S-transformations for CFT${}_2$ as linear mappings from closed to open sector linear spaces

We make the first attempt to define S-transformations for CFT${}_2$ as linear mappings from closed to open sector linear spaces. The definition is based on closed-open sector linear space isomorphisms and boundary condition completeness. Diagonal RCFTs can be applied to our definition straight-forwardly, while more classes of CFT${}_2$ are expected to be applicable. An unconventional open sector sewing, not among open sector sewing introduced by Lewellen, rises naturally and generalizes the definition. Our geometrical approach, partially inspired by string field theory, reveals the relationship between algebraic information in CFT${}_2$ and curvature on surfaces.