A natural basis for intersection numbers

Bertrand Eynard; Danilo Lewański

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arXiv · 2108.00226

A natural basis for intersection numbers

Bertrand Eynard·Danilo Lewański

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Abstract

We advertise elementary symmetric polynomials $e_i$ as the natural basis for generating series $A_{g,n}$ of intersection numbers of genus g and n marked points. Closed formulae for $A_{g,n}$ are known for genera $0$ and $1$ -- this approach provides formulae for $g = 2,3,4$, together with an algorithm to compute the formula for any g. The claimed naturality of the e_i basis relies in the unexpected vanishing of some coefficients with a clear pattern: we conjecture that $A_{g,n}$ can have at most $g$ factors $e_i$, with $i>1$, in its expansion. This observation promotes a paradigm for more general cohomology classes. As an application of the conjecture, we find new integral representations of $A_{g,n}$, which recover expressions for the Weil-Petersson volumes in terms of Bessel functions.

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Bertrand Eynard, Danilo Lewański. 2021-07-31. A natural basis for intersection numbers. https://arxiv.org/abs/2108.00226

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