Abstract
AbstractWe propose a class of models which generate three-dimensional random volumes, where each configuration consists of triangles glued together along multiple hinges. The models have matrices as the dynamical variables and are characterized by semisimple associative algebras \( \mathcal{A} \). Although most of the diagrams represent configurations which are not manifolds, we show that the set of possible diagrams can be drastically reduced such that only (and all of the) three-dimensional manifolds with tetrahedral decompositions appear, by introducing a color structure and taking an appropriate large N limit. We examine the analytic properties when \( \mathcal{A} \) is a matrix ring or a group ring, and show that the models with matrix ring have a novel strong-weak duality which interchanges the roles of triangles and hinges. We also give a brief comment on the relationship of our models with the colored tensor models.
Highlights
M-theory [1] is a description of string theory, where membranes are believed to play an important role.1 The worldvolume theory of membranes is equivalent to a threedimensional gravity theory, where the target space coordinates of an embedded membrane are expressed as scalar fields in three-dimensional worldvolume
Most of the diagrams represent configurations which are not manifolds, we show that the set of possible diagrams can be drastically reduced such that only three-dimensional manifolds with tetrahedral decompositions appear, by introducing a color structure and taking an appropriate large N limit
Most of the diagrams represent configurations which are not manifolds,3 we show that the set of possible diagrams can be drastically reduced such that only three-dimensional manifolds with tetrahedral decompositions appear, by introducing a color structure and taking an appropriate limit of parameters existing in the models
Summary
M-theory [1] is a description of string theory, where membranes are believed to play an important role. The worldvolume theory of membranes is equivalent to a threedimensional gravity theory, where the target space coordinates of an embedded membrane are expressed as scalar fields in three-dimensional worldvolume (see [3] for a review). Tensor models [20,21,22] or group field theory [23, 24] are natural generalizations of matrix models to three (and higher) dimensions. Most of the diagrams represent configurations which are not manifolds, we show that the set of possible diagrams can be drastically reduced such that only (and all of the) three-dimensional manifolds with tetrahedral decompositions appear, by introducing a color structure and taking an appropriate limit of parameters existing in the models. We show that our models have a novel strong-weak duality which interchanges the roles of triangles and hinges when A is a matrix ring This duality may suggest the analytic solvability of the models
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