Witten solution of the Gelfand-Dikii hierarchy

TL;DR

This paper explores Witten's solution to the Gelfand-Dikii hierarchy, deriving recurrence relations for its coefficients, which connect to the intersection numbers of moduli spaces and potentially simplify calculations related to Witten's conjecture.

Contribution

It introduces recurrence relations between the coefficients of Witten's solution, linking them to the Mumford-Morita-Muller numbers and providing an algorithm for their computation.

Findings

Derived recurrence relations for Witten's solution coefficients.

Connected these relations to Mumford-Morita-Muller intersection numbers.

Provided an algorithm for calculating n-spin Mumford-Morita-Muller numbers.

Abstract

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,...).