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

TL;DR

This paper introduces a novel mathematical framework for S-transformations in two-dimensional conformal field theories, connecting closed and open sectors through linear mappings and geometric insights.

Contribution

It defines S-transformations as linear maps from closed to open sectors in CFT$_2$, expanding applicability beyond diagonal RCFTs and incorporating a new open sector sewing method.

Findings

Established a geometric approach linking algebraic data to surface curvature

Defined S-transformations using sector isomorphisms and boundary conditions

Generalized open sector sewing beyond Lewellen's framework

Abstract

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.