Elliptic b-Hurwitz theory and Jack heat trace

Abstract

We study a deformation of the central heat trace on U(N) obtained by deforming Schur polynomials to Jack polynomials, and prove that it admits an asymptotic expansion to arbitrary order. Its coefficients are governed by elliptic b-Hurwitz numbers, which are genus-one counterparts of the b-Hurwitz theory of Chapuy and Dołęga \cite{ChapuyDolega22}. We construct the associated elliptic b-Hurwitz theory by means of generalized coverings on a torus and identify it with the genus-one closure of the genus-zero simple b-Hurwitz theory. At b=0 the construction recovers ordinary elliptic Hurwitz theory, while at b=1 it gives an automorphism-weighted geometric interpretation of the connected and disconnected twisted elliptic Hurwitz numbers of Hahn--Markwig \cite{HahnMarkwig26}. Our results extend the topological expansion obtained in the classical case in \cite{LemMai25,LM2} and take the form of a coupling between chiral and antichiral elliptic b-Hurwitz generating functions.

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BibTeX RIS

BibTeX

@misc{lemoine2026elliptic,
  title = {{Elliptic $b$-Hurwitz theory and Jack heat trace}},
  author = {Thibaut Lemoine},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12256},
  eprint = {2609.12256},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12256},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Elliptic b-Hurwitz theory and Jack heat trace
AU  - Thibaut Lemoine
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.12256
UR  - https://arxiv.org/abs/2609.12256
AB  - We study a deformation of the central heat trace on U(N) obtained by deforming Schur polynomials to Jack polynomials, and prove that it admits an asymptotic expansion to arbitrary order. Its coefficients are governed by elliptic b-Hurwitz numbers, which are genus-one counterparts of the b-Hurwitz theory of Chapuy and Dołęga \cite{ChapuyDolega22}. We construct the associated elliptic b-Hurwitz theory by means of generalized coverings on a torus and identify it with the genus-one closure of the genus-zero simple b-Hurwitz theory. At b=0 the construction recovers ordinary elliptic Hurwitz theory, while at b=1 it gives an automorphism-weighted geometric interpretation of the connected and disconnected twisted elliptic Hurwitz numbers of Hahn--Markwig \cite{HahnMarkwig26}. Our results extend the topological expansion obtained in the classical case in \cite{LemMai25,LM2} and take the form of a coupling between chiral and antichiral elliptic b-Hurwitz generating functions.
ER  -

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