The \(n\)-point functions for intersection numbers on moduli spaces of curves. (English) Zbl 1263.14034

Liu, Kefeng; Xu, Hao

Adv. Theor. Math. Phys. 15, No. 5, 1201-1236 (2011).

In 1990s, Faber proposed a series of remarkable conjectures on the tautological rings of moduli spaces of curves. Faber’s conjecture was partly inspired by the Witten-Kontsevich theorem which asserts that the integrals of \(\psi\) classes on moduli spaces of curves are governed by the KdV hierarchy. As pointed out by Faber, the explicit form of \(n\)-point functions, a formal power series whose coefficients encode integrals of \(\psi\) classes, is useful in studying the tautological ring. Such explicit formulas were obtained by Witten, Dijkgraaf and Zagier respectively when \(n=1,2,3\). This paper under review shows that the Witten-Kontsevich theorem can be equivalently formulated as a recursive formula of \(n\)-point functions, which can be effectively used to obtain some closed formulas of integrals of \(\psi\) classes. These observations are used to give a direct proof of Faber’s intersection number conjecture in [J. Differ. Geom. 83, No. 2, 313–335 (2009; Zbl 1206.14079)].

Reviewer: Hao Xu (Cambridge)

MSC:

14H10	Families, moduli of curves (algebraic)
14D21	Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

Citations:

Zbl 1206.14079