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Divergences in spinfoam quantum gravity

In this thesis we study the flat model, the main buidling block for the spinfoam approach to quantum gravity, with an emphasis on its divergences. Besides a personal introduction to the problem of quantum gravity, the manuscript consists in two part. In the first one, we establish an exact powercounting formula for the bubble divergences of the flat model, using tools from discrete gauge theory and twisted cohomology. In the second one, we address the issue of spinfoam continuum limit, both from the lattice field theory and the group field theory perspectives. In particular, we put forward a new proof of the Borel summability of the Boulatov-Freidel-Louapre model, with an improved control over the large-spin scaling behaviour. We conclude with an outlook of the renormalization program in spinfoam quantum gravity.

In this preparatory chapter, we introduce the Ponzano-Regge model, give a first account of its bubble divergences, discuss its relationship with three-dimensional quantum gravity, and introduce a cellular generalization coined the flat spinfoam model.

The Ponzano-Regge model
The Ponzano-Regge model was put forward in 1968 as a simplicial model of quantum gravity in three dimensions. Its influence over the later developments of theoretical and mathematical physics cannot be overestimated: the spinfoam and group field theory approaches to quantum gravity, as well as the mathematical theory of state-sum topological invariants, 1 are direct offsprings of the Ponzano-Regge model.

The Ponzano-Regge asymptotic formula
At the roots of these developments is a most surprising result: the Wigner {6j} symbol (a key object in SU(2) recoupling theory [136], with applications in nuclear, atomic and molecular physics) contains, in the large spin asymptotics, the three-dimensional Regge action for a tetrahedron with edge lengths l e = j e + 1/2, where j e are half-integers. Explicitly, if V (l e ) is the volume of a Euclidean tetrahedron with edge lengths l e , and S R (l e ) the corresponding Regge action (1.1), one has in the uniform j e → ∞ limit 2 Moreover, when the spins j i do not define a Euclidean tetrahedron, the {6j} symbol is exponentially suppressed in this limit. Ponzano and Regge conjectured this asymptotic behavior on the basis of numerical simulations [99]. An influential proof thereof, using geometric quantization, was given by Roberts in [111]; see [5,26] for the most recent advances on the semiclassics of Wigner symbols. (Higher orders in the large-spin asymptotic expansion were also investigated using saddle point techniques in my paper [27] and the subsequent [38].) 1 Part of my work with Bonzom has been concerned with the topological features of the (generalized) Ponzano-Regge model, and in particular with its relationship to twisted Reidemeister torsion. This aspect of our work will not be addressed in this thesis. 2 Meaning that the spins are scaled as j e → λj e with a single λ → ∞. In this regime, the "shape" of the tetrahedron, defined by the ratios j e /j e ′ with e = e ′ , is preserved in the limit.
The remarkable character of the asymptotic formula (2.1) lies in the fact that, not only it contains the Regge action S R (hence it is related to three-dimensional gravity), but contains it within a cosine. This is reminiscent of the semiclassical (WKB) approximation of quantum amplitudes (with the only difference that the eikonal exponential e iS , where S is the Hamilton function, is replaced by the cosine. 3 ) In other words, the {6j} symbol appears to be related to semiclassical quantum gravity.
This striking connection between SU(2) representation theory and threedimensional quantum gravity prompted Ponzano and Regge to put forward the first spinfoam model ever written.

A simplicial partition function
Since a {6j} symbol appears to represent a quantum tetrahedron, 4 with quantized edge lengths and Regge semiclassics, it is not unreasonable to surmise that a model of (Riemannian) quantum gravity on a triangulated 3-manifold ∆ can be obtained by • associating one {6j} symbol to each tetrahedron, and • summing over the spins, in the spirit of Feynman's path integral.
Indeed, Ponzano and Regge defined in [99] a partition function Z PR (∆) for the triangulation ∆ by Here, ∆ i denotes the set of i-simplices of ∆, {6j} t is the {6j} symbol based on the spins coloring the 6 edges of the tetrahedron t ∈ ∆ 3 , and χ is a linear function of the spins j e , left unspecified by Ponzano and Regge (except for very special cases) but fixed in [19]. Note that, if ∆ has a boundary, this expression defines a transition amplitude for a two-dimensional Regge geometry with fixed edge lengths.
The key input in this heuristic quantization of three-dimensional geometry is the quantization of edge lengths. This follows from the quantization of irreducible representations (spins) j i of SU (2), and the identification l i = j i + 1/2 suggested by the Ponzano-Regge asymptotic formula (2.1). As pointed out by Rovelli [115], this quantization hypothesis has a compelling justification in loop quantum gravity. We refer to Rovelli's book [119] for more details on this aspect.

Bubble divergences
It did not escape Ponzano and Regge's attention that the multi-series in the definition (2.2) is unlikely to converge for a general triangulation ∆ [99]. Unfortunately, it is difficult to turn this concern into a precise mathematical statement. Even absolute convergence of (2.2) is hard to check, mainly for two reasons.
• First, not all the values of the spins j e actually contribute to the sum, because of the triangle constraints which enter the definition of {6j} symbols. For instance, a necessary condition for the {6j} symbol in (2.1) is that |j 2 − j 3 ≤ j 1 ≤ j 2 + j 3 . Describing the domain D ∆ of admissible spins for a general triangulation ∆ is by no means straightforward.
• Even if D ∆ could be described explicitly, as in the elementary cases studied in [96], we would not know how the {6j} symbols behave there: the Ponzano-Regge asymptotic formula only prescribes its behavior in the homogeneous limit, where all the spins grow at the same rate. 5 This is only a very restricted subdomain of D ∆ .
Because of these difficulties, simple estimates based on a balance between the divergent edge factors (2j e + 1) and the convergent tetrahedra factors V (j e ) −1/2 ∼ j Worse still, even assuming that we can resolve these issues satisfactorily, the most likely outcome is be that the Ponzano-Regge sum (2.2) is not absolutely convergent. One would then ask whether it is perhaps semi-convergent, like the alternating series (−1) n /n. Unfortunately, the complicated oscillation pattern induced by the factors (−1) χ and the Ponzano-Regge cosines cos S R (l i ) + π 4 make this question almost impossible to answer in full generality.
This notwithstanding, one observation made in an attempt by Perez and Rovelli [95] to describe the domain D ∆ deserves to be mentioned. First, consider the dual cell complex ∆ * of the triangulation ∆, whose k-cells are in one-to-one correspondence with the (3 − k)-simplices of ∆. (Since the partition function (2.2) only involves the edges and tetrahedra of ∆, and not its vertices, we can actually restrict to the 2-skeleton of ∆ * .) Within ∆ * , the triangle constraints can be localized on the edges e * , where three spin-colored faces f * meet. A moment of reflection then shows that, for one the spins to be free, namely to have admissible values from zero to infinity, a necessary condition is that there exists a closed set of faces within ∆ * : a bubble.
This insight has two immediate consequences. First, it suggests that the number of bubbles of ∆ * is somehow the driving parameter in the divergence degree of Z PR (∆), if such a thing can be defined. Second, it gives a field-theoretic flavor to the issue of divergences in the Ponzano-Regge model: bubbles are higher-dimensional analog of loops -and we know that loops support the divergences of Feynman amplitudes. This circumstance calls for an interpretation of the Ponzano-Regge model as a "higher categorical" version of perturbative quantum field theory [13].
We shall see in this thesis that both intuitions are relevant.

A discrete gauge theory
So far, the only evidence we have given for the connection between the Ponzano-Regge ansatz and three-dimensional quantum gravity is the Ponzano-Regge asymptotic formula. There is however another argument to this effect, relying on the Palatini-Cartan formulation of general relativity. The latter sheds a new light on the Ponzano-Regge model: it is a discrete gauge theory. This new perspective turns out to be the key for understanding its bubble divergences, as we will see in the second part of this thesis.

Palatini-Cartan gravity
Recall that, in a "first order" formulation of (Riemannian) three-dimensional general relativity, the gravitational field can be represented by a pair of fields (e, ω), where ω is an SU(2) connection 1-form on spacetime M, and e is an su(2)-valued 1-form over M, or equivalently a triad field on M. 6 The so-called Palatini-Cartan 7 action is then [14] S PC (e, ω) : Here the bracket is the Killing form in the Lie algebra su(2). It is not hard to see that the Euler-Lagrange equations for this action, namely F (ω) = 0 and d ω (e) = 0, (2.4) are indeed the vacuum Einstein equation R ab = 0 in disguise: the first states that ω is flat, and the second that it is the unique spin connection associated to e, see [14]. The Palatini-Cartan action is an instance of the BF models considered by Horowicz [74], defined in dimension d by the analogue of (2.3) with e replaced by a (d − 2)-form B. Like the latter, it is invariant under an extended gauge group which, besides the diffeomorphisms and SU(2) gauge transformations, includes a shift symmetry generated by e → e + d ω λ, (2.5) where λ is an su(2)-valued 0-form over M and ω is a flat connection.
Let us now perform a formal path-integral "quantization" of the Palatini-Cartan, according to Integrating out the e field gives The tilde and quotes here are meant to remind us that no gauge-fixing has been performed for the on-shell gauge symmetry (2.5) [51,53], and so we cannot speak of a genuine quantization in the usual sense of gauge theory [145,74]. For clarity, the operation performed in (2.6)-(2.7) should perhaps be called a "half-quantization" of Palatini-Cartan gravity. Before we move on to explain the connection between this procedure and the Ponzano-Regge model, one more comment is in order. In quantizing a classical theoryà la Feynman, it is not uncommon that the space of histories has to be enlarged with respect to its classical counterpart. For instance, the quantum histories of a free particle contains rough (viz. non-differentiable) paths in addition to the smooth ones considered in the classical theory. Similarly, the quantum histories of a free field include distributional fields as well as smooth ones. Now, the path-integral half-quantization of the Palatini-Cartan action relies on a similar extension of the classical notion of a gravitational history, including torsion and degenerate metrics. Witten has argued [145] that such configurations are essential in three-dimensional quantum gravity (showing that they permit a renormalizable perturbative extension), but later changed his mind [147]. All in all, it is fair to say that the status of torsion and degenerate metrics is quantum gravity remains to be understood. 8

