Orthogonal Schurs for Classical Gauge Groups

TL;DR

This paper demonstrates that orthogonality of Schur operators in half-BPS sectors is a universal property across classical gauge groups, explained through Lie algebra embeddings, unifying the understanding of these operators in different gauge theories.

Contribution

It provides a unified framework for understanding Schur operator orthogonality across classical gauge groups using Lie algebra embeddings, resolving previous puzzles.

Findings

Orthogonality of Schur operators is gauge group-independent.

Lie algebra embeddings explain the diagonalization of two-point functions.

Unified treatment applies to SO(N), Sp(N), and U(N) gauge groups.

Abstract

Finite N physics of half-BPS operators for gauge groups SO(N) and Sp(N) has recently been studied[1, 2]. Among other things they showed that, alike U(N), Schur operators (but in the square of their eigenvalues) diagonalize the free field two-point function of half-BPS operators for SO(N) and Sp(N) gauge groups. This result was unexpected since Wick contractions behave differently. In this paper we solve the puzzle by treating all gauge groups in a unified framework and showing how orthogonality of Schur operators emerges naturally from the embedding structure of classical Lie algebras g(N) -> g(M). We go further and we state that orthogonality of Schurs is a gauge group-independent property for classical gauge groups.