Flavour singlets in gauge theory as permutations

Abstract

Gauge-invariant operators can be specified by equivalence classes of permutations. We develop this idea concretely for the singlets of the flavour group SO(Nf) in U(Nc) gauge theory by using Gelfand pairs and Schur-Weyl duality. The singlet operators, when specialised at Nf = 6, belong to the scalar sector of N= 4 SYM. A simple formula is given for the two-point functions in the free field limit of gY M 2 = 0. The free two-point functions are shown to be equal to the partition function on a 2-complex with boundaries and a defect, in a topological field theory of permutations. The permutation equivalence classes are Fourier transformed to a representation basis which is orthogonal for the two-point functions at finite Nc, Nf. Counting formulae for the gauge-invariant operators are described. The one-loop mixing matrix is derived as a linear operator on the permutation equivalence classes.