Topological recursion for spin enumeration of hypermaps
Abstract
We establish topological recursion for the spin enumeration of hypermaps, or, in the dual language, spin dessins d'enfants. This problem has remarkable features distinguishing it from other applications of topological recursion to enumerative combinatorics. First, the spectral curve is elliptic rather than rational, as in most cases studied before. Second, it contains degenerate ramification points, and one has to apply a version of degenerate topological recursion. Finally, the meromorphic correlator differentials have "moving" poles on the antidiagonals between pairs of distinct arguments, and one has to adjust the loop equations to this kind of degeneracy as well. The computations involve concepts of different kinds, such as BKP integrability, Gaussian integrals, the x-y swap, and symplectic duality.
Related
- Enumeration of plane hypermaps with a mixed boundary II
- Quantum modularity in refined topological recursion
- Eynard-Orantin Topological Recursion, Moduli-Space Virasoro Constraints, and Semisimple CohFTs
- Topological Recursion on Spectral Curves with Infinite Ramification Loci
- Folding to the Bessel-Clifford system, and Equivariant Topological Recursion
BibTeX
@misc{alexandrov2026topological,
title = {{Topological recursion for spin enumeration of hypermaps}},
author = {Alexander Alexandrov and Boris Bychkov and Petr Dunin-Barkowski and Maxim Kazarian and Sergey Shadrin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20464},
eprint = {2610.20464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20464},
}
RIS
TY - UNPB
TI - Topological recursion for spin enumeration of hypermaps
AU - Alexander Alexandrov
AU - Boris Bychkov
AU - Petr Dunin-Barkowski
AU - Maxim Kazarian
AU - Sergey Shadrin
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20464
UR - https://arxiv.org/abs/2610.20464
AB - We establish topological recursion for the spin enumeration of hypermaps, or, in the dual language, spin dessins d'enfants. This problem has remarkable features distinguishing it from other applications of topological recursion to enumerative combinatorics. First, the spectral curve is elliptic rather than rational, as in most cases studied before. Second, it contains degenerate ramification points, and one has to apply a version of degenerate topological recursion. Finally, the meromorphic correlator differentials have "moving" poles on the antidiagonals between pairs of distinct arguments, and one has to adjust the loop equations to this kind of degeneracy as well. The computations involve concepts of different kinds, such as BKP integrability, Gaussian integrals, the x-y swap, and symplectic duality.
ER -