Discrete gauge theory
Let us now consider the half-quantized expression (2.7), and discretize it. To this aim, we must 1. introduce a triangulation ∆ of M, and its dual cell complex ∆ * , and 2. discretize the connection ω (resp. the curvature F ) on the edges (resp. the faces) of the 2-skeleton Γ of ∆ * 3. Choose orientations for the edges and faces of Γ.
Recall that, given this data, discrete gauge theory is formulated in terms of parallel transport operators and holonomies. For a given cell decomposition ∆ * of M, these are defined respectively as assignments of group elements g e ∈ SU(2) to each edge e, and of the products to each face f , with ǫ f e = ±1 depending on the relative orientation of e and f . Thus, a discrete connection A = (g e ) e is an element of SU(2) E and the curvature of A is the family of holonomies H f f defined by (2.8).
Using these definitions, we can make the following ansatz for the discretization of (2.7) on Γ: Here, dg is the Haar measure on SU(2) and Γ i denotes the set of i-cells of Γ. Just like the formal path integral (2.7) described a system of flat SU(2) connections, the expression (2.9) can be seen as the path-integral for a system of flat discrete connections. Now comes the main result of this section: the half-quantized, discretized path integral (2.9) is formally identical to the Ponzano-Regge partition function (2.2). To understand how this comes about, we must recall the basis of the Peter-Weyl theory of Fourier analysis on compact groups.
Let f : G → C be a smooth function on a compact group G, and ρ j : G → GL(V j ) be the sequence of unitary irreducible representations of G, labelled by a discrete index j. 9 For each j, pick a basis (v j m ) m=0,...,dim V j of V j , and define the representative functions Then the Peter-Weyl theorem states that f can be decomposed over the representative functions as When f is a class function, viz. when it is conjugation invariant, this decomposition reduces to where χ j = m D j mm is the character of the representation ρ j . These decompositions extend to square-integrable functions and Schwartz distributions. In particular, one computes that 14) The second ingredient we need to derive the Ponzano-Regge model from the discrete path-integral (2.9) is the recoupling identity Let us now come back to the integral (2.9), and use (2.14) to decompose the delta functions, noting that j is now a spin. This gives Now, since ∆ is the triangulation of a three-dimensional manifold, every edge e of the dual decomposition is shared by exactly three faces. Therefore the recoupling identity (2.15) can be applied along each edge e. Associating the vectors |ι and ι| to the source and target of each e, this results in a contraction pattern whereby four invariant vectors |ι are contracted at each vertex of ∆ * . Examination of this contraction pattern shows that the corresponding invariant is nothing but the Wigner {6j} symbol. All in all, we find that This result clearly vindicates the relevance of the Ponzano-Regge for quantum gravity. But it also provides us with a new interpretation of the bubble divergences discussed in sec. 2.1.3 [53]: they are the discrete remnants of the (non-compact) shift symmetry of the Palatini-Cartan action. Building on this insight, these authors developed a regularization scheme based on a tentative gauge-fixing of this symmetry. It appears, however, that their ansatz is successful only in limited cases. Recent work by Barrett and Naish-Guzman [19], and also by Bonzom and myself [28], has clarified the reasons of this failure and the way to overcome it.

Boulatov's group field theory
The presentation of the Ponzano-Regge model as a discrete gauge theory, with group elements instead of spins as basic variables, has another interesting payoff: it puts us on the right track to understand its bubble divergences as genuine field-theoretic divergences.
To proceed in this direction, let us follow Boulatov [31] and introduce an auxiliary field φ, defined over the configuration space SU(2) ×SU(2) ×SU(2) and invariant under the right diagonal action of SU(2) and cyclic permutations c of its arguments, i.e.: Let us assume furthermore that the "dynamics" of this field is given by the non-local action where dg i denotes the SU(2) Haar measure. Note that, in the quartic term, the six integration variables are repeated twice, following the pattern of the edges of a tetrahedron.
(The factor 8 is only for simpler final formulas.) Consider now the perturbative quantization of this model, by mean of the formal partition function Examination of the corresponding Feynman rules reveals that the propagator is threestranded, with integral kernel Likewise, the interaction vertex can be written a product of six delta functions, matching twelve group elements pair-wise, as in fig. 2.1: For a given stranded graph Γ, these Feynman rules correspond to a contraction pattern which gives, after integration over all h variables, the Feynman amplitude (for a vacuum diagram, viz. with no external leg) where V is the number of vertices of the stranded graph Γ, e denotes an edge Γ, and f a face, i.e. a closed strand. Up to a multiplicative constant, this is nothing but the Ponzano-Regge partition function (2.9) evaluated on the 2-complex defined by the edges and faces of the stranded Feynman graph. If we interpret furthermore this 2-complex as dual to a three-dimensional space, as in [35], the connection with the Ponzano-Regge model becomes transparent: Boulatov's model is a generating function for Ponzano-Regge amplitudes. (Note however that not all such 2-complexes define a three-dimensional manifold : there can be point-like singularities, yielding what is known as a pseudo-manifold. We refer the interested reader to [65,128,67] for a discussion on this point.)

The flat spinfoam model
The upshot of the above discussion of the Ponzano-Regge model can be summarized as follows: 1. Although originally defined on a triangulation ∆ of the spacetime manifold, the structure of the partition function (and in particular its divergence) is better understood in terms of the dual cell complex ∆ * , and in fact of the 2-skeleton Γ of ∆ * .
2. The analytic structure of the Ponzano-Regge partition function is more transparent when written as a group integral (2.9); its gauge-theoretic underpinning, and in particular its connection to the Palatini-Cartan formulation of gravity, its then manifest.
This motivates a generalization of the Ponzano-Regge model defined on more general 2-complexes: a kind of cellular gauge theory, with no reference to a particular spacetime manifold. This is concomitant with Boulatov's approach, which, as we already mentioned, departs from the strict setup of three-dimensional manifolds.

Foams
Before we introduce this generalization, let us spell out what we mean by the expression "2-complex" above. The most general possible definition one can think of is a finite, connected twodimensional CW complex. Recall that a d-dimensional CW-complex X is the last of a nested sequence of topological spaces X 1 ⊂ X 2 ⊂ · · · ⊂ X d = X defined inductively as follows. The 0-skeleton X 0 is just a discrete set of points. For each integer k > 0, the k-skeleton X k is the result of attaching a set of k-dimensional balls B k to X k−1 by gluing maps of the form σ : ∂B k → X k−1 . For each k, the subspace X k is called the k-skeleton of X, and the interiors of the balls B k attached to X k−1 are the k-cells of X.
Contrary to what one might expect (perhaps on the basis of the easy classification of two-dimensional manifolds), two-dimensional CW complexes are very subtle objects, about which several outstanding conjectures remain open. 10 In order to stay away from such mathematical difficulties, in this thesis we will stick to a more restricted context, where the gluing maps are themselves cellular : we will assume that each ball B k has a cellular decomposition such that the (k − 1)-cells on the boundary of B k are glued onto (k − 1)-cells of X k−1 . This condition introduces a form of rigidity in the gluing patterns, thereby removing most of the topological difficulties associated to general CW complexes.
The most useful examples of such CW complexes can be found in the polyhedral category, where the balls B k are in fact convex polyhedra. Explicitly, a polyhedral complex, or piecewise-linear cell complex, is a CW complex where each ball B k is in fact a k-dimensional convex polytope P k in some R n , and the gluing maps σ : ∂P k → X k−1 are cellular homeomorphisms onto their images. When all the polytopes are simplices, one speaks of a ∆-complex [70].
One reason which makes two-dimensional polyhedral complexes, from now on called foams, particularly convenient to deal with is that their structure can be described purely combinatorially: attaching a polygon onto the 1-skeleton X 1 amounts to choosing a cycle in X 1 , i.e. a finite sequence of vertices (0-cells) such that, from each of its vertices there is an edge (1-cell) to the next vertex in the sequence, and the start and end vertices are the same. In other words, a two-dimensional polyhedral complex Γ can be equivalently defined as a triple Γ = (V Γ , E Γ , F Γ ) with V Γ a finite set of vertices, E Γ a set of ordered pairs of vertices e = (v, v ′ ), or edges, and F Γ a finite set of cycles, or faces. This data uniquely defines a two-dimensional polyhedral complex up to homeomorphism.
Note that this definition includes the stranded Feynman diagrams of the Boulatov model, but is also much more general: unlike the former, the number of faces adjacent to any edge can be arbitrary. It is in this sense that the flat spinfoam model, being defined on general foams, generalizes the Ponzano-Regge model to "arbitrary dimensions".
Finally, observe that there is a notion of boundary for foams. Indeed, let us call the edges of E Γ appearing exactly once in only one face its links, and the other ones its interior edges. Similarly, let us call the vertices appearing exactly once in an interior edge its nodes, and the other ones its interior vertices. The sets of nodes and links of a foam Γ generally do not form a graph, but when they do, and moreover the orientation of each link matches the one induced by the unique face passing through it, we say that The flat spinfoam model Γ is a proper foam. We then define the boundary ∂Γ as the subcomplex of Γ defined by those faces of Γ which contain at least one link. The underlying graph of ∂Γ is the boundary graph of Γ. In this language, foams with the same boundary graphs can still have different boundaries, because the boundary faces can be different.

