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

TL;DR

This paper explores the algebraic structures and trace functions associated with supersymmetric boundary conditions in 4d $ abla=4$ super Yang-Mills theory, providing a classification and solutions for specific cases.

Contribution

It identifies algebraic and trace structures for known boundary conditions and solves the associated Ward identities, advancing understanding of boundary correlators in supersymmetric gauge theories.

Findings

Algebras and traces are classified for Dirichlet and Nahm pole boundary conditions.

Ward identities constrain the trace structures, enabling complete solutions in many cases.

Explicit examples include universal enveloping algebras and finite W-algebras.

Abstract

We study supersymmetric sectors at half-BPS boundaries and interfaces in the 4d $\mathcal{N}=4$ 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 $U(\mathfrak{g})$ with the trace describing the Dirichlet boundary conditions; and the finite W-algebra $\mathcal{W}(\mathfrak{g},t_+)$ with the trace describing the Nahm pole boundary conditions.