JGRAV Journal of Gravity 2314-6907 Hindawi Publishing Corporation 549824 10.1155/2013/549824 549824 Review Article Spin Foam Models with Finite Groups http://orcid.org/0000-0003-0487-9684 Bahr Benjamin bbahr@aei.mpg.de 1 http://orcid.org/0000-0002-0072-5242 Dittrich Bianca bdittrich@perimeterinstitute.ca 2 Ryan James P. james.ryan@aei.mpg.de 2 Bamba Kazuharu 1 DAMTP University of Cambridge Wilberforce Road Cambridge CB3 0WA UK cam.ac.uk 2 MPI for Gravitational Physics Albert Einstein Institute Am Mühlenberg 1 14476 Potsdam Germany mpg.de 2013 2472013 2013 30 03 2013 03 06 2013 2013 Copyright © 2013 Benjamin Bahr et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Spin foam models, loop quantum gravity, and group field theory are discussed as quantum gravity candidate theories and usually involve a continuous Lie group. We advocate here to consider quantum gravity-inspired models with finite groups, firstly as a test bed for the full theory and secondly as a class of new lattice theories possibly featuring an analogue diffeomorphism symmetry. To make these notes accessible to readers outside the quantum gravity community, we provide an introduction to some essential concepts in the loop quantum gravity, spin foam, and group field theory approach and point out the many connections to the lattice field theory and the condensed-matter systems.