Definition of the model
We are now in the position to define the flat spinfoam model. Let Γ be a foam, and G be a compact Lie group. Denote A(Γ, G) := G E the space of discrete G-connections A = (g e ) e∈Γ 1 over Γ, and H = (H f f ∈Γ 2 its curvature as in (2.8). The flat model is then defined formally by the expression where dA = e∈Γ 1 dg e is the Haar measure on A(Γ, G). When Γ is the 2-skeleton of the cell complex dual to a triangulated 3-manifold, this expression of course coincides with the Ponzano-Regge partition function. But this is only a very special case of (2.24): there are many foams which cannot be constructed in this way. Note in particular that, when there are more than three faces adjacent to each edge, the Peter-Weyl transform of (2.24) does not consist of {6j}-symbols. For instance, if Γ is the 2-skeleton of the dual cell complex of a triangulated 4-manifold, these are replaced by {15j}-symbols: this is the so-called Ooguri model [91].
Another particular case of the flat spinfoam model is when Γ is the cellular decomposition of a closed, orientable surface: the partition function (2.24) then corresponds to the weak coupling limit of two-dimensional Yang-Mills theory, considered notably by Witten in [146]. When the genus of Γ is greater or equal to 2, extensive studies of (2.24) after [146] have showed that Z(Γ) is finite. 11 We will re-derive this result in sec. 4.3.2.

Heat kernel regularization
Just like its Ponzano-Regge specialization, the flat spinfoam model is ill-defined. In the definition (2.24), this is manifest from the fact that delta distributions supported at coincident points are multiplied -a famously illicit operation.
Very roughly speaking, we can say that a troublesome situation will arise every time the number of delta functions (F , the number of faces of Γ) does not match the number of integrals in (2.24) (E, the number of edges). Thus, when F > E, we might expect that the a possible regularization of Z(Γ, G) consists in removing some redundant delta functions in (2.24). Indeed, this is the idea followed by Freidel and Louapre in [51,53], and later by Barrett and Naish-Guzman [19]. We will comment on this approach in sec. 3.3.
For now, we will content ourselves with a more straightforward regularization, consisting simply in replacing the delta functions by smooth functions, peaked on the unit ½ ∈ G. One way to construct such functions is by letting the delta diffuse over G for some small time τ > 0. This gives the so-called heat kernel 12 K τ , viz. the solution of the heat equation with initial condition lim τ →0 K τ = δ. Here ∆ is the Laplace-Beltrami operator, defined using some bi-invariant metric on G (e.g. the one induced by the Killing form on its Lie algebra, when G is semi-simple).
Using the heat kernel on G, we thus define the regularized flat spinfoam model by This ansatz turns the problem of understanding the bubble divergences of the flat spinfoam model (and in particular of the Ponzano-Regge and Ooguri models) into a well-posed analytical question: what is the τ → 0 asymptotic behavior of Z τ (Γ, G)? Answering this question is the purpose of the next chapters of this thesis.

Homological powercounting of bubble divergences
In this part of the thesis, we address the problem of computing the divergence degree of the regularized flat spinfoam model Z τ (Γ, G) in the τ → 0 limit. The approach we take has two ingredients, detailed in chapters 3 and 4 respectively: 1. study the τ → 0 asymptotics of (2.26) by means of a (generalized) Laplace approximation 2. use (twisted) cellular homology to disentangle the interplay between the topology of Γ and the non-Abelian structure of G within Ω(Γ, G) We will see that the divergence degree is given by the number of "bubbles" of Γ only in very special cases; in general, the combinatorial structure of the foam does not fix Ω(Γ, G) alone -the rôle played by the non-Abelian structure of G is crucial. Applications of our result will be presented in chapter 5.

.1 The amplitude as a Laplace integral
In this section, we define the divergence degree of a foam, and show that, in principle, it can be computed by means of the Laplace method.

Divergence degree
The small-time asymptotic behaviour of the heat kernel on G is well-known, and easy to understand: when τ → 0, a Brownian particle on G has no time to explore the geometry of G and confuses the latter with the Euclidean space T ½ G. In this limit, the heat kernel therefore takes a Euclidean form: where |g| is the Riemannian distance between g and the unit ½ of G. In particular, we playing the rôle of a large-spin cutoff. This motivates the definition of the divergence degree of Γ as the number Ω(Γ, G) such that the limit is finite and non-vanishing. Let us emphasize that we do not assume a priori that Ω(Γ, G) is a multiple of dim G, as would be implied by the notion that to regularize the flat spinfoam model, one should remove redundant delta functions [53].

A Laplace integral
Denote V, E, F the number of vertices, edges and faces of the foam Γ. Using the asymptotics (3.1), we can write the regularized amplitude Z τ (Γ, G) in the τ → 0 limit as the integral The function S : A(Γ, G) → R thus defined is smooth, non-negative, and vanishes on flat connections where H f (A) = ½ for every face f of Γ. This suggests that the integral on the right-hand side of (3.5) can be estimated by means of Laplace's method, which we now review.

Degenerate Laplace asymptotics
Let M be a (closed) Riemannian manifold, dx the canonical volume form on M, and f a smooth function.
Here y * i runs over the minima of f . This formula can be proved by Taylor-expanding f to second order about y * i , and performing the resulting Gaussian integral. It is tempting to try to apply this well-know asymptotic formula to the amplitude (3.5). Unfortunately, one sees easily that the function f defined by (3.6) does have the required property. Indeed, the Morse lemma 1 implies that non-degenerate critical points are isolated. But we know that topologically non-trivial spaces admit a continuum of flat connections, hence that F (Γ, G) consists of degenerate critical points. To deal with this situation, we need a generalized Laplace method.

The generalized Laplace method
Let us consider the case where the critical set of f is a disjoint union of submanifolds of M. Denote C the (possibly disconnected) critical manifold with maximum dimension. Using the metric on M, we can split the tangent bundle of M along C into its tangential and normal sub-bundles, as Let us then define the normal Hessian ∇ 2 ⊥ f of f as the restriction of ∇ 2 f to the normal fibers N y C, and assume that ∇ 2 ⊥ f is non-degenerate (one says that f is a Morse-Bott function). Laplace's formula (3.7) then generalizes to where dy is the induced Riemannian volume form on C. Indeed, the orthogonal decomposition (3.8) implies that, along C, the volume form on M factorizes as where dz is the Riemannian volume form on the fibers normal to C. Using the exponential decay e −f /τ away from C to restrict the integral over M to a tubular neighborhood U of C, in which x ∈ U is parametrized by (y, z) ∈ C × N y C, and then applying Fubini's theorem, we obtain The usual Laplace formula (3.7) can then be applied to the normal integrals NyC dz y e −f (y,zy)/τ ; this gives immediately (3.9).

Tame singularities
What if f is not a Morse-Bott function? Define a singularity of f as a critical point were either C is not a manifold, or the normal Hessian ∇ 2 ⊥ f is degenerate. Following [55], say that the singularities of f tame if for every ǫ-neighborhood U ǫ of the set D of singularities, (Unfortunately, there is no useful sufficient condition on f for this relation to hold; it must be checked explicitly case by case.) Assuming (3.12), we can apply the generalized Laplace method by simply excluding the singularities from the integral: .
In other words, for what concerns the τ → 0 asymptotic behaviour of a Laplace-type integral, tame singularities are like no singularities at all. 2 2 That not all singularities are tame, however, can be seen on the two-dimensional example (3.14) Here, the critical set is the 'cross' {x = 0} ∪ {y = 0}, and has a singularity at (x, y) = (0, 0). A naive application of the generalized Laplace method would give z τ ∝ τ 1/2 as τ → 0, since the critical set is of codimension one except at the origin. However, this is not the correct estimate, which turns out to be z τ ∝ τ 1/2 ln τ . In this case, indeed, integrating along the normal fibers yields a Hessian which is not integrable at the singularity: formally, It can be shown that isolated degenerate critical points always yield an asymptotic behaviour of the form τ a/2 ln b τ with a, b two integers, although computing the power of the logarithm is a difficult task, which requires to expand f to orders higher than two. For extended degenerate critical sets, there is no general result.

A powercounting formula
Let us now come back to the regularized amplitude 16) and assume that S has tame singularities. Applying the generalized Laplace method described above then gives where F 0 (Γ, G) is the set of flat connections F (Γ, G) minus its singularities, and dφ is the induced Riemannian volume form on F 0 (Γ, G). Since the remaining integral does not contain τ anymore, we can read off from (3.17) an explicit for the sought-for divergence degree: Recall that from the formal expression (2.24), we inferred that a foam is potentially divergent if it has more faces than edges, because then some of the delta functions are redundant. This is confirmed by the first term in the above formula. But the latter also shows how to sharpen this naive hunch, namely by studying the geometry of the space of flat connections F (Γ, G). We will address this problem in some detail in the next section. For now, let us ask: is the intuition of Perez and Rovelli [95] and Freidel and Louapre [51,53] that the divergence of a foam can be traced back to the presence of bubbles vindicated by the formula (3.18)?

