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A finite generating set for the genus g (p,q,n)-dipole series from perturbative Yang-Mills theory

There is an emerging class of permutation factorization questions that cannot be expressed wholly in terms of the centre of the group algebra of the symmetric group. We shall term these non-central. A notable instance appears in recent work of Constable et al. [1] in perturbative Yang-Mills theory on the determination of a 2-point correlation function of the Berenstein-Maldacena-Nastase operators by means of Feynman diagrams. In combinatorial terms, this question relates to (p, q, n)-dipoles: loopless maps with exactly two vertices and n edges, with two distinguished edges, separated by p edges at one vertex and q edges at the other. By the introduction of join and cut operators, we construct a formal partial differential equation which uniquely determines a generating series from which the (p,q,n)-dipole series may be obtained. Moreover, we exhibit a set of functions with the property that the genus g solution to this equation may be obtained recursively as an explicit finite linear combination of these. These functions have explicit expressions as sums indexed by elementary combinatorial objects, and we demonstrate how the recursion can be used to give series solutions for surfaces of low genera.