Critical dimensions and small cycle dominance from all-orders asymptotics of d-matrix theory

A bstract Supersymmetric sectors of $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory motivate the study of the partition function for the counting of gauge-invariant functions of d = 2, 3 matrices transforming under the adjoint action of U ( N ). The partition function $$ {\mathcal{Z}}_d(x) $$ Z d x in the large N limit has a known Hagedorn phase transition at x = d −1 which provides a simple model for the phase structure of the thermal partition function of SYM. We study the all-orders asymptotic expansion of $$ {\mathcal{Z}}_d(x) $$ Z d x based on a geometric picture of concentric circles of poles in the complex plane accumulating in a natural boundary at | x | = 1. We find that the order by order structure has a precise combinatorial interpretation organized in terms of increasing cycle size of permutations arising in the enumeration of the invariants. We refer to this organization as small-cycle dominance, and find that it extends to refined versions of the partition functions depending on several complex variables. An analysis of the coefficients in the asymptotic expansion of $$ {\mathcal{Z}}_d(x) $$ Z d x using the modular property of the Dedekind eta function reveals that the asymptotic expansion is actually convergent for d ≥ d crit = 13. A fermionic version of $$ {\mathcal{Z}}_d(x) $$ Z d x has an analogous critical dimension of d crit = 7. This distinction indicates that the partition functions of the matrix models can be completely reconstructed from their high-energy (UV) limit for d ≥ d crit whereas additional input is required to reconstruct the exact coefficients of the low-energy (IR) expansion for 2 ≤ d ≤ d crit − 1.