Spin Hurwitz Numbers and the Gromov-Witten Invariants of Kähler Surfaces

The classical Hurwitz numbers which count coverings of a complex curve have an analog when the curve is endowed with a theta characteristic. These “spin Hurwitz numbers”, recently studied by Eskin, Okounkov and Pandharipande, are interesting in their own right. By the authors’ previous work, they are also related to the Gromov-Witten invariants of Kähler surfaces. We prove a recursive formula for spin Hurwitz numbers, which then gives the dimension zero GW invariants of Kähler surfaces with positive geometric genus. The proof uses a degeneration of spin curves, an invariant defined by the spectral flow of certain anti-linear deformations of ∂, and an interesting localization phenomenon for eigenfunctions that shows that maps with even ramification points cancel in pairs. The Hurwitz numbers of a complex curve D count covers with specified ramification type. Specifically, consider degree d (possibly disconnected) covering maps f : C → D with fixed ramification points q, . . . , q ∈ D and ramification given by m, . . . ,m where each m = (m1, · · · ,mi`i) is a partition of d. The Euler characteristic of C is related to the genus h of D and the partition lengths `(m) = `i by the Riemann-Hurwitz formula χ(C) = 2d(1− h) + k ∑ i=1 ( `(m)− d). (1.1) In this context, there is an ordinary Hurwitz number ∑ 1 |Aut(f)| (1.2) that counts the covers f satisfying (1.1) mod automorphisms; the sum depends only on h and {m}. Now fix a theta characteristic N on D, that is, a holomorphic line bundle with an isomorphism N = KD where KD is the canonical bundle of D. The pair (D,N) is called spin curve. By a well-known theorem of Mumford and Atiyah, the deformation class of the spin curve is completely characterized by the genus h of D and the parity p = (−1) (D,N) (1.3) Now consider degree d ramified covers f : C → D for which • each partition m is odd, i.e. each mj is an odd number. (1.4) In this case, the ramification divisor Rf of f is even and the twisted pullback bundle Nf = f ∗N ⊗O ( 1 2Rf ) (1.5) ∗partially supported by the N.S.F.