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 \int \mathrm{dh} the integral with respect to the Haar measure of the group.

In 3 dimensions we introduce two fields, {\stackrel{\u0304}{\psi}}^{i} and *ψ*^{i}, *i*=0,1,2,3 be four couples of complex scalar (or Grassmann) fields over three copies of *G*, {\psi}^{i}:G\times G\times G\to \u2102. The index *i* runs from=0 to *n*+1, where *n* is the number of dimensions, and the *ψ* and \stackrel{\u0304}{\psi} 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 *δ*^{Θ}(*h*) the regularized delta function over *G* with some cutoff *Θ* such that *δ*^{Θ}(*e*) is finite, but diverges when *Θ* goes to infinity. A feasible regularization is given, for instance for the group *G*=*SU*(2), by

{\delta}^{\Theta}\left(h\right)=\sum _{j=0}^{\Theta}(2j+1){\chi}^{j}\left(h\right).

(1)

where *χ*^{j}(*h*) is the character of *h* in the representation *j*. The path integral for the colored Boulatov model over *G* is:

\begin{array}{lcr}Z(\lambda ,\stackrel{\u0304}{\lambda})& =& {e}^{-F(\lambda ,\stackrel{\u0304}{\lambda})}\\ =& \int \prod _{i=0}^{4}d{\mu}_{P}({\stackrel{\u0304}{\psi}}^{i},{\psi}^{i})\phantom{\rule{2.77695pt}{0ex}}{e}^{-{S}^{\mathrm{int}}({\stackrel{\u0304}{\psi}}^{i},{\psi}^{i})}\phantom{\rule{2.77695pt}{0ex}},\end{array}

(2)

where the Gaussian measure *P* is chosen such that:

\int \prod _{i=0}^{4}d{\mu}_{P}({\stackrel{\u0304}{\psi}}^{i},{\psi}^{i})=1\phantom{\rule{1em}{0ex}},

(3)

and:

\begin{array}{l}{P}_{{h}_{0}{h}_{1}{h}_{2};{h}_{0}^{\u2033}{h}_{1}^{\u2033}{h}_{2}^{\u2033}}=\phantom{\rule{2em}{0ex}}\\ =\int \phantom{\rule{0.3em}{0ex}}d{\mu}_{P}({\stackrel{\u0304}{\psi}}^{i},{\psi}^{i})\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0304}{\psi}}_{{h}_{0}{h}_{1}{h}_{2}}^{i}{\psi}_{{h}_{0}^{\u2033}{h}_{1}^{\u2033}{h}_{2}^{\u2033}}^{i}=\phantom{\rule{2em}{0ex}}\\ =\int \phantom{\rule{0.3em}{0ex}}\mathrm{dh}\phantom{\rule{2.77695pt}{0ex}}{\delta}^{\Theta}\left({h}_{0}h{\left({h}_{0}^{\u2033}\right)}^{-1}\right){\delta}^{\Theta}\left({h}_{1}h{\left({h}_{1}^{\u2033}\right)}^{-1}\right){\delta}^{\Theta}\left({h}_{2}h{\left({h}_{2}^{\u2033}\right)}^{-1}\right),\phantom{\rule{2em}{0ex}}\end{array}

(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:

\begin{array}{ll}{S}^{\mathrm{int}}& =\frac{\lambda}{\sqrt{{\delta}^{\Theta}\left(e\right)}}\int {\left(\mathrm{dh}\right)}^{6}{\psi}^{0}{\psi}^{1}{\psi}^{2}{\psi}^{3}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\frac{\stackrel{\u0304}{\lambda}}{\sqrt{{\delta}^{\Theta}\left(e\right)}}\int {\left(\mathrm{dh}\right)}^{6}{\stackrel{\u0304}{\psi}}^{0}{\stackrel{\u0304}{\psi}}^{1}{\stackrel{\u0304}{\psi}}^{2}{\stackrel{\u0304}{\psi}}^{3}\phantom{\rule{2em}{0ex}}\end{array}

(5)

where we omitted the internal structure of the group elements of the fields *ψ*^{i} and {\stackrel{\u0304}{\psi}}^{i}. 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 \stackrel{\u0304}{\psi}’s. Thus any line coming out of a cGFT vertex has a color *i*.

The group elements *h*_{ij} 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!), {\mathcal{\mathcal{F}}}_{\Gamma}, as any closed strand in the Feynman graph of a GFT. Thus a generic vacuum Feynman amplitude of the theory can be written as:

\mathcal{\ud49c}=\frac{{\left(\lambda \stackrel{\u0304}{\lambda}\right)}^{\frac{n}{2}}}{{\left[{\delta}^{N}\right(e\left)\right]}^{\frac{n}{2}}}\int \prod _{l\in \Gamma}d{h}_{l}\prod _{f\in {\mathcal{F}}_{\Gamma}}\underset{f}{\overset{\Theta}{\delta}}\left(\overrightarrow{\prod _{{l}_{0}\in f}}\underset{{l}_{0}}{\overset{\sigma ({l}_{0},f)}{h}}\right),

(6)

where *l*_{0} is a line associated to a face *f* and *σ*(*l*_{0},*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 {\mathcal{\mathcal{B}}}_{{i}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{i}_{k}} 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).