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 $SO(N_f)$ 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 $g_{YM}^2=0$. 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.

50 pages, v2: typos corrected, v3: to appear in JHEP

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Cited by in corpus (9)

• Structure Constants in $\mathcal{N}=4$ SYM at Finite Coupling as Worldsheet $g$-Function
• BPS operators in $\mathcal{N}=4$ $SO(N)$ super Yang-Mills theory: plethysms, dominoes and words
• Quarter-BPS states, multi-symmetric functions and set partitions
• Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras
• Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM
• Hidden symmetries and Large N factorisation for permutation invariant matrix observables
• Noncommutative Frobenius algebras and open-closed duality
• Three-point Functions in $\mathcal{N}=4$ SYM at Finite $N_c$ and Background Independence
• Multi-trace Correlators from Permutations as Moduli Space