Towards large genus asymptotics of intersection numbers on moduli spaces of curves

Abstract

We explicitly compute the diverging factor in the large genus asymptotics of the Weil-Petersson volumes of the moduli spaces of n-pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the WeilPetersson volumes in the inverse powers of the genus with coefficients that are polynomials in n. This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied.

JT gravity, KdV equations and macroscopic loop operators

“…(2.87) For d = 0 this agrees with the result in [23,37]. Plugging (2.87) into the definition of V g,1 (b) in (2.11), we find the large genus asymptotics of V g,1 (b)…”

“…(2.89) f 0 (b) agrees with the result in [10]. Note that the above f n (b) vanishes at b = 2πi which is consistent with the property V g,1 (2πi) = 0 [37]. The large genus asymptotics in (2.88) implies that there is a non-perturbative correction of the form…”

“…The (2g)! growth comes from the opposite end κ 3g−2−d ψ d 1 g,1 with fixed small d [23,37]. One can also see that the sum over genus of (2.84) is convergent, which we will study in detail in section 2.6.…”

A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

“…Mirzakhani considered analogues of these well-studied questions for Weil-Petersson random Riemann surfaces [Mir13, MZ15,MP17], and devoted her 2010 talk at the International Congress of Mathematicians to this topic [Mir10].…”

Large genus asymptotics for volumes of strata of abelian differentials

“…Expression (14) for c (1) cyl (H(m)) and bound (15) for ε cyl (m) imply existence of a universal constant B such that the ratio c (1) cyl (H(m)) dim C H(m) − 1 can be represented in the form (16) with ε area (m) satisfying bound (17).…”