Flavour singlets in gauge theory as permutations

Gauge-invariant operators can be specified by equivalence classes of permutations. we develop this idea concretely for the singlets of the flavour group so(n f ) in u(n c ) gauge theory by using gelfand pairs and schur-weyl duality. the singlet operators, when specialised at n f = 6, belong to the scalar sector of $$ mathcal{n}=4 $$ sym. a simple formula is given for the two-point functions in the free field limit of g 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 n c , n f . 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.