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The central heat trace on large compact classical groups

Thibaut Lemoine, Mylène Maïda

TL;DR

The paper proves a comprehensive large-$N$ expansion for the central heat trace on all compact classical groups, providing explicit coefficients that depend on the group type and revealing a parity structure: even powers for unitary cases and all powers for orthogonal/symplectic cases. It develops a random-partition framework and a highest-weights-to-partitions dictionary to derive the asymptotics, then translates the results into random-surface representations via Hurwitz spaces and into a Gromov–Witten picture on the torus. Two gauge/string dualities follow: a Yang–Mills/Hurwitz duality that expresses YM observables in terms of ramified coverings, and a Yang–Mills/Gromov–Witten duality that links YM data to GW invariants on elliptic curves, with coefficients given by derivatives of GW-generated functions. The work thus provides a rigorous bridge between 2D YM theory and both Hurwitz theory and GW theory, yielding explicit, computable formulas for the asymptotic expansion coefficients and clarifying the role of group-type couplings and chirality in the dualities.

Abstract

We establish the large-$N$ asymptotic expansion of the (central) trace of the heat kernel on any compact classical group $G_N\subset\mathrm{GL}_N(\mathbb{C})$, which extends a previous result known only for $\mathrm{U}(N)$ \cite{LM2}. It admits two new interpretations of the trace: in terms of ramified coverings of the torus, and Gromov-Witten invariants on elliptic curves. These connections allow us to explore several aspects of the gauge/string duality in two dimensions: a Yang-Mills/Hurwitz duality, and Yang-Mills/Gromov-Witten duality.

The central heat trace on large compact classical groups

TL;DR

The paper proves a comprehensive large- expansion for the central heat trace on all compact classical groups, providing explicit coefficients that depend on the group type and revealing a parity structure: even powers for unitary cases and all powers for orthogonal/symplectic cases. It develops a random-partition framework and a highest-weights-to-partitions dictionary to derive the asymptotics, then translates the results into random-surface representations via Hurwitz spaces and into a Gromov–Witten picture on the torus. Two gauge/string dualities follow: a Yang–Mills/Hurwitz duality that expresses YM observables in terms of ramified coverings, and a Yang–Mills/Gromov–Witten duality that links YM data to GW invariants on elliptic curves, with coefficients given by derivatives of GW-generated functions. The work thus provides a rigorous bridge between 2D YM theory and both Hurwitz theory and GW theory, yielding explicit, computable formulas for the asymptotic expansion coefficients and clarifying the role of group-type couplings and chirality in the dualities.

Abstract

We establish the large- asymptotic expansion of the (central) trace of the heat kernel on any compact classical group , which extends a previous result known only for \cite{LM2}. It admits two new interpretations of the trace: in terms of ramified coverings of the torus, and Gromov-Witten invariants on elliptic curves. These connections allow us to explore several aspects of the gauge/string duality in two dimensions: a Yang-Mills/Hurwitz duality, and Yang-Mills/Gromov-Witten duality.

Paper Structure

This paper contains 20 sections, 21 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Large-$N$ expansion of the trace
  3. Random surface interpretation
  4. Expansion of the central heat trace
  5. Heat kernel on compact classical groups
  6. Random partitions
  7. Highest weights/partitions correspondence
  8. Proof of the asymptotic expansion
  9. Random surface representation
  10. Integration over Hurwitz spaces
  11. Cutoff on the Euler characteristic
  12. Cutoff on degree
  13. Proof of Theorems \ref{['thm:mainbis']} and \ref{['thm:main_alt']}
  14. Applications to gauge/string duality
  15. Yang--Mills/Hurwitz duality
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(G_N)_{N\geqslant 1}$ be a sequence of compact classical groups with $G_N\subset{\mathrm{GL}}_N(\mathbb{C})$. For any $t>0$, there is an explicit family of coefficients $(a_k(t))_{k\geqslant 0}$ (depending on the group type) such that for any $p\geqslant 1$, the following asymptotic expansion h

Figures (1)

  • Figure 1: The Young diagram representing the partition $\alpha=(8,6,5,5,4,2,1)$.

Theorems & Definitions (52)

  • Theorem 1.1: See Theorem \ref{['thm:asympt_expansion_detailed']}
  • Theorem 1.2: See Theorem \ref{['thm:mainbis']}
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 42 more