Abstract
A systematic study of holomorphic gauge invariant operators in general $$ \mathcal{N} $$ = 1 quiver gauge theories, with unitary gauge groups and bifundamental matter fields, was recently presented in [1]. For large ranks a simple counting formula in terms of an infinite product was given. We extend this study to quiver gauge theories with fundamental matter fields, deriving an infinite product form for the refined counting in these cases. The infinite products are found to be obtained from substitutions in a simple building block expressed in terms of the weighted adjacency matrix of the quiver. In the case without fundamentals, it is a determinant which itself is found to have a counting interpretation in terms of words formed from partially commuting letters associated with simple closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foatamonoid. For finite ranks of the unitary gauge groups, the refined counting is given in terms of expressions involving Littlewood-Richardson coefficients.
Highlights
The study of giant gravitons [2] in the context of AdS/CFT [3] has instigated detailed investigations of BPS operators in four dimensional N = 4 super-Yang-Mills theory (SYM) with U(N ) gauge group [4,5,6,7,8,9,10,11]
We extend this study to quiver gauge theories with fundamental matter fields, deriving an infinite product form for the refined counting in these cases
We have revisited the counting of local holomorphic operators in general quiver gauge theories with bi-fundamental fields, which was started in [1], focusing on the infinite product formula obtained for the limit of large N
Summary
This uncovers a new link between gauge invariant operators of quiver theories and the mathematics of CartierFoata monoids [23, 24] The latter is expressed here in terms of a word counting problem where the letters correspond to loops on a graph, with partial commutation relations. This generating function depends on chemical potentials, one for each of the bifundamental fields in the theory, i.e. one for each edge in the quiver joining gauge nodes.
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