2021/07/31 by Eynard, Bertrand, Lewański, Danilo
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2108.00226
We advertise elementary symmetric polynomials ei as the natural basis for generating series Ag,n of intersection numbers of genus g and n marked points. Closed formulae for Ag,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 ei basis relies in the unexpected vanishing of some coefficients with a clear pattern: we conjecture that Ag,n can have at most g factors ei, 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 Ag,n, which recover expressions for the Weil-Petersson volumes in terms of Bessel functions.