  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.

High Energy Physics - Theory;Strongly Correlated Electrons;General Relativity and Quantum Cosmology