Enhanced symmetries of gauge theory and resolving the spectrum of local operators

TL;DR

This paper explores how enhanced non-abelian symmetries in zero-coupling Yang-Mills theory help diagonalize multi-matrix operator two-point functions, introducing generalized Casimirs and connecting to various bases like Schur and Brauer.

Contribution

It introduces a framework using generalized Casimirs from enhanced symmetries to resolve multiplicities in operator bases, extending diagonalization techniques to arbitrary global symmetry groups.

Findings

Generalized Casimirs resolve multiplicity labels in operator bases.

Different Casimir choices lead to known bases like restricted Schur and Brauer.

Schur-Weyl duality links enhanced symmetries with diagonal bases.

Abstract

Enhanced global non-abelian symmetries at zero coupling in Yang Mills theory play an important role in diagonalising the two-point functions of multi-matrix operators. Generalised Casimirs constructed from the iterated commutator action of these enhanced symmetries resolve all the multiplicity labels of the bases of matrix operators which diagonalise the two-point function. For the case of U (N) gauge theory with a single complex matrix in the adjoint of the gauge group we have a U(N)^{\times 4} global symmetry of the scaling operator at zero coupling. Different choices of commuting sets of Casimirs, for the case of a complex matrix, lead to the restricted Schur basis previously studied in connection with string excitations of giant gravitons and the Brauer basis studied in connection with brane-anti-brane systems. More generally these remarks can be extended to the diagonalisation for any global symmetry group G. Schur-Weyl duality plays a central role in connecting the enhanced symmetries and the diagonal bases.