Spin Foam Models with Finite Groups

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. 1. Introduction Spin foam models [1–8] arose as one possible quantization method for gravity. These models can be seen in the tradition of lattice approaches to quantum gravity [9], in which gravity is formulated as a statistical system on either a fixed or fluctuating lattice. In a rather independent fashion, lattice methods are also ubiquitous in condensed-matter and (Yang-Mills) gauge theories. It is interesting to note that the lattice structures appearing in the strong coupling expansion of QCD or in the high-temperature expansion for the Ising gauge [10] systems are of spin foam type. In any lattice approach to gravity, one has to discuss the significance of the (choice of) lattice for physical predictions. Although this point arises also for other lattice field theories, it carries extreme importance for general relativity. When it comes to matter fields on a lattice, we can interpret the lattice itself as providing the background geometry and therefore background space time. This could not contrast more with the situation in general relativity, where geometry, and therefore space time itself, is a dynamical variable and has to be an outcome rather than an ingredient of the theory. This is one aspect of background independence for quantum gravity [11, 12]. One possibility, to alleviate the dependence of physical results on the choice of lattice, is to turn the lattice itself into a dynamical variable and to sum over (a certain class of) lattices. This is one motivation for the dynamical triangulation approach [13–15], causal dynamical triangulations [16, 17], and tensor model/tensor group field theory [18–26]. See also [27, 28] for another possibility to introduce a dynamical lattice. Once one decides to sum over lattice structure, one must provide a prescription to do so. The class of triangulations to be summed over can be restricted, for instance, in order to