Bubbles from cellular homology: the simply connected case
The notion of bubble has a precise mathematical definition: it is a 2-cycle in cellular homology (see the appendix for a review of cellular homology). On a 2-complex Γ, the space of 2-cycles coincides with the second homology group H 2 (Γ). Thus, a tentative definition of the "number of bubbles" of Γ is b 2 (Γ) := dim H 2 (Γ), the second Betti number of Γ. Is the divergence degree Ω(Γ, G) related to b 2 (Γ)? Consider the case when the foam is simply connected, i.e. when its fundamental group π 1 (Γ) is trivial. Then we have that • the flat connections on Γ are all gauge-equivalent to the trivial one, hence • the first homology group of Γ vanishes 3 , In this case, the result (6.6) gives therefore Ω(Γ, G) = (dim G)(F − E + V − 1). Furthermore, the equations b 0 (Γ) = 1 (Γ connected) and b 1 (Γ) = 0 (Γ simply connected), together with the Euler-Poincaré theorem, give Hence That is, for simply connected foams, the divergence degree is -indeed -given by the number of bubbles of Γ, times the dimension of G.

Chapter 4
The (co)homology of flat connections Unfortunately, this identity between divergence degree and number of bubbles (second Betti number) breaks down when Γ is not simply connected, because then the term dim F 0 (Γ, G) in the divergence degree intertwines the topological structure of Γ with the non-Abelian structure of G. (In particular, it does not factorize as dim G times a function of Γ as in (3.19).) We will see in this section that, this notwithstanding, the divergence degree can still be seen as counting the number of bubbles -only these are not subsets of Γ. Rather, they are twisted bubbles, in the sense of twisted (co)homology.
To develop this perspective, we need to introduce more concepts from discrete gauge theory. This is the goal of this chapter.

More on discrete gauge theory
Discrete gauge theory has been studied extensively in two dimensions, especially after the seminal works of Atiyah and Bott [11], Goldman [59] and Witten [146]. In this preliminary section, we extend this framework from triangulated 2-manifold to general foams.

Further definitions
In the previous chapter, we called an element of A(Γ, G) := G E a (discrete) Gconnection on Γ, and defined the curvature map H : Let us now complete these definitions.

Flat connections Let us say that
(From now on, we shall write φ for a generic element of F (Γ, G).) Moreover, in differential-geometric parlance we say that a connection A is regular if the differential dH A has maximal rank, i.e.
else we say that A is critical.
Parallel transport Let A = (g e ) e∈Γ 1 be a connection on Γ. If v and w are two vertices of Γ, and γ = (e η 1 1 , . . . , e ηn n ) is an edge-path connecting them, with n i = ±1 to take into account the orientations of the edges, we define the parallel transport operator from v to w by If A is flat, two paths in the same homotopy class yield the same parallel transport operator.
Gauge transformations Connections are by nature acted upon by gauge transformations. In this discrete setup, these correspond to the action of G at the vertices of Γ, according to where h = (h v ) v∈Γ 1 ∈ G V and s(e) and t(e) denote the source and target vertices of e. For each connection A, the relation (4.4) defines a map γ A from the gauge group G := G V to A(Γ, G), the local gauge map. The set of gauge transformations Finally, two connections A and A ′ belonging to the same orbit are called gauge equivalent.
Moduli space The space of gauge orbits of flat connections is a very important object in two-dimensional gauge theory: it is the semiclassical limit of the Yang-Mills phase space. This space is usually called the moduli space of flat connections. Here, we denote it (4.5)

Reducing gauge symmetries
The moduli space of flat connections is best described after gauge transformations have been reduced, in the following sense. Since gauge transformations multiply the elements g e of a connection A = (g e ) e∈Γ 1 independently at each vertex, every connection A is gauge equivalent to one which has g e = ½ on a subset T of Γ 1 touching all the vertices of Γ without forming loops -a spanning tree. In particular, the orbit of a flat connection on Γ is the same as the orbit of a flat connection on Γ/T , and we can show that 1 (4.6) The advantage of this reduction is that the action of gauge transformations on Γ/T is simply the conjugation of each element g e of A ∈ A(Γ/T, G) by a single group element h ∈ G: Understanding the orbits of this action is usually much simpler than that of G V with V > 1. Moreover, since the integrand in Z τ (Γ, G) is gauge-invariant, we can safely replace Γ by the foam Γ/T to compute the amplitude: (4.8) For this reason, hereafter (and unless explicitly stated) we shall only consider foams with one vertex, understanding that the reduction of the gauge symmetry has already been performed.

Representations of the fundamental group
There is another benefit to this reduction of gauge transformations: it provides a very useful algebraic interpretation of the space of flat connections. The latter is based on the one-to-one relationship between foams with a single vertex and group presentations, as follows.
If Γ is a foam with a single vertex, one can immediately read off a presentation of the fundamental group π 1 (Γ) by associating a generator a e to each edge of Γ and one relation per face: Reciprocally, a finite presentation of a group π unambiguously determines a foam Γ. From a single vertex, draw an edge for each generator, and attach the faces according to the relators. 2 Furthermore, the obvious translation of the flatness conditions e∈∂f g ǫ f e e = ½ into relations in a presentation of π 1 (Γ) shows that a flat connection on Γ can be seen as a homomorphism from π 1 (Γ) to G, hence that F (Γ, G) ≃ Hom π 1 (Γ), G . First, an algebraic set consists of several irreducible components, which by definition do not decompose as the union of smaller algebraic sets. Each irreducible component, also called an algebraic variety V, has a well-defined dimension, in spite of the fact that it is not a manifold, viz. contains singular points: dim V is simply the dimension of a suitable neighborhood of any point where it is a manifold.
Second, the singularities of an algebraic variety V can be identified by computing the Zariski tangent space T v V at v ∈ V. This tangent space is defined as the joint kernel of the differentials of all the polynomials which vanish on V. It is naturally a vector space, and we have in general (4.12) The smooth, or non-singular points of V are those saturating this inequality. 3 To get a feel for this definition, one can think of the cross {xy = 0} ⊂ R 2 in the plane considered in sec. 3.2.2. At each point different from (0, 0), there is one tangent direction to the variety: the Zariski tangent space is one-dimensional. But at the origin, there are two independent tangent vectors, spanning a two-dimensional Zariski tangent space: (0, 0) is the unique singular point of the cross. The set of smooth points V 0 of a variety V, its smooth locus, always forms a manifold that is dense 4 in V. For instance, the smooth locus of the cross in R 2 is the (disconnected) union of four half-lines.
In the case of the space of flat connections F (Γ, G), one sees immediately that the smooth locus F 0 (Γ, G) contains all the regular connections φ: the map H is submersive at φ, hence by the inverse function theorem φ has a manifold neighborhood. In fact, the smooth locus is precisely the subset of F (Γ, G) where H has maximal rank, and the Zariski tangent space is just T φ F (Γ, G) = ker dH φ .
Note also that the singularities of F (Γ, G) in the algebraic-geometric sense coincide with the analytic singularities of the function S defined in sec. 3.2.2. To see this, it is enough to check that hence that the degeneracy of ∇ 2 ⊥ S corresponds to points where dim ker dH φ , the dimension of the Zariski tangent space, is larger than expected.
Observe that the smooth locus is stable under the action of G by conjugation: if φ is non-singular and φ ′ is gauge-equivalent to φ, then φ is also non-singular. This fact turns F 0 (Γ, G) into a G-manifold, and allows to describe the quotient M 0 (Γ, G) := F 0 (Γ, G)/G as consisting of a disjoint union of manifolds characterized by their orbit type

Twisted (co)homology
We are now ready to introduce the theory of twisted cohomology. 5

Curvature as a coboundary operator
To understand the relationship between the space of flat connections and cohomology in more general terms than in sec. 3.3.2, it is illuminating to compute explicitly the differential of the curvature map H at φ = ½, the trivial connection. For (4.14) where ∂f denotes the family of edges of Γ lying on the boundary of the face f , and as before ǫ ef is a sign encoding the relative orientation of e and f . Now, this map dH ½ coincides with the first coboundary operator δ 1 in the cellular cohomology C * (Γ, g) of Γ with coefficients in g, the Lie algebra of G (see the appendix for a reminder on cellular cohomology): Here δ 0 vanishes because Γ has a single vertex. This observation provides further insight into the fact that for simply connected foams (which after gauge reduction admit a single flat connection, the trivial one), we have Indeed, from H 1 (Γ) = ker δ 1 / im δ 0 = 0 we get ker δ 1 = ker dH ½ = 0, hence the flat connection ½ is non-degenerate. The standard Laplace method then gives That is, the corank of the curvature map H is the second Betti number of the cochain complex (4.15), hence counts the number of "bubbles with coefficients in g". This explains the homological nature of the divergence degree Ω(Γ, G) for simply connected foams.

