In this section we introduce the colored Boulatov model Boulatov ([1992]) and Gurau ([2009]). Let us consider a compact Lie group H, denote h its elements, e the unit element, and the integral with respect to the Haar measure of the group.
In 3 dimensions we introduce two fields, and ψi, i=0,1,2,3 be four couples of complex scalar (or Grassmann) fields over three copies of G, . The index i runs from=0 to n+1, where n is the number of dimensions, and the ψ and are functions of n copies of the group. In the fermionic version of the theory the indices i can be seen as the dependence of the field from a (global) gauge group SU(N), where N=n + 1. We denote δΘ(h) the regularized delta function over G with some cutoff Θ such that δΘ(e) is finite, but diverges when Θ goes to infinity. A feasible regularization is given, for instance for the group G=SU(2), by
(1)
where χj(h) is the character of h in the representation j. The path integral for the colored Boulatov model over G is:
(2)
where the Gaussian measure P is chosen such that:
(3)
and:
(4)
The fermionic colored model has two types interactions, a “clockwise” and an “anti-clockwise”, and one is obtained from the other one by conjugation in the internal group color SU(N), where N is 4 in 3 dimensions, one for each face of the 3-simplex a. For convenience we denote ψ(h,p,q)=ψ
hpq
. Invariance under global rotations in the internal color group require at least two interactions:
(5)
where we omitted the internal structure of the group elements of the fields ψi and . In order to make the notation clearer (already the orientation of the colors is sufficient to distinguish the two vertices), we call “red” the vertex involving the ψ’s and “black” the one involving the ’s. Thus any line coming out of a cGFT vertex has a color i.
The group elements hij in eq. (3) are associated to the propagators (represented as solid lines), and glue two vertices with opposite orientation. The vertex can be seen as the dual of a tetrahedron and its lines represent the triangles which form the tetrahedron. Each propagators is decomposed into three parallel strands which are associated to the three arguments of the fields, i.e. the 1-dimensional elements of the 1-skeleton of the tetrahedron which bound every face. These are associated to the edges of the tetrahedron. A colored line represents the gluing of two tetrahedra (of opposite orientations) along triangles of the same color as in Figure (1).
It is easy to understand that a cGFT graph can be seen either as a stranded graph (using the vertex and the propagators as depicted in Figure 2) or as a “colored graph” with (colored) solid lines, and two classes of oriented vertices. In this paper we consider only vacuum graphs, i.e. all the vertices of the graphs are 4-valent and we deal only with connected graphs (thus with the logarithm of the partition function (2)). The lines of a vacuum cGFT graph Γ have two natural orientations given by the fact that only vertices of opposite orientations can be glued. It is easy to see that a vacuum cGFT graph must have the same number of black and red vertices. For any graph Γ, we denote n as the number of vertices, l as the lines of Γ, and we define as faces (not to be confused with the faces of the tetrahedron!), , as any closed strand in the Feynman graph of a GFT. Thus a generic vacuum Feynman amplitude of the theory can be written as:
(6)
where l0 is a line associated to a face f and σ(l0,f) is alternatively +1 or -1 depending on the orientation. In the following we will assume that an orientation is fixed. Because of the properties of δ ″ s the orientation does not affect the amplitude. To each colored graph associated to an amplitude of the colored Boulatov model it is possible to associate bubbles by removing all the edges of one color. We call the set of k-bubbles associated to the deletion of n-k colors. In 3-dimensions, for instance, 3-bubbles have 3-colors (surfaces), 2-bubbles have 2 colors (lines) and so on and so forth. Bubbles play a special role in the theory, since they discriminate manifold from pseudo-manifolds (see next section for the same result in the theory of 3-gems).