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Recursion Formulas for Spin Hurwitz Numbers

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 Kahler surfaces. We prove a recursive formula for spin Hurwitz numbers, which then gives the dimension zero GW invariants of Kahler 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 the d-bar operator, and an interesting localization phenomenon for eigenfunctions that shows that maps with even ramification points cancel in pairs.