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

Yang Lei, Sanjaye Ramgoolam

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 $.

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

Abstract

Supersymmetric sectors of super-Yang-Mills theory motivate the study of the partition function for the counting of gauge-invariant functions of matrices transforming under the adjoint action of . The partition function in the large limit has a known Hagedorn phase transition at which provides a simple model for the phase structure of the thermal partition function of SYM. We study the all-orders asymptotic expansion of based on a geometric picture of concentric circles of poles in the complex plane accumulating in a natural boundary at . 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 using the modular property of the Dedekind eta function reveals that the asymptotic expansion is actually convergent for . A fermionic version of has an analogous critical dimension of . This distinction indicates that the partition functions of the matrix models can be completely reconstructed from their high-energy (UV) limit for whereas additional input is required to reconstruct the exact coefficients of the low-energy (IR) expansion for .

Paper Structure

This paper contains 18 sections, 5 theorems, 165 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. All orders asymptotic expansion of $\mathcal{Z}_2(x)$
  3. Asymptotics of unrefined $2$-matrix partition function
  4. Analytic asymptotic formula for $Z_2(K)$ coefficients
  5. Critical dimensions in $d$-matrix theory
  6. Bosonic $d$-matrix theory
  7. Fermionic $d$-matrix theory
  8. Small cycle dominance and asymptotics of multi-traces
  9. Counting multi-traces: contraction and equivalence permutations
  10. From permutations to infinite product
  11. Asymptotics and small cycles of equivalence permutations
  12. Unrefined partition function
  13. Refined partition function
  14. Circles of poles and asymptotics of weighted partition numbers
  15. All-orders asymptotics for meromorphic functions
  16. ...and 3 more sections

Key Result

Proposition 3.1

For $2 \le d \le 12$ the asymptotic behaviour, the series expansion eq:general-d-subleading-ZK is divergent.

Figures (2)

  • Figure 1: A meromorphic function with singularities (marked by red and blue points) of the simple poles at radius $r_1,r_2$ satisfying $r_1<r_2$. Singularity subtraction defined by remainder function $\mathcal{R}_1$ in \ref{['eq:def-remainder-truncation']} is regular within the disk of radius $r_2$.
  • Figure 2: The vertical axis is log of the error, i.e. $\ln|Z_2(K)- Z_2(K;M)|$, while the horizontal axis is the energy level $K$. The green, blue and red correspond to $M=1,2,3$ respectively.

Theorems & Definitions (10)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof