The Hurwitz Enumeration Problem of Branched Covers and Hodge Integrals

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group Sn. We also find a generating function for Hodge integrals on the moduli space Mg,2 of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point. ∗E-mail: [email protected]. †E-mail: [email protected]. Research supported in part by an NSF Graduate Fellowship and the U.S. Department of Energy under cooperative research agreement #DE-FC02-94ER40818. ‡E-mail: [email protected]. Research supported in part by an NSF Graduate Fellowship and the U.S. Department of Energy under contract DE-AC03-76SF00515.