Operator formulation of riemann problem and $Z_n$-symmetric conformal algebra from KdV-type equation

Abstract

We treat two types of Riemann boundary value problem, circular and line branch cuts, in a complex plane with the operator formulation of free fermion fields in order to see more precise meaning of the universal Grassmann mainfold method. Thus the GL($\infty$) group transform on the fermion Fock space is directly parameterized by the Cauchy index and the boundary condition in the circular case, and by infinite singular structure in the line branch cuts. Also we apply these to Zn-symmetric fonformal system. We obtain Zn-symmetric conformal algebra from the KdV-type equation by Miura transform.