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The geometry of unitary 2representations of finite groups and their 2characters
, 2008
"... Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2representations of finite groups on 2Hilbert spaces, and their 2characters. We show how the basic ideas of geometric quantization are ‘categorified ’ in this context: just as representations of groups ..."
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Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2representations of finite groups on 2Hilbert spaces, and their 2characters. We show how the basic ideas of geometric quantization are ‘categorified ’ in this context: just as representations of groups correspond to equivariant line bundles, 2representations of groups correspond to equivariant gerbes. We also show how the 2character of a 2representation can be made functorial with respect to morphisms of 2representations. Under the geometric correspondence, the 2character of a 2representation corresponds to the geometric character of its associated equivariant gerbe. This enables us to show that the complexified 2character is a unitarily fully faithful functor from the complexified ho
Modular Theory, NonCommutative Geometry and Quantum Gravity
, 2010
"... This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita–Takesaki modular theory and A. Connes noncommutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of state ..."
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This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita–Takesaki modular theory and A. Connes noncommutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in noncommutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.
2VECTOR SPACES AND GROUPOIDS
, 810
"... Abstract. This paper describes a relationship between essentially finite groupoids and 2vector spaces. In particular, we show to construct 2vector spaces of Vectvalued presheaves on such groupoids. We define 2linear maps corresponding to functors between groupoids in both a covariant and contrav ..."
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Abstract. This paper describes a relationship between essentially finite groupoids and 2vector spaces. In particular, we show to construct 2vector spaces of Vectvalued presheaves on such groupoids. We define 2linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation— a weak functor—from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail. It has applications in constructing quantum field theories, among others. 1.