Algebras, traces, and boundary correlators in $$ \mathcal{N} $$ = 4 SYM

Abstract

We study supersymmetric sectors at half-BPS boundaries and interfaces in the 4d $$ \mathcal{N} $$ N = 4 super Yang-Mills with the gauge group G, which are described by associative algebras equipped with twisted traces. Such data are in one-to-one correspondence with an infinite set of defect correlation functions. We identify algebras and traces for known boundary conditions. Ward identities expressing the (twisted) periodicity of the trace highly constrain its structure, in many cases allowing for the complete solution. Our main examples in this paper are: the universal enveloping algebra $$ U\left(\mathfrak{g}\right) $$ U g with the trace describing the Dirichlet boundary conditions; and the finite W-algebra $$ \mathcal{W}\left(\mathfrak{g},{t}_{+}\right) $$ W g t + with the trace describing the Nahm pole boundary conditions.

Superspin chains from superstring theory

“…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 fourdimensional 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 fourdimensional N = 4 super Yang-Mills theory [15], to name a few.…”

Integrable domain walls in ABJM theory

“…As for the AdS 5 /CF T 4 system a double scaling parameter naturally appears, but in the present case it does not allow for a precise comparison of the gauge and string theory results. One could hope that a version of supersymmetric localization makes it possible to compute overlaps exactly for particular chiral primaries as it was the case in N = 4 SYM [7,61,62]. Interestingly, the operator algebra in the presence of the defect carries a natural integrability structure [63].…”

Integrable domain walls in ABJM theory

“…As for the AdS 5 CF T 4 system a double scaling parameter naturally appears, but in the present case it does not allow for a precise comparison of the gauge and string theory results. One could hope that a version of supersymmetric localization makes it possible to compute overlaps exactly for particular chiral primaries as it was the case in N = 4 SYM [51,7,52]. Interestingly, the operator algebra in the presence of the defect carries a natural integrability structure [53].…”