Enumeration of maps with tight boundaries and the Zhukovsky transformation

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.

💡 Research Summary

This paper investigates the enumeration of maps (graphs embedded on surfaces) whose boundaries are “tight”, meaning that each boundary has minimal length within its homotopy class. The authors introduce generating functions (T^{(g)}_{\ell_1,\dots,\ell_n}) that count maps of fixed genus (g) with (n) tight boundaries of prescribed lengths (\ell_1,\dots,\ell_n), while controlling the degrees of inner faces.

A central combinatorial construction is the trumpet decomposition. A “trumpet” is a genus‑0 map with two boundary faces: one rooted face of arbitrary length and a second, unrooted “mouthpiece” that is strictly tight. Lemma 2.1 shows that the mouthpiece’s contour is a simple closed curve, a property that relies on the cylindrical topology of the trumpet. By attaching a suitable trumpet to each tight boundary of a map, any map with arbitrary boundaries can be uniquely decomposed into a map with only tight boundaries together with a collection of trumpets.

The authors demonstrate that this decomposition exactly implements the Zhukovsky transformation used in the Eynard‑Orantin topological recursion. In the recursion, the basic meromorphic function (x(z)) on the spectral curve is traditionally taken as
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