New Properties of the Intersection Numbers on Moduli Spaces of Curves

We present certain new properties about the intersection numbers on moduli spaces of curves Mg,n. In particular we prove a new identity, which together with a conjectural identity implies the famous Faber’s conjecture about certain values of intersection numbers [1]. These new identities are much simpler than Faber’s identity and clarified the mysterious constant in Faber’s conjecture. We also discuss some numerical properties of Hodge integrals which have provided numerous inspirations for this work. 1. Two new identities of intersection numbers We denote by Mg,n the moduli space of stable n-pointed genus g complex algebraic curves. We have the forgetting the last marked point morphism π : Mg,n+1 −→ Mg,n, We denote by σ1, . . . , σn the canonical sections of π, and by D1, . . . , Dn the corresponding divisors in Mg,n+1. We let ωπ be the relative dualizing sheaf and set ψi = c1(σ ∗ i (ωπ)) K = c1 ( ωπ (∑ Di )) κi = π∗(K ) E = π∗(ωπ) λl = cl(E), 1 ≤ l ≤ g. Where E is the Hodge bundle. We adopt Witten’s notation, 〈τd1 . . . τdn〉g = ∫ Mg,n ψ1 1 · · ·ψ dn n . The famous Faber’s conjecture [1] is the following identity (2g − 3 + n)! 22g−1(2g − 1)! · 1 ∏n j=1(2dj − 1)!! = 〈τd1 . . . τdnτ2g〉 − n ∑ j=1 〈τd1 . . . τdj−1τdj+2g−1τdj+1 . . . τdn〉