Uniqueness of open/closed rational CFT with given algebra of open states
Jens Fjelstad, Jurgen Fuchs, Ingo Runkel, Christoph Schweigert
TL;DR
This work proves a uniqueness result for open/closed RCFT correlators with given open-state boundary data by embedding the problem in a modular tensor category and its associated 3-d TFT. It shows that, under natural nondegeneracy conditions (a unique closed-state vacuum, invertible disk and sphere two-point correlators, and nonzero open-state dimension), any solution to the sewing constraints is equivalent to a TFT-construction $\mathsf{S}(\mathcal{C},A)$ determined by a normalised open-state Frobenius algebra $A$ in $\mathcal{C}$. Consequently, all RCFT correlators can be obtained from the data of $A$ (up to Morita equivalence), with the full center $Z(A)$ encoding closed-state information; the equivalence is established via a monoidal natural transformation linking the given correlators to those derived from $A$ and its center. The result provides a conceptually clean, model-independent link between open/closed RCFT data and Frobenius algebra structures in a modular tensor category, clarifying how boundary and bulk sectors determine the full theory. The approach also highlights the role of categorical structures, such as the full center and Morita classes, in the classification of consistent RCFTs.
Abstract
We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly non-empty boundary. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: For a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two,- and three-point correlators on the disk. Thus for any such theory every consistent collection of correlators can be obtained by the TFT approach of hep-th/0204148, hep-th/0503194. As morphisms of the category of world sheets we include not only homeomorphisms, but also sewings; interpreting the correlators as a natural transformation then encodes covariance both under homeomorphisms and under sewings of world sheets.