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In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by , and generalized in . Witten's original conjecture was proved by . Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy. ==Statement== Suppose that ''M''''g'',''n'' is the moduli stack of compact Riemann surfaces of genus ''g'' with ''n'' distinct marked points ''x''1,...,''x''''n'', and ''g'',''n'' is its Deligne–Mumford compactification. There are ''n'' line bundles ''L''''i'' on ''g'',''n'', whose fiber at a point of the moduli stack is given by the cotangent space of a Riemann surface at the marked point ''x''''i''. The intersection index 〈τ''d''1, ..., τ''d''''n''〉 is the intersection index of Π ''c''1(''L''''i'')''d''''i'' on ''g'',''n'' where Σ''d''''i'' = dim''g'',''n'' = 3''g'' – 3 + ''n'', and 0 if no such ''g'' exists, where ''c''1 is the first Chern class of a line bundle. Witten's generating function : encodes all the intersection indices as its coefficients. Witten's conjecture states that the partition function ''Z'' = exp ''F'' is a τ-function for the KdV hierarchy, in other words it satisfies a certain series of partial differential equations corresponding to elements ''L''''i'' for ''i''≥–1 of the Virasoro algebra. 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「Witten conjecture」の詳細全文を読む スポンサード リンク
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