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

TL;DR

This paper analyzes the all-orders asymptotic expansion of partition functions in matrix theories related to supersymmetric gauge theories, revealing a critical dimension where the expansion becomes convergent and uncovering small-cycle dominance in their combinatorial structure.

Contribution

It introduces a geometric and combinatorial framework for understanding the asymptotics of matrix model partition functions, identifying a critical dimension for convergence.

Findings

Asymptotic expansion converges for dimensions d ≥ 13.

Small-cycle dominance organizes the combinatorial structure of invariants.

Critical dimension differs for fermionic versions, affecting reconstructibility.

Abstract

Supersymmetric sectors of $\mathcal{N}=4$ 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) $ 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)$ 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) $ using the modular property of the Dedekind eta function reveals that the asymptotic expansion is actually convergent for $d\ge d_{ \rm crit } = 13$. A fermionic version of $\mathcal{Z}_d (x)$ has an analogous critical dimension of $ d_{ \rm crit} = 7$. This distinction indicates that the partition functions of the matrix models can be completely reconstructed from their high-energy (UV) limit for $d\ge d_{ \rm crit}$ whereas additional input is required to reconstruct the exact coefficients of the low-energy (IR) expansion for $2\le d \le d_{ \rm crit } -1 $.