1998/08/22 by Shigeki Matsutani, Matsutani, Shigeki, Yoshihiro Onishi +2
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #math.DG
paper · pdf · doi:10.48550/arxiv.math/9808099
64 pages, to appear Rev. Math. Phys
openalex publication_date 1998/08/22 · arxiv created 2003/05/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \CMeP of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere \PP. 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 \CMeP. Using the topology, \CMeP 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 \bf 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 \CC, this paper devotes relations between hyperelliptic curves and a quantized elastica on \PP as an extension of Euler's perspective of elastica.