A natural basis for intersection numbers
Bertrand Eynard, Danilo Lewański
TL;DR
By expressing the generating series of intersection numbers $A_{g,n}$ in the basis of elementary symmetric polynomials $e_i$, the paper identifies a sharp vanishing pattern for the coefficients $C_g(\lambda)$ and proves that $\ell(\lambda)\le g$ governs the expansion. It derives Virasoro-based recursions and computes explicit formulas for genus $2$–$4$, with verification up to $g\le7$ and $n\le3$, and provides $g$-dimensional integral representations that connect to Weil–Petersson volumes via Bessel functions. The authors also recast the conjecture in terms of $\Omega$-CohFTs and extend the framework to ELSV-type formulae for one-part Hurwitz numbers, as well as to other cohomological classes such as $\lambda$-classes and monotone Hurwitz numbers, illustrating a broad structural pattern. These results offer both practical computational tools for high-genus intersection numbers and deeper conceptual insight into the role of a natural symmetric-basis description in moduli-space cohomology.
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.