Group field theories generating polyhedral complexes

Group field theories are a generalization of matrix models which provide both a second quantized reformulation of loop quantum gravity as well as generating functions for spin foam models. While states in canonical loop quantum gravity, in the traditional continuum setting, are based on graphs with vertices of arbitrary valence, group field theories have been defined so far in a simplicial setting such that states have support only on graphs of fixed valency. This has led to the question whether group field theory can indeed cover the whole state space of loop quantum gravity. In this contribution based on [1] I present two new classes of group field theories which satisfy this objective: i) a straightforward, but rather formal generalization to multiple fields, one for each valency and ii) a simplicial group field theory which effectively covers the larger state space through a dual weighting, a technique common in matrix and tensor models. To this end I will further discuss in some detail the combinatorial structure of the complexes generated by the group field theory partition function. The new group field theories do not only strengthen the links between the mentioned quantum gravity approaches but, broadening the theory space of group field theories, they might also prove useful in the investigation of renormalizability.Comment: accepted for publication in PoS, Frontiers of Fundamental Physics 14 (AMU Marseille

Embracing <i>n</i>-ary Relations in Network Science

Most network scientists restrict their attention to relations between pairs of things, even though most complex systems have structures and dynamics determined by n-ary relation where n is greater than two. Various examples are given to illustrate this. The basic mathematical structures allowing more than two vertices have existed for more than half a century, including hypergraphs and simplicial complexes. To these can be added hypernetworks which, like multiplex networks, allow many relations to be defined on the vertices. Furthermore, hypersimplices provide an essential formalism for representing multilevel part-whole and taxonomic structures for integrating the dynamics of systems between levels. Graphs, hypergraphs, networks, simplicial complex, multiplex network and hypernetworks form a coherent whole from which, for any particular application, the scientist can select the most suitable

Recent progress on the combinatorial diameter of polytopes and simplicial complexes

The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25% is known. This paper reviews several recent attempts and progress on the question. Some work in the world of polyhedra or (more often) bounded polytopes, but some try to shed light on the question by generalizing it to simplicial complexes. In particular, we include here our recent and previously unpublished proof that the maximum diameter of arbitrary simplicial complexes is in $n^{Theta(d)}$ and we summarize the main ideas in the polymath 3 project, a web-based collective effort trying to prove an upper bound of type nd for the diameters of polyhedra and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter of simplicial complexes and abstractions of them, in preparation

Group field theories for all loop quantum gravity

Group field theories represent a 2nd quantized reformulation of the loop quantum gravity state space and a completion of the spin foam formalism. States of the canonical theory, in the traditional continuum setting, have support on graphs of arbitrary valence. On the other hand, group field theories have usually been defined in a simplicial context, thus dealing with a restricted set of graphs. In this paper, we generalize the combinatorics of group field theories to cover all the loop quantum gravity state space. As an explicit example, we describe the GFT formulation of the KKL spin foam model, as well as a particular modified version. We show that the use of tensor model tools allows for the most effective construction. In order to clarify the mathematical basis of our construction and of the formalisms with which we deal, we also give an exhaustive description of the combinatorial structures entering spin foam models and group field theories, both at the level of the boundary states and of the quantum amplitudes.Comment: version published in New Journal of Physic