2-Group Representations for Spin Foams

2-Group Representations for Spin Foams

Aristide Baratin    Derek K. Wise
Abstract

Just as 3d state sum models, including 3d quantum gravity, can be built using categories of group representations, ‘2-categories of 2-group representations’ may provide interesting state sum models for 4d quantum topology, if not quantum gravity. Here we focus on the ‘Euclidean 2-group’, built from the rotation group and its action on the translation group of Euclidean space. We explain its infinite-dimensional unitary representations, and construct a model based on the resulting representation 2-category. This model, with clear geometric content and explicit ‘metric data’ on triangulation edges, shows up naturally in an attempt to write the amplitudes of ordinary quantum field theory in a background independent way.

Spin foam models, higher-dimensional algebra, representation theory
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04.60.Pp, 04.60.Nc, 02.20.Qs
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address=Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1, 14467 Golm, Germany, email=Aristide.Baratin@aei.mpg.de

address=Department of Mathematics, University of California, Davis, 95616, USA, email=derek@math.ucdavis.edu

1 Introduction

The success of combinatorial and algebraic methods in 3d quantum gravity [PonzanoRegge, TuraevViro] has long been an inspiration for analogous 4d models, including spin foam models of quantum gravity. A mathematically elegant approach to getting 4d models from 3d ones uses so called ‘higher-dimensional algebra’. Our aim here is not only to explain an instance of this approach, but also to present evidence that the resulting models may be relevant for real-world physics. Indeed, as we shall explain, they have already shown up in an unexpected way, in an attempt to understand a certain ‘limit’ of quantum gravity.

The reason for the term ‘higher-dimensional algebra’ is easily explained using the example most relevant to this paper: ‘2-groups’ [BaezLauda-2gps]. Whereas a group might consist of symmetry transformations of some ‘object’ , drawn as ‘arrows’:

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