A note on Marino-Vafa formula

Hodge integrals over moduli spaces of curves appear naturally during the localization procedure in computation of Gromov-Witten invariants. A remarkable formula of Marino-Vafa expresses a generation function of Hodge integrals via some combinatorial and algebraic data seemingly unrelated to these apriori algebraic geometric objects. We prove in this paper by directly expanding the formula and estimating the involved terms carefully that except a specific type all the other Hodge integrals involving up to three Hodge classes can be calculated from this formula. This implies that amazingly rich information about moduli spaces and Gromov-Witten invariants is encoded in this complicated formula. We also give some low genus examples which agree with the previous results in literature. Proofs and calculations are elementary as long as one accepts Mumford relations on the reductions of products of Hodge classes.

作 者： LU Wenxuan   作者單位： Department of Mathematics, Tsinghua University, Beijing 100084, China  刊 名： 中國(guó)科學(xué)A輯（英文版）  SCI 英文刊名： SCIENCE IN CHINA (MATHEMATICS)  年，卷(期)： 2006 49(1)  分類號(hào)： O1  關(guān)鍵詞： Hodge integrals   Gromov-Witten invariants   Marino-Vafa formula   Mumford relations  

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