2 2 A ug 1 99 8 On the Moduli of a quantized loop in P and KdV flows :

On quantization of a loop on a Riemannian sphere P with an energy functional, we must not evaluate its stationary points with respect to the energy but also all states. Thus in this paper, we have investigated moduli M of loops (a quantize loop) on P. Then we proved that its moduli is decomposed to equivalent classes determined by flows of the KdV hierarchy. Since the flows of the KdV hierarchy have a natural topology, we can induce the topology to the loop space from the moduli of flows. Using the topology, we have considered the moduli space. Then we conjectured that the cohomology of the moduli is ring homomorphic to that of the loop space over S. In other words, we proposed a functor from loop space in category of differential geometry Geom to that in category of topology Top. In the content, we reviewed Baker’s construction of hyperelliptic ℘ function in order to show an one-to-one correspondence from moduli of hyperelliptic curves to solution spaces of the KdV equation, which differs from Krichever’s method: we showed that for any hyperelliptic curve, we can obtain an explicit function form of solution of the KdV equation. In this article, as Euler investigated elliptic integral and its moduli by observing a shape of classical elastica on C, we have considered hyperelliptic curves by studying a quantized loop on P.