Enumeration of maps with tight boundaries and the Zhukovsky transformation

TL;DR

This paper links the enumeration of maps with tight boundaries to the Eynard–Orantin topological recursion by showing that the generating functions $T^{(g)}_{oldsymbol\ell}$ appear as coefficients in $\omega^{(g)}_n(z_1,...,z_n)$ after a Zhukovsky substitution. The trumpet decomposition provides a combinatorial interpretation of the Zhukovsky transform, splitting arbitrary-boundary maps into tight-boundary cores plus trumpet pieces. The authors derive recursion relations that add a tight boundary, establish explicit formulas in the planar bipartite case via Collet–Fusy polynomials, and prove a parity-dependent quasi-polynomial structure for $T^{(g)}_{oldsymbol\ell}$ in terms of $\ell_i^2$ with coefficients that depend rationally on derivatives of $R$ and $S$. They also extend Norbury–Scott’s lattice-count quasi-polynomial framework to the non-bipartite setting and provide a bijective derivation for $(g,n)=(0,3)$. Overall, the work deepens the bridge between map enumeration and topological recursion, highlighting the combinatorial meaning of Zhukovsky and revealing rich algebraic structure in tight-boundary genera.

Abstract

We consider maps with tight boundaries, i.e. maps whose boundaries have minimal length in their homotopy class, and discuss the properties of their generating functions $T^{(g)}_{\ell_1,\ldots,\ell_n}$ for fixed genus $g$ and prescribed boundary lengths $\ell_1,\ldots,\ell_n$, with a control on the degrees of inner faces. We find that these series appear as coefficients in the expansion of $ω^{(g)}_n(z_1,\ldots,z_n)$, a fundamental quantity in the Eynard-Orantin theory of topological recursion, thereby providing a combinatorial interpretation of the Zhukovsky transformation used in this context. This interpretation results from the so-called trumpet decomposition of maps with arbitrary boundaries. In the planar bipartite case, we obtain a fully explicit formula for $T^{(0)}_{2\ell_1,\ldots,2\ell_n}$ from the Collet-Fusy formula. We also find recursion relations satisfied by $T^{(g)}_{\ell_1,\ldots,\ell_n}$, which consist in adding an extra tight boundary, keeping the genus $g$ fixed. Building on a result of Norbury and Scott, we show that $T^{(g)}_{\ell_1,\ldots,\ell_n}$ is equal to a parity-dependent quasi-polynomial in $\ell_1^2,\ldots,\ell_n^2$ times a simple power of the basic generating function $R$. In passing, we provide a bijective derivation in the case $(g,n)=(0,3)$, generalizing a recent construction of ours to the non bipartite case.

Figures (7)

• Figure 1: A toric map, with three distinguished boundary-faces of degree 6, admitting an automorphism group of order $3$. There are only $2$ rooted maps resulting from marking one of the $6$ corners incident to the cyan face, and we must account for this fact by weighing this map by the inverse automorphism group order factor $1/3$.
• Figure 2: An illustration of the decomposition of a map of genus $3$ with three external faces into a map with tight boundaries and with the same topology, and a brassband of three trumpets. We emphasize in this picture that, while the mouthpieces of the trumpets always have simple boundaries by Lemma \ref{['sec:decomp-theor-1']}, the (tight) external faces of the central map need not have simple boundaries.
• Figure 3: An illustration of the decomposition of a trumpet with mouthpiece of length $\ell$ endowed with an extra marked vertex, viewed as a boundary vertex. It results in two pieces: a trumpet with mouthpiece of length $m>\ell$ and a map with three boundaries counted by $m\, T^{(0)}_{m,0|\ell}$.
• Figure 4: A sketch of why $m_{i_1}<\ell_{i_1}$ and $m_{i_2}<\ell_{i_2}$ are mutually exclusive: indeed, it would imply $\mathcal{L}_{i_1}<\ell_{i_1}$ or $\mathcal{L}_{i_2}<\ell_{i_2}$, in contradiction with the fact that the boundary-faces of length $\ell_1$ and $\ell_2$ are both tight.
• Figure 5: A summary representation of the bijection of triskell between maps with three tight boundaries and quintuples $(D_1,D_2,D_3,T_1,T_2)$ made of three bigeodesic diangles $D_1,D_2,D_3$ and two bigeodesic triangles $T_1,T_2$. The solid lines indicate the identification between attachment points for type I. For type II, the two outer identification lines have to be replaced by the dotted lines.

Theorems & Definitions (64)

• Lemma 2.1
• proof
• Theorem 2.2: Trumpet decomposition of maps
• Corollary 2.3
• Proposition 2.4
• Theorem 2.5: Combinatorial interpretation of $\omega_n^{(g)}$
• proof
• Remark 2.6
• Theorem 2.7
• Corollary 2.8