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In mathematics, the ELSV formula, named after its four authors Torsten Ekedahl, Sergei Lando, Michael Shapiro, Alek Vainshtein, is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves. Several fundamental results in the intersection theory of moduli spaces of curves can be deduced from the ELSV formula, including the Witten conjecture, the Virasoro constraints, and the -conjecture. == The formula == Define the ''Hurwitz number'' : as the number of ramified coverings of the complex projective line (Riemann sphere, P1(C)) that are connected curves of genus ''g'', with ''n'' numbered preimages of the point at infinity having multiplicities ''k1'', ..., ''kn'' and ''m'' more simple branch points. Here if a covering has a nontrivial automorphism group ''G'' it should be counted with weight 1/''|G|''. The ELSV formula then reads : Here the notation is as follows: * ''g'' ≥ 0 is a nonnegative integer; * ''n'' ≥ 1 is a positive integer; * ''k1'', ..., ''kn'' are positive integers; * ; * is the moduli space of stable curves of genus ''g'' with ''n'' marked points; * ''E'' is the Hodge vector bundle and ''c(E *)'' the total Chern class of its dual vector bundle; * ψ''i'' is the first Chern class of the cotangent line bundle to the ''i''-th marked point. The numbers : in the left-hand side have a combinatorial definition and satisfy properties that can be proved combinatorially. Each of these properties translates into a statement on the integrals on the right-hand side of the ELSV formula . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ELSV formula」の詳細全文を読む スポンサード リンク
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