On the moduli of a quantized elastica in P and KdV flows: Study of hyperelliptic curves as an extention of Euler’s perspective of elastica I

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 Melas of a quantized elastica, a quantized loop on with an energy functional associated with the Schwarz derivative, on a Riemannian sphere P. Then we proved that its moduli is decomposed to equivalent classes determined by flows of the Korteweg-de Vries (KdV) hierarchy which conserve the energy. Since the flows of the KdV hierarchy have a natural topology, it induces topology in the moduli space Melas. Using the topology, we classified Melas. Studies on a loop space in category of topological space Top are well-established and its cohomological properties are well-known. As the moduli of a quantized elastica can be regarded as a loop space in category of differential geometry DGeom, we also proved existence of a functor from loop space in Top to that in DGeom using the induced topology. In the content, we, further, reviewed Baker’s construction of hyperelliptic ℘ function in order to show a 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. As Euler investigated elliptic integral and its moduli by observing a shape of classical elastica on C, in this paper, we have considered relations between hyperelliptic curves and a quantized elastica on P as an extension of Euler’s perspective of elastica.