Interlude: twisted deRham cohomology
If M is a smooth d-manifold, it is well known that the exterior derivative d i : is also a cochain complex. The corresponding cohomology H * φ M, Ad(P ) is the twisted de Rham cohomology of M. 7

Twisted cellular cohomology
Just like de Rham cohomology, there is a twisted version of cellular cohomology. Here is one possible description of it. (The mathematically-oriented reader can consult Turaev's book [134] for a more algebraic presentation.) Let K be a finite-dimensional CW complex, with cochain groups C i (K), and let (Since the connection is flat, the operator is independent of the chosen path.) Via the adjoint representation of G, these operators act on the cochain spaces C i (K, g). Use them to define the twisted coboundary operators δ i φ : for X ∈ g. Here, [e i+1 β , e i α ] is the incidence number of the pair of cells (e i+1 β , e i α ) (the higher-dimensional analogue of the numbers ǫ ef considered so far).
It can then be checked that δ i+1 φ • δ i φ = 0, and that the φ-twisted cellular cohomology groups are well-defined, i.e. do not depend on the choice of reference vertices. Like the usual cohomology groups, they are homotopy invariants of K; their dimensions b i φ (K, g) := dim H φ (K, g) are the φ-twisted Betti numbers of K.
Let us emphasize that, in general, these are not simply related to the standard Betti numbers b i (K). There is such a simple relationship, however, for the twisted Euler characteristic, which satisfies

Twisted cohomology in the flat model
Twisted cellular cohomology arises naturally in the flat spinfoam model. Thus, a moment of reflection shows that the 0-th twisted coboundary operator δ 0 φ is nothing but the differential of the local gauge map γ φ at the unit, That is to say that the space of 1-coboundaries im δ 0 φ is the tangent space of the gauge orbit of φ, and the space of 0-cocycles ker δ 0 φ is the Lie algebra of its stabilizer ζ(Γ, G) φ . In particular, H 0 φ vanishes if and only if φ is irreducible. Similarly, we can check that Hence, the space of 1-cocycles ker δ 1 φ is the tangent space to F (Γ, G) at φ, and the first cohomology group that φ is regular. 8 Note that, by definition of non-singular connections, the second twisted Betti number b 2 φ has a constant value b 2 0 := min on F 0 (Γ, G), given by Comparing with the result (3.18), we see that In words: (provided the singularities of the curvature map are tame) the divergence degree of a foam is given by the value of its second twisted Betti number on nonsingular connections. This is the main result of this chapter, and the precise sense in which divergences in the flat model are counted by "bubbles".

Examples
Let us now illustrate this result with several examples of interest.

Abelian structure group
Consider first the case where the structure group G is Abelian. In this case the curvature map H is a group homomorphism G E → G F , hence • the space of flat connections F (Γ, G) = ker H is a Lie subgroup of A(Γ, G) = G E , and in particular has no singularity, • the relation (4.14) holds for any flat connection φ ∈ F (Γ, G), i.e.
where X = (X e ) e∈Γ 1 ∈ T φ G E and µ : T φ G → g is the Maurer-Cartan form on G.
In other words, Both observations imply that b 2 φ = corank δ 1 φ is constant on F (Γ, G), equal to its value on the trivial connection ½. But we saw in the previous section that the twisted cochain complex of Γ at φ = ½ is just the cellular cochain complex of Γ with coefficients in g, hence Thus, for Abelian groups like for simply connected foams, the relevant "bubbles" are the cellular 2-cycles of Γ. The initial intuition that, in the flat spinfoam model, powercounting amounts to the chasing of "closed surfaces" within Γ is correct is this case too.

Surface groups with G = SU(2)
Assume now that Γ is the cellular decomposition Γ k of a (closed, orientable) surface with genus k, and that G = SU (2). This is the case studied by Witten in the context of two-dimensional Yang-Mills theory [146], and by Goldman [59] and Atiyah and Bott [11] from the mathematical perspective.
As well-known, Γ k can always be reduced to a flower graph with 2k edges supporting only one face. This corresponds to the following presentation of the fundamental group: Here the square brackets denote the group commutator in π 1 (Γ g ), [a, b] := aba −1 b −1 .
The space F k := F (Γ k , SU(2)) of discrete SU(2)-connections on Γ k can therefore be described as Geometrically, the group elements a i and b i can be thought of as rotations with the same axis of rotationn i ∈ S 2 , but a i and b j can have different axes for i = j. Computing the action of SU(2) on F k by conjugation, we find that a gauge transformation is a rotation of these axes.
It is not difficult to convince oneself that there are singular connections in F k , as the differential of the flatness relation is clearly not of constant rank. Yet, one circumstance greatly simplifies the description of F k with respect to the general case: its singularities are classified by the orbit type ζ φ . This follows from Poincaré duality, which states that H 2 φ ≃ H 0 φ . Hence, in this case a flat connection φ is non-singular (H 2 φ = 0) if and only if it is irreducible (H 0 φ = 0). To make this observation more concrete, we now describe explicitly the Zariski tangent space ker dH φ , for φ ∈ F k . We treat the cases k = 1 (the torus) and k ≥ 2 separately (the sphere, which is simply connected, was already treated in sec. 4

.2.1).
Higher genera (k ≥ 2) Identifying the tangent space T φ A(Γ, G) with its Lie algebra g E by means of the Maurer-Cartan form, we have in this case for algebra elements u := (u i ) i=1,...,k ∈ g E and v := (v i ) i=1,...,k ∈ g E . Considering u i and v i as vectors in three-dimensional space, the adjoint action Ad a i is a rotation around the axisn i . Thus, for each i, it is natural to decompose u i and v i into orthogonal and parallel components to the directionn i , viz.
and similarly for v i . All parallel components u i drop out from (4.35 Thus, we find that the regularized amplitude has a finite limit as τ → 0. This is consistent with Witten's formula [146] lim τ →0 We refer to the work of Sengupta [127] for details on why singularities are tame in this case. Let us also say a word about the singular connections. In this case, they are of two kinds: • Abelian connections. All group elements have the same axisn. These form a submanifold of dimension 2k + 2, on which rk δ 1 φ = 2. Such flat connections have a U(1) stabilizer, corresponding to rotations aroundn. These observations are consistent Poincaré duality: the more a flat connection is reducible, the more it is singular.
The torus (k = 1) The case of the torus is more subtle, because all flat connections are critical, and also more interesting, because Ω Γ k=1 , SU (2) is not a multiple of dim G = 3.
To see this, recall first that the curvature map in this case is a single group commutator, (a, b) → [a, b] = aba −1 b −1 , hence flat connections φ consist in pairs of rotations with angles 0 ≤ ψ a , ψ b < π and the same axisn ∈ S 2 . Moreover, the coboundary operator δ 1 φ (again pulled back to the Lie algebra g 2 ) reads As before, let us distinguish between the Abelian and central connections: and therefore fixes 2 real components of (u a , u b ) ∈ R 3 × R 3 . It expresses the condition that the connection remains flat under variations of the directions of a and b. It follows that rk δ 1 φ = 2, hence that b 2 φ = 3 − 2 = 1. This confirms that the torus is divergent, as suggested by (4.40).

A foam with wild singularities
We close this section by an example of a foam presenting wild singularities, still with G = SU(2). Consider the group 9  This set has two irreducible components.
• If h is in the center of SU(2), i.e. h = ±½, then, a, b can be arbitrary: These are the irreducible flat connections. 9 We do not know if this is the fundamental group of a 3-manifold. (But it is that of a 4-manifold, like any other finitely generated group.) • If h is not in the center, then a, b and h have to lie in a common U(1) subgroup of SU (2). These are the Abelian representations. If we write a generic element g = exp(iψn · σ), withn ∈ S 2 the direction of the rotation and 0 ≤ ψ ≤ π its class angle, then Quite obviously, F irred is of dimension 6, while F red is 5-dimensional. 10 It follows that b 2 irred (Γ, SU(2)) = 3, and b 2 red (Γ, SU(2)) = 2. Applying Laplace's method as in sec. 3.3 therefore suggests that (2)). (4.51) In other words, it suggests that the divergence of the amplitude is controlled by irreducible connections, and Ω(Γ, SU(2)) = 3. However, it so happens that reducible connections actually spoil this result, because A ′ red (Γ, SU(2)) = ∞. This was showed in [29] by direct computation of the contribution of F red to the Laplace asymptotics of (4.52) Explicitly, we showed in this reference that The factor Vol(S 2 ) corresponds to the integral over gauge orbits, which here correspond to rotations of the common directionn h of the three group elements, the class angles remaining fixed; the class angles ψ a , ψ b and ψ h on the other hand parametrize the moduli space. This integral is manifestly divergent. This is strictly analogous to the wild singularity discussed in sec. 3.2.2, and shows that the divergence degree of Γ cannot be estimated by Laplace's method.

Applications of homological powercounting
We found in the previous chapter that, except for a caveat related to possibly wild singularities in the space of flat connections F (Γ, G), the divergence degree of a foam Γ is the second twisted Betti number of Γ, evaluated on the non-singular elements of F (Γ, G). We commented on the interpretation of this result as the actual realization of the idea that the divergences of Z(Γ, G) can be traced back to the presence of "bubbles". In this chapter, we discuss another aspect of this result: the clarification, unification and generalization of all the earlier results on divergences in the Ponzano-Regge and Ooguri models.

Sorting out topology from cell structure
One such clarification concerns the extent to which the divergence degree of a foam is determined by the topology of spacetime. We now derive a decomposition of Ω(Γ, G) which completely settles this question.

A useful decomposition
Part of the interest of the powercounting result obtained in the previous chapter is that it holds for arbitrary foams. Suppose, however, that Γ is actually the 2-skeleton of a cell decomposition K M of a n-pseudomanifold M. 1 Then the result Ω(Γ, G) = b 2 0 can be sharpened considerably.
Consider indeed the twisted Euler-Poincaré formula for the 2-skeleton Γ, From the discussion in sec. 4.2.4, we know that b 0 φ is the dimension of the stabilizer ζ φ , and that b 1 φ is the local dimension of the moduli space M(Γ, G). Combining this with (4.23), we get where ζ 0 is the common stabilizer of non-singular flat connections φ ∈ F 0 (Γ, G). Moreover, by definition of the Euler characteristic, we have This fact makes the computation of the divergence degree much easier in the form (5.5): in practice, it reduces the problem of computing I(M, G) for a given (pseudo)manifold M to that of identifying the representation variety of its fundamental group Hom π 1 (M), G and computing the the orbits of φ ∈ Hom π 1 (M), G under the conjugation action. Since these only depend on π 1 (M), any presentation of π 1 (M) can be used to that effect; in particular, from the standard presentation (4.9) obtained by retracting a tree T in Γ, as in sec. 4.1.2.
One more comment about the computation of I(M, G). It may happen that the character variety M(Γ, G) = Hom π 1 (Γ), G /G has several irreducible components. These lead to different values of I(M, G); in τ → 0 limit, the leading behaviour of the amplitude Z τ (Γ, G) is of course given by the largest one. This phenomenon will be illustrated in sec. 5.3.

