Harer–Zagier formula via Fock space

Let $\epsilon_g (d)$ be the number of ways of obtaining a genus $g$ Riemann surface by identifying in pairs the sides of a $(2d^{\prime})$-gon. The goal of this note is to give a short proof of the following theorem (Harer–Zagier formula, 1986):\[\epsilon_g(d^{\prime}) =\dfrac{(2d^{\prime}-1) !! 2^{d^{\prime}-2g}}{(d^{\prime} - 2g + 1)!}[u^{2g}]{\biggl[ \frac{u}{\mathrm{sinh}(u)} \biggr ]}^2 {\biggl[ \frac{u}{\mathrm{tanh}(u)} \biggr ]}^{d^{\prime}}\]

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Received 12 September 2018

Accepted 17 June 2019

Published 8 August 2022