Gelfand-Dikii族的Witten解
Witten solution of the Gelfand-Dikii hierarchy
摘要：
在n-Gelfand-Dikii族的解中，存在一个满足弦方程的显著解W.我们称之为威滕解，因为根据威滕猜想，函数F（x_1，x_2，x_3，.）= W（x_1，（x_2）/2，（x_3）/3，.）是n-自旋Riemann曲面的模空间的Mumford-Morita-Muller上同调类的交数的母函数。这一猜想证明了孔采维奇为n=2和威滕自己的表面属0。本文给出了W.这就把Witten猜想简化为Mamford-Morita-Muller数满足相同关系的猜想。在Witten猜想成立的条件下，这些关系也给出了计算$n$ -spin Mamford-Morita-Muller数的算法.并且证明了F（x_1，x_2，.，x_{n-1}，0，0，.）= W（x_1，（x_2）/2，.，（x_{n-1}）/（n-1），0，0，.）。
英文摘要：
Among solutions of n-Gelfand-Dikii's hierarchy there exists a remarkable solution W, which satisfies the string equation. We call it Witten's solution because according to the Witten conjecture the function F(x_1, x_2, x_3,...) = W(x_1,(x_2)/2, (x_3)/3,...) is the generating function for intersection nambers of Mumford-Morita-Muller cohomological classes of the moduli space of n-spin Riemann surfaces. This conjecture was proved by Kontsevich for n=2 and by Witten himself for surfaces of genus 0. In this paper we find recurrence relations between coefficients of Taylor series of W. This reduces the Witten's conjecture to conjecture that the Mamford-Morita-Muller numbers satisfy to the same relations. These relations give also an algorithm for calculation of $n$-spin Mamford-Morita-Muller numbers in assuming that the Witten conjecture is true. Moreover we prove that F(x_1,x_2,...,x_{n-1},0,0,...)= W(x_1, (x_2)/2,...,(x_{n-1})/(n-1),0,0,...).
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