Summary of Remarks by Aeyal M. Gross

Group field theory as the second quantization of loop quantum gravity

“…Each graph-based Hilbert space H γ can be given in the connection representation as an L 2 space of cylindrical functions over a group manifold G 4 (usually, SU (2)) or, equivalently, in the flux representation [21,22] as non-commutative functions on the Lie algebra of the same group. Moreover, each graph-based state is assumed to be gauge invariant; that is, one imposes a gauge invariance under simultaneous translation (left or right, depending on the orientation of the edges) of the group elements associated to edges of the graph, by a single group element associated to the vertex to which the same edges are incident; in the flux representation, the same condition is imposed by including in the definition of the states a non-commutative delta function for each vertex, imposing that the fluxes associated to the incident edges sum to zero 5 . This means that there exist a realization of the graph-based Hilbert space as H γ = L 2 G E /G V .…”

“…INTRODUCTIONGroup field theories [1,2] are a generalization of matrix models and an enrichment of tensor models [3] through the addition of group-theoretic data interpreted as pre-geometric data, 'seeds' of the continuum geometry that these models should generate in a continuum approximation. Indeed, they have been first introduced [4] in parallel to tensor models [5] in the early 90's in the context of topological field theories and in particular to give a tensorial formulation to lattice models like the Ponzano-Regge model and its 4d generalization. The Ponzano-Regge model and its higher dimensional extensions were linked, soon afterwards [6], to canonical loop quantum gravity [7], in that their boundary states were of the same type: spin networks.…”

Renormalization of a SU(2) Tensorial Group Field Theory in Three Dimensions

“…The Feynman diagrams of GFTs are cellular complexes, and the perturbative GFT dynamics is defined by the sum over them, in principle extended to include arbitrary topologies. Recently, work on (colored) tensor models [11,12,13], generalizing matrix models to define a perturbative sum over cellular complexes of arbitrary dimension, have led to a detailed understanding of the combinatorial features, statistical properties and universality aspects of such sums. The progress has been remarkable, leading for example to: 1) the definition of a large-N expansion [14] (where N is the size of the tensor index set), and the identification of the dominant configurations in this expansion, which turn out to be special types of spherical complexes called "melons"; 2) the proof that random un-symmetric rank-d tensors have natural polynomial interactions based on U (N ) ⊗d invariance 1 .…”

Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions

“…In the simplest, purely combinatorial models of this type, referred to as tensor models and first introduced in the early 90's [15], the indices of the tensors take value in finite sets of dimension N . More structure to quantum states, action and dynamical amplitudes is the result of endowing the tensors with more interesting domain spaces.…”