A Note On Hodge Integral And Hurwitz Number

A relation between Hurwitz number and Hodge integrals is got via virtual localization method. A Hodge integral identity is proofed from the relation. The definition and some property of Hurwitz number were introduced.In the Introduction chapter,I mentioned several theorem and their relations. In the second chapter I review the virtual localization formula in Gromov-Witten theory in the special case of degree 1 maps to P~1. then we derived Hodge relations via localization. In the third chapter,I first introduce the definition of Hurwitz number, then by using the localization tecnique with branch divisor,I proved the ELSV formula, finally I proved the main theorem mentioned in the introduction.