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Quantization of strings and branes coupled to BF theory. (English) Zbl 1147.83015
The paper presents a generalization to higher dimensions of the quantum dynamics of a spinless point particle coupled to three-dimensional Riemannian gravity based on the classical action proposed by P. de Sousa Gerbert [Nucl. Phys. B 346, No. 2–3, 440–472 (1990)]. This generalization consists of a BF theory in $$d$$ dimensions coupled to $$(d-3)$$-branes, whose connection is flat except along the membrane worldsheet where it has a conical singularity whose strength is proportional to the membrane tension. The canonical quantization of this model is performed as in loop quantum gravity by constructing first an auxiliary Hilbert space, then a kinematic Hilbert space spanned by “membrane spin networks” (spin networks equipped with extra data where their edges end on a brane), and finally a physical Hilbert space which solves the curvature constraint that was not imposed in the previous steps. The construction rests on the compactness of the gauge group and hence on the Riemannian character of the manifold.

##### MSC:
 83E15 Kaluza-Klein and other higher-dimensional theories 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T45 Topological field theories in quantum mechanics
##### Keywords:
BF theory; topological field theories; spin networks
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