On the Moduli of a Quantized Elastica in ℙ and KdV Flows: Study of Hyperelliptic Curves as an Extension of Euler's Perspective of Elastica I

Abstract

Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli [Formula: see text] of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere ℙ. Then it is proved that its moduli space is decomposed to a set of equivalent classes determined by flows obeying the Korteweg-de Vries (KdV) hierarchy which conserve the energy. Since the flow obeying the KdV hierarchy has a natural topology, it induces topology in the moduli space [Formula: see text]. Using the topology, [Formula: see text] is classified. Studies on a loop space in the category of topological spaces Top are well-established and its cohomological properties are well-known. As the moduli space of a quantized elastica can be regarded as a loop space in the category of differential geometry DGeom, we also proved an existence of a functor between a triangle category related to a loop space in Top and that in DGeom using the induced topology. As Euler investigated the elliptic integrals and its moduli by observing a shape of classical elastica on [Formula: see text], this paper devotes relations between hyperelliptic curves and a quantized elastica on ℙ as an extension of Euler's perspective of elastica.