Tensorial methods and renormalization in Group Field Theories by Sylvain Carrozza

This week I am going to look at a the PhD thesis, Tensorial methods and renormalization in Group Field Theories by  Sylvain Carrozza .

The thesis looks at the two main ways of understanding the construction of GFT models. One way stems from the quantization program for quantum gravity, in the form of loop quantum gravity and spin foam models. Here GFTs are generating functionals for spin foam amplitudes, in the same way as quantum field theories are generating functionals for Feynman amplitudes. They complete the definition of spin foam models by assigning canonical weights to the different foams contributing to a same transition between boundary
states i.e. spin networks. See the post

A second route for GFTs is given by  discrete approaches to quantum gravity. Starting from  matrix models, which allowed us to define random two dimensional surfaces, and so achieve a quantization of two-dimensional quantum gravity. The natural extensions of matrix models are tensor models. From this perspective GFTs appear as enriched tensor models, which allow to the definition of  finer notions of discrete quantum geometries such as the emergence of the continuum.

The thesis then looks at recent aspects of GFTs and tensor models, particularly those following the introduction of colored models. The main results and tools include the combinatorial and topological properties of coloured graphs.

The thesis has two main results, the first set of results concerns the so-called 1/N expansion of topological GFT models. This applies to GFTs with cut-off, given by the parameter N, in which a particular scaling of the coupling constant allows them to reach an asymptotic many-particle regime at large N. The second set of results concerns full-fledge renormalization. Tensorial Group Field Theories (TGFTs), are refined versions of the cut-off models with new non trivial propagators. They have a built-in notion of scale, which generates a well-defined renormalization group flow, and gives rise to dynamical versions of the 1/N expansions.

The thesis looks at a renormalizable TGFT based on the group SU(2) in three dimensions, and incorporating the closure constraint of spin foam models. This TGFT can be considered a field theory realization of the original Boulatov model for three-dimensional
quantum gravity .

Let’s look at the relationship between  the quantum tetrahedron, GFT and the Boulatov model. If the GFT field ϕ is assumed to represent an elementary building block of geometry, then the geometric data should refer to this building block.The Boulatov model generates Ponzano-Regge amplitudes. In its simplicial version, the boundary states of the Ponzano-Regge model are labeled byclosed graphs with three-valent vertices, whose analogue in the field theory formalism are convolutions of the fields ϕ. Following general QFT procedure, we encode the boundary states of the model into functionals of  a single scalar field ϕ. Using the Boulatov model as an example ϕ(g1, g2, g3) is to be interpreted as a flat triangle, and the variables gi label its edges. It is the role of the constraint to introduce an SU(2) flat discrete connection at the level of the amplitudes, encoded in the elementary line holonomies hℓ. The natural interpretation of the variable gi is as the holonomy from a reference point inside the triangle, to the center of the edge i. Thanks to the flatness assumption, this holonomy is independent of the path one chooses to compute gi. The constraint  encodes the freedom in the choice of reference point. From the discrete geometric perspective, the Boulatov model can therefore naturally be called a second quantization of a flat triangle: the GFT field ϕ is the wave-function of a quantized flat triangle, and the path integral provides an interacting theory for such quantum geometric degrees of freedom.


In four dimensions, the correspondence between group and Lie algebra representation can also be put to use. There, the GFT field represents a quantum tetrahedron, and Lie algebra elements correspond to bivectors associated to its boundary triangles. In addition to the closure constraint – equivalent to the Gauss constraint in group space, additional geometricity conditions have to be imposed to guarantee that the bivectors are built from edge vectors of a geometric tetrahedron. These additional constraints are nothing but the simplicity constraints, and non-commutative δ-functions can be used to implement them.

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