Comparison with previous results
Prior to our introduction of twisted cohomology in the discussion of powercounting in the flat model, several results had been presented in the literature, mostly within the context of the Boulatov and Ooguri models, where foams are "simplicial": in dimension d, exactly (d + 1) edges (resp. d faces) are incident on each vertex (resp. edge) of Γ, corresponding to the number of faces of a d-simplex (resp. (d − 1)-simplex).
These results fall in four classes: • Powercounting from Pachner moves. If one assumes that Γ arises from a triangulation T M of a closed n-manifold M, as the 2-skeleton of its dual cell complex, one can see that Pachner moves on T M generate divergences in the partition function. Ponzano and Regge [99], Boulatov [31] and Ooguri [91] relied on this observation to relate the divergence degree to the combinatorics of Γ.
• Powercounting from bubble counting. A different approach to the divergences of the flat spinfoam model was initiated by Perez and Rovelli [95], who realized that they are related to the presence of "bubbles" within Γ. Freidel and Louapre [51,53], and later Freidel, Gurau and Oriti [49], pushed this intuition further in three-dimensions and obtained a powercounting estimate for certain special complexes, coined "type 1". Within Gurau's colored tensor models, this result was then extended to higher dimensions by Ben Geloun et al. [58].
• Powercounting from vertex counting. Yet another approach relies on the fieldtheoretic notion that the divergence degree of a Feynman diagram can be bounded by the number of its vertices. Together with Magnen, Noui and Rivasseau, I obtained such bounds in the Boulatov model [81]; these were adapted to the colored tensor models in [21].
• Powercounting from jackets. The notion of "jacket" for a (colored) Boulatov-Ooguri complex was introduced by Ben Geloun et al. in [58], and used by Gurau and Rivasseau [68,64] to obtain an upper bound on the divergence degree improving the one obtained by vertex counting mentioned above.
In this section, we explain how our main formula (5.5) encompasses all the results quoted above. One important should be stressed: except for the perturbative bounds obtained from vertex counting, these results are all based on the (usually implicit) assumption that Ω(Γ, G) is an integer multiple of dim G. An immediate consequence of (5.6) and (5.7), however, is that this is not true in general, because neither dim M(Γ, G) nor dim ζ(Γ, G) in the topological part I(M, G) are multiples of dim G.

Powercounting from Pachner moves
When Ponzano and Regge introduced their model in 1968, they considered a 3-manifold M equipped with a triangulation T M , G = SU (2), and conjectured that the divergence degree was given by three (i.e. dim SU(2)) times the number of vertices V (T ) in the triangulation. They were guided in making this conjecture by the following consideration: because of the Biedenharn-Elliot identity for {6j}-symbols, the formal amplitudes of two triangulations T M and P 14 T M of M related by a 1 − 4 Pachner move satisfy Since Λ j=0 (2j + 1) 2 scales as Λ 3 when Λ → ∞, and since the 1 − 4 move introduces a single new vertex in the triangulation, the conclusion that each vertex contributes a factor of Λ 3 in the amplitude appears tantalizing. Indeed, Ponzano and Regge proposed to cut off the sums in their state-sum to a maximal value Λ to obtain a finite value Z PR,Λ T for the amplitude, and tentatively defined a regularized amplitude by The origin of the intuition that divergences in the Ponzano-Regge model are related to the vertices of T M , but also the reason why this naive regularization is bound to fail, is completely elucidated by our decomposition (5.5). Indeed, taking K M as the dual cell complex to a triangulation T M , and noting that in three dimensions, a vertex is dual to a 3-cell, we see that

Powercounting from bubble counting
Freidel, Gurau and Oriti defined in [49] a restricted class of foams Γ, which they called "type 1", for which they could show that the divergence degree is simply related to the number B(Γ) of "bubbles" in Γ, Taking their cues from the Turaev-Viro model, they speculated that the "type 1" condition should correspond to "topologically trivial manifolds", i.e. to the 3-sphere. Given the decomposition (5.5), the identity (5.14) is equivalent to In other words, the "type 1" complexes are, for a given number of 3-cells in the corresponding complex K M , are minimally divergent. Moreover, they are such that their flat connections up to gauge are isolated and completely reducible. We will see in the next section that this last condition alone does not single out the 3-sphere (it is also satisfied e.g. by the real projective space RP 3 ).
In [58], Ben Geloun et al. considered Gurau's colored model, and used the corresponding "bubble homology" to obtain the following formula for the divergence degree in dimension d ≥ 3 for the Abelian structure groups G = R: 3 Here, G is a (d + 1)-colored graph, c k (G) is for 0 ≤ n ≤ d the number of k-bubbles in G , c d+1 (G) := 1 and b k (G) is the k-th "bubble" Betti number of G. All the details are given in the appendix. The relationship with our result is based on the following correspondence. A (d+1)colored graph G naturally defines a d-dimensional CW complex ∆ G with the following property: • the underlying graph of G is the 1-skeleton of ∆ G , • the number of k-bubbles of G equals the number of k-cells of ∆ G , • the "bubble" homology of G coincides with the cellular homology of ∆ G .
With this correspondence, the relationship between (5.16) and our result (5.4) is straightforward. First, note that what is denoted c d+1 (G) in (5.16) is really b d (G) = 1, so that it can (and should) be rewritten Then, use the identities c k (G) = c k (∆ G ) and b k (G) = b k (∆ G ) and the Euler-Poincaré relation This is the same as (5.4) in this particular case. Our result can therefore be described as the generalization of (5.16) to general cell complexes, and non-Abelian groups.

Powercounting from vertex and jacket counting
Using tools from perturbative quantum field theory, and notably Cauchy-Schwarz inequalities, an upper bound on the divergence degree in the Boulatov model was obtained in [81]. It is formulated in terms of the number of vertices rather than 3-cells in the complex, and reads for a Boulatov complex Γ without generalized tadpole [21] Ω(Γ) ≤ 3 2 c 0 (Γ) + 6. (5.20) Within the colored model, this bound was then generalized to higher dimensions in [21]. In dimension d, it becomes for a colored graph G with V (G) vertices Using the notion of "jacket" introduced in [58], Gurau and Rivasseau further improved this bound in [68,64], obtaining where g(J ) is the genus of (orientable surface dual to) the jacket J . Remarkably, the jacket bound (5.22) follows easily from our exact result (5.4). Indeed, denoting E(G) the number of edges and F (G) the number of faces of G, the formula (5.4) reads Since a colored graph has no tadpole, we have 2E(G) = (d + 1)V (G), hence where is the number of jackets of G. Now it is well-known that the dimension of the SU(2) representation variety of a genus g surface is 6g − 3. Thus, Using the Euler relation and that each face of G belongs to (d − 1)! jackets, we have [69] .

Three-dimensional examples
In this section, we illustrate how the divergence degree, and more specifically its topological part I(M, G), can be computed for certain three-dimensional manifolds M, always with G = SU (2). (Hence, hereafter we will drop the reference to G in the notation.) Most of them cannot be handled by the previous methods and do not saturate the corresponding bounds [53,19,49,58]. Closed 3-manifolds have χ(M) = 0, so I(M) = dim M − dim ζ and it is completely determined by the fundamental group. It is also well-known that M is a finite set of points, and hence dim M = 0, when the fundamental group π 1 (M) is finite (but we will re-derive this feature in specific examples for illustrative purpose).
The method to compute I(M) for a cellular pseudomanifold (and in fact for any cell complex) has been presented in the section 5.1.2. The steps are: • Choose a presentation of π 1 (M) (e.g. from a deformation retract of the 2-skeleton Γ).
• Identify the representation variety Hom π 1 (M), G using the chosen presentation. In general, it has several irreducible components.
• Identify the orbit of the G-action on each irreducible component.

The 3-torus
The case of the 3-torus T 3 was discussed in Appendix C of [53]. Its fundamental group has the presentation withn ∈ S 2 their common direction of rotation, ψ a,b,c ∈ [0, π] three class angles, and τ the 3-vector formed by a set of (anti-Hermitian) generators of the algebra SU(2)(2). They form a 5-dimensional manifold F . The group action by conjugation rotates the directionn. For each representation there is a stabilizer ζ isomorphic to U(1) (and larger if the three rotations are in the center of SU(2)) which leaves it invariant: the subgroup generated by the directionn. Hence, the dimension of the stabilizer is dim ζ = 1. Thus Hence, Note that this implies that the divergence degree of a cell decomposition of T 3 cannot be a multiple of dim SU(2) = 3, as the procedure of [53] assumed implicitly.

