Simplicity condition and boundary-bulk duality

Abstract

In the first-order formulation, general relativity could be formally viewed as the topological $BF$ theory with a specific constraint, the Plebanski constraint. $BF$ theory is expected to be the classical limit of the Crane-Yetter~(CY) topological state sum. In the Euclidean case, the Plebanski constraint could be lifted in an elegant way to a quantum version in the CY state sum, called the simplicity condition. The constrained state sum is known as the Barrett-Crane~(BC) model. In this note we investigate this condition from the topological field theory side. Since the condition is in fact imposed on the faces, we want to understand it from the viewpoint of the surface theory. Essentially this condition could be thought of as resulting from the boundary-bulk dualtiy, or more precisely from the recent "bulk=center" proposal. In the language of topological phases, it corresponds to the diagonal anyon condensation, with the BC model being the condensed phase. The CY state sum, and correspondingly the BC model, is usually constructed with a modular tensor category~(MTC) from representations of a quantum group at roots of unity. The category of representations of quantum group and that of modules of the corresponding affine Lie algebra are known to be equivalent as MTCs. This equivalence, together with the simplicity condition in the BC model, guarantees the construction of a full 2d rational conformal field theory through the Fuchs-Runkel-Schweigert formalism. We thus obtain a full-fledged 4d quantum geometry, which we name as "stringy quantum geometry". Some attracting features of such a new geometry are briefly discussed.

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