [3,5,25] and the Q-operators of the spin chain [13,26,27].…”

“…These parallel lines are aligned along a direction of R 2 and are simultaneously crossed by a perpendicular magnetic 't Hooft line at z = z. The 't Hooft line defect plays an important role in this modeling as it was interpreted in terms of the transfer (monodromy) matrix [3,5,25] and the Q-operators of the spin chain [13,26,27]. In this setup, the nodes' spin states of the quantum chain are identified with the weight states of the Wilson lines, which in addition to the spectral parameter z, are characterised by highest weight representations R of the gauge symmetry G [7].…”