Lens spaces
Lens spaces L p,q are standard spherical manifolds, with Z p as their fundamental group (we exclude the case of L(0, 1) = S 1 × S 2 which can be understood separately). They include as a particular case the real projective space ÊP 3 , which is the first example in the appendix of [66]. The standard presentation of Z p is of course If 0 < r < π, let S 2 r be the subset of SU(2) defined by S 2 r = exp rn. τ ;n ∈ S 2 , (5. 36) consisting in those rotations of fixed angle r. In the topological picture induced by the identification SU(2) ≃ B 3 /∂B 3 , S 2 r is a sphere of radius r centered on the origin. (In this picture, the central elements ±½ of SU(2) are the center and boundary of B 3 respectively.) Hence, • If p = 1 (the 3-sphere), there is only one representation, the trivial one.
• If p = 2 (the real projective space), the representations of Z 2 send its generator to an element of the center of SU(2), i.e. {±½}.
• If p ≥ 3, the set of representations decomposes as Hom(Z p , SU(2)) = F p is a union of 2-spheres.
The next step consists in finding the orbits generated by conjugation in SU (2). It turns out that all of the above representations are reducible.
• In the cases p = 1, 2, and on F (0) p when p ≥ 3, the representations are left invariant by the group action (since they commute with the whole group): they are central. Thus, the centralizer is ζ = SU(2), which is 3-dimensional.
• On F  p . This means that the relevant value is the one computed on the less reducible representations, i.e. those with the smallest stabilizer.
In conclusion

Prism spaces
Prism manifolds P m,n , with m ≥ 1, n ≥ 2, form a different class of spherical manifolds. They are characterized by their fundamental group Let us first give the irreducible representations φ in the case of m even. The first relation imposes the direction of the rotation φ(x) to be orthogonal to that of φ(y), and its class angle to be π 2 . Then the second relation reduces to y n = ½, which only constrains the class angle of φ(y) to be ψ y = 2kπ/n for k = 1, . . . , ⌊ n 2 ⌋ (the case k = 0 gives a reducible representation). This way the set of irreducible representations is identified as Clearly this space is of dimension 3. Moreover, simultaneous conjugation of φ(x), φ(y) by some SU(2) element induces a simultaneous rotation onn x ,n y . Thus, the orbit is 3-dimensional, isomorphic to SU(2)/{±½} = SO(3) and the centralizer is just the center of SU (2). This leads to dim M = 0, dim ζ = 0, and hence: when m is even, and evaluated on irreducible representations. The situation is similar for m odd, only the specific values of the class angle of φ(y) are changed.
As for reducible representations, they are obtained by taking φ(y) = ±½ ∈ ζ(SU(2)). Then, the image of the generator x lives on a sphere of the type S 2 r (5.36) or is ±½. One easily sees that dim M = 0 again, but dim ζ = 1 or dim ζ = 3. This produces a topological part for the divergence degree which is negative. Thus, the highly divergent contribution to the amplitude comes from the irreducible representations of the fundamental group.
The purpose of the third part of this thesis is to present two preliminary results on the continuum limit of spinfoam models, viz. the problem of describing a quantum spacetime with infinitely many dynamical degrees of freedom from the truncated amplitudes A(Γ).
In chap. 6, we address a long-standing question in the spinfoam community: to construct this continuum limit, should one sum over all foams Γ, or infinitely refine a foam? We give a mathematical definition for each alternative, and an argument to the effect that, as it happens, they could be two facets of the same solution.
In chap. 7, we consider a specific framework for the sum over all foams, in three dimensions: Boulatov's group field theory. Using a tool of constructive field theory -Magnen's and Rivasseau's cactus expansion -we improve Freidel and Louapre's proof of Borel summability of the Boulatov series with a high-spin cutoff, and establish a remarkable scaling limit.
The two sections are completely independent.
What is the mathematical definition of the operation of refining a foam to increase the number of gravitational degrees of freedom? Does this process have a well-defined limit? How is it related to the sum over foams sometimes advocated in the spinfaom community?
These questions were addressed in collaboration with Carlo Rovelli [122]. Some aspects of the arguments presented there, such as the emphasis on the structure of directed sets and the rôle of zero-spin colorings, were already discussed in Zapata's earlier work. 1

The refinement limit of spinfoams
We sketched in the introduction the Ashtekar-Lewandoswki construction of the Hilbert space of loop quantum gravity as the inductive limit of truncated spaces H γ labelled by graphs. This defines the quantum kinematics of the gravitational field as a continuum limit of lattice gauge fields.
It turns out that little work has been done to study the analogue of this construction for spinfoams, viz. at the dynamical level. (The notable exception is Zapata's papers [83,149].) This is surprising: such a hierarchical scheme, whereby more and more degrees of freedom of the field are taken into account according to some notion of scale, lies at the core of our modern understanding of quantum field theory, in the form of the renormalization group.
The obvious stumbling block to apply Wilsonian techniques to study the dynamics of quantum gravity is the absence of a background metric, which makes the notion the scale difficult to assess. 2 But there is a way out of this dead end: to think of the foams themselves as generalized scales. Let us elaborate on this idea. 1 We came to know Zapata's work on the continuum limit of spinfoams on the occasion of the Loops 2011 conference, after the paper [122] was completed; the similarities between his approach and ours are thus only fortuitous.
2 Of course, one can run the functional renormalization equation with respect to a fixed, arbitrary, background metric: this is the strategy underlying Weinberg's and Reuter's asymptotic safety scenario [89]. The problem with this approach is the difficulty to extract physics from the truncated flow equation.

The directed set of generalized scales
Lattice field theory [112] is based on the discretization of (Euclidean) spacetime by means of regular lattices: lattices where the distance between neighboring vertices is constant, given by the lattice spacing a. To refine the lattice, one introduces new vertices at the midpoints of edges and plaquettes, and constructs the corresponding regular lattice with lattice spacing a/2. Note that both the choice of the initial lattice and the refinement step rely on the metric structure of Euclidean space. (For instance, the midpoint of an edge is singled out by the fact that it is located at a distance a/2 from the endpoints of the edge.) The prominent feature of this metric notion of scale is its one-dimensional character, which leads to sequences of discretizations. This can be expressed mathematically by the statement that the set of distance scales S = R + is a totally ordered set: for any two scales s 1 and s 2 , we have either s 1 ≤ s 2 or s 2 ≤ s 1 , viz. s 1 and s 2 are mutually comparable.
Non-metric discretizations generally do not share this one-dimensional character. Consider for instance the set T M of triangulations of a given manifold M. There is a natural order relation on this set, defined by T 1 T 2 when all the simplices of T 2 are subsets of the simplices of T 1 ; this corresponds to the intuitive notion that T 2 is finer than T 1 . It is obvious that not all triangulations are comparable with respect to this relation: T M is only a partially ordered set, or poset.
The limit of a family of objects indexed by a poset is ill-defined, because non-unique. Yet, in the case of triangulations of a manifold, we have the intuition that there is only one continuum limit: roughly speaking, it corresponds to a "triangulation" with infinitely many simplices. The reason why this intuition is meaningful is because the set of triangulations is directed : for any two triangulations T 1 , T 2 , there is a triangulation T 3 which is finer than both T 1 and T 2 , viz. T 1 T 3 and T 2 T 3 . A poset satisfying this condition is called a directed set. Roughly speaking, a directed order relation has a single 'asymptotic direction': here, the continuum limit.
This observation provides a sound definition of the notion of generalized scale: it is simply an element of some directed set, be it totally ordered or not. In this sense, a triangulation is indeed a generalized scale.

Kinematical and dynamical continuum limits
Generally speaking, a quantum system can be described by an algebra of observables A acting on a Hilbert space H, its kinematics, and a state over this algebra ω : A → C, its dynamics. In particular, a truncation of the degrees of freedom by means of a directed set S associates to each generalized scale s ∈ S an algebra A s and a state ω s , corresponding respectively to the truncated kinematics and dynamics of the system. Thus, to obtain the continuum limit of a truncated system, we need two limits: an algebraic one, for the family of algebras (A s ) s∈S and Hilbert spaces H γ (kinematics), and a topological one, for the family of states (ω s ) s∈S (dynamics). The corresponding mathematical notions are that of inductive limits and limit of nets. The definitions are as follows.
• Let S be a directed set, and (A s ) s∈S a family of algebraic objects indexed by S.
Suppose that, for each s 1 s 2 , we have a homomorphism f s 1 s 2 : These conditions ensure that the identification of a s 1 ∈ A s 1 with its images f s 1 s 2 (a s 1 ) for all s 2 s 1 defines an equivalence relation over the disjoint union s∈S A s . The inductive limit A ∞ of (A s ) s∈S is then the set of equivalence classes for this relation. It naturally comes equipped with the same algebraic structure as the A ′ s . For instance, if the latter are inner product spaces, then the maps f s 1 s 2 are isometric embeddings, and the inner product on A ∞ is given by a 1 , a 2 A∞ = f s 1 s 3 (a s 1 ), |f s 2 s 3 (a s 2 ) A 3 , where a s 1 ∈ A s 1 and a s 2 ∈ A s 2 are representatives of a 1 and a 2 respectively, and s 3 s 1 , s 2 .
• Let X be a topological space, and N : S → X a function over a directed set S valued in X, viz. a net over S. Then N converges to N ∞ ∈ X, or that N ∞ is the limit of N, if for any neighborhood U of N ∞ there exists s U ∈ S such that N(s) ∈ U for any s s U . It is easy to check that, thanks to the directedness of S, the limit of a net over S, when it exists, is unique (assuming X is Hausdorff). 3 These definitions clarify the general notion of refinement: in general terms, a set of generalized scales is given in the form a directed set; kinematics (resp. dynamics) defines an inductive system (resp. a net) over the set of generalized scales; the continuum limit is the corresponding limit. No metric is involved in this process.
Although little known outside the mathematical community, the structure of directed set plays a fundamental rôle in the conceptual architecture of physics. Indeed, it captures in its general form the concept of approximation scheme: a hierarchical structure directed towards a unique limit. Since physics is the science of approximations, we could say (paraphrasing Rivasseau on the renormalization group) that directed sets are the soul of physics.

Loop quantization as a continuum limit
The kinematical truncation scheme used in loop quantum gravity is precisely of the above type, with the notion of generalized scale provided by graphs. 4 For each graph γ, one considers the algebra A γ of gauge-invariant functions on the cotangent bundle T * SU(2) E(γ) and its standard quantization over the Hilbert space H γ := L 2 (SU(2) E(γ) /SU(2) V (γ) ). This defines the quantum kinematics of a system of discrete connections on γ.
The kinematics of the quantum gravitational field is then constructed as the continuum limit of this system of discrete connections. To this effect, one uses the natural order relation on the set of graphs G defined by γ 1 γ 2 if γ 1 is a subgraph of γ 2 . The latter clearly equips G with a directed set structure: given any two graphs γ 1 and 3 Standard examples of nets are provided by Riemann sums. Let P denote the set of partitions of the unit interval [0, 1] directed by inclusion, and f : [0, 1] → C a (continuous) function. The map N f : P → C associating to each partition p the corresponding Riemann sum of f is a net over P. Its limit is the Riemann integral 1 0 dtf (t). 4 Along the lines of the canonical formulation of general relativity, these are usually defined as embedded in a three-dimensional Cauchy surface in some globally hyperbolic spacetime. Rovelli has recently considered a purely combinatorial version of this construction, but it is not clear that the inductive limit procedure is meaningful in this case. γ 2 , it is easy to construct γ 3 γ 1 , γ 2 by adding all the vertices and edges of γ 1 to γ 2 . Moreover, each embedding γ 1 ֒→ γ 2 naturally induces maps A γ 1 → A γ 2 and H γ 1 → H γ 2 which turn the family of algebras and Hilbert spaces into inductive systems. Their inductive limits A ∞ and H ∞ form the kinematical arena of loop quantum gravity, where the action of (some extension of) the diffeomorphism group can be implemented.
In other words, the image of i γ 1 γ 2 in H γ 2 is the set of gauge-invariant functions on SU(2) E(γ 2 ) which are constant on those edges of γ 2 that are not edges of γ 1 ; equivalently, it is the subspace of H γ 2 spanned by spin-network functions with zero spins on the edges of γ 2 \ γ 1 . It follows from this definition that the inductive limit H ∞ can be written as where H * γ denotes the subspace of H γ spanned by spin-network functions with non-zero spins. This is an interesting observation: at the kinematical level, the continuum limit of loop quantum gravity consists of superselection sectors with non-zero spin colorings. Pictorially, we can say that "refining the graphs" is the same as "summing over the graphs with zero-spins excluded".

The spinfoam continuum limit
To discuss the analogue construction for spinfoam models, we must begin by identifying the relevant directed set. One possibility (explored by Zapata in [149]) is to work with embedded foams and their diffeomorphisms classes. Let us use instead the combinatorial framework sketched in sec. 2.3.1, in which a foam Γ is defined as a triple of sets (V Γ , E Γ , F Γ ) enumerating its vertices, edges and faces.
In a given spinfoam model, to each foam Γ with boundary γ is associated a vector Z(Γ) in the boundary Hilbert space H γ . Now, this map Γ → Z(Γ) defines a net over the set F γ of foams with fixed boundary γ.
Consider indeed the natural order relation defined by Γ 1 Γ 2 if there is an embedding ι : Γ 1 ֒→ Γ 2 , namely a triple of injective maps ι v : preserving the relations between vertices, edges and faces. It is easy to see that any pair of foams with the same boundary has an upper bound Γ Γ 1 , Γ 2 with respect to this relation, given essentially by attaching the vertices, edges and faces of Γ 2 to Γ 1 . Thus, a spinfoam model is a net over the set of proper foams with a given boundary.
Hence, the set F γ is naturally a directed set, and Γ → Z(Γ) a net over it. We can therefore define formally, for each boundary γ, the spinfoam continuum limit as the limit W ∞ γ of this net: Z ∞ γ : .

Refining = summing?
The spinfoam continuum limit defined above, if compelling, is at variance with Baez's, Rovelli's and Reisenberger's original idea that foams are "quantum histories" of the gravitational field, like Feynman diagrams are "quantum histories" of systems of particles. Indeed, this latter conception suggests that the proper way to unfreeze the truncated degrees of freedom is not by refining a foam, as in lattice field theory, but rather by summing over the set of all foams. Is there a relationship between refinement and sum over foams, similar to kinematical relation between refinement and sum over graphs?

Spinfoam cylindrical consistency
From now, let us assume that the foam amplitudes Z(Γ) are given as sums over colorings of the faces of Γ by irreducible representations of a compact Lie group G (typically G = SU (2)). More specifically, we assume that for each coloring σ of Γ, we have a spinfoam amplitude Z(Γ, σ). This assignement defines two spinfoam models, depending whether the trivial representations are included or not: Here, for a fixed foam Γ, we denoted Col(Γ) the set of all colorings σ, and Col * (Γ) the subset consisting of colorings by non-trivial (j f = 0) representations. Since trivial representations play no rôle in the kinematics of loop quantum gravity, it is natural to impose a cylindrical consistency condition relating Z(Γ) to Z * (Γ). To state it, we must first observe that each coloring σ of a foam Γ comes with a multiplicity, related to the symmetries of Γ. The multiplicity |σ| Γ of a coloring σ ∈ Col(Γ) is the number of colorings σ ′ such that σ = σ ′ • φ, with φ an automorphism of Γ, i.e. a foam embedding of Γ into itself. Then we say that the amplitude A Γ (σ) := |σ| Γ Z(Γ, σ), (6.5) is cylindrically consistent if A(Γ, σ) = A(Γ ′ , σ ′ ) when (Γ ′ , σ ′ ) is a trivial extension of (Γ, σ), that is when Γ is a subfoam of Γ ′ , σ and σ ′ coincide on the faces of Γ and σ ′ is trivial on the other faces of Γ ′ .

Summing is refining
Under this cylindrical consistency condition, the relationship between the continuum limit Z ∞ γ and the sum of Z(Γ) over all the foams Γ with fixed boundary is the following: If we assume furthermore that the net Z is convergent, then we have by passing to the limit in the identity above Thus, refining foams in the model Z is the same as summing over foams in the model W * . Note that, due to the peculiar directed set structure, all foams appear in (6.7) the following sense: every foam Γ with boundary ∂Γ = γ appears in one finite sum (6.6) whose value can be chosen arbitrarily close to W ∞ γ . The proof of (6.6) is easy. First, observe that the subfoams of Γ index a partition of Col(Γ), in which each each class is made of the trivial extensions of a given subfoam Γ ′ ⊂ Γ: This implies that Second, check that we have 10) with N Γ ′ ,Γ the number of subfoams of Γ isomorphic to Γ ′ . Third, use cylindrical consistency to get and conclude to (6.6).

Discussion
But is there any rationale for assuming the cylindrical consistency condition and expecting it to be part of the quantum theory of gravity? There is. The spinfoam sum is meant to be an implementation of the Misner-Hawking sum over geometries Z =

Metrics/Diff
Dg µν e i S[gµν ] . (6.12) Here the integral is not over metrics, but over equivalence classes of metrics under diffeomorphisms. In the truncation induced by the choice of a foam, the diffeomorphisms are reduced to the automorphisms φ of the foam. Therefore the colorings σ and σ • φ have a natural interpretation as the discrete residual equivalent of diffeomorphismrelated metrics. This interpretation is reinforced by the fact that the amplitude is in fact invariant. If we want to integrate over geometries, then, the contributions of σ and σ • φ represent an overcounting, and we must divide by the number of them, namely by |σ| Γ . The same conclusion can be reached by interpreting colored spinfoams as histories of nontrivial spin networks. Then colorings related by foam automorphisms clearly represent the same history. For both these reasons, it is physically interesting to consider the amplitudes modified as in (6.5) and the cylindrical consistency condition. Then the sum over foams is equal to the continuum limit.
In this section, we address the issue of the continuum limit from the angle of group field theory (GFT). In this scheme, the sum over foams is nothing but a Feynman series, and the issue of convergence is a problem of constructive field theory. Is the sum over foams the asymptotic expansion of a well-defined GFT correlation functions?
The first results in this direction were obtained in the Boulatov model by Freidel and Louapre [52]. Here, using more advanced constructive techniques, we improve their proof of Borel summability, notably by establishing the constructive scaling behaviour of the correlation functions in the large cutoff limit. These results were obtained under the lead of Rivasseau, in collaboration with Magnen and Noui [81].

Constructive field theory
Constructive field theory [103] is the name of a mathematical physics program launched around 1970 by Wightman. It aims at defining rigorously certain field theory models, in increasing order of difficulty, by checking that they obey the "Wightman axioms" [130]. The strategy is clear-cut: introduce as many cutoffs as necessary for the correlation functions of these models to be well-defined, and then develop the necessary methods to lift these cutoffs.

Current status of constructive field theory
Early constructive field theory succeeded in building rigorously super-renormalizable field theories such as the emblematic φ 4 model in two and three dimensions. It also elucidated their relationship to perturbation theory: the Schwinger functions of these models are the Borel sum of their perturbation theory [40,80]. But four-dimensional φ 4 theory itself could not be built, because its coupling constant does not remain small in the ultraviolet regime. Being asymptotically free, non-Abelian gauge theories do not have this problem. Nevertheless, although some partial results were obtained, they could not be built in the full constructive sense either, due to technical difficulties such as Gribov ambiguities; neither could the interesting infrared confining regime of the theory be understood rigorously. Probably the first four-dimensional field theory to be built completely through constructive methods will be the Grosse-Wulkenhaar model [63], a non-commutative field theory which, ironically, should not satisfy the Wightman axioms of the initial constructive program.
The constructive program is largely unknown in the quantum gravity community, with the notable exception of [52]. However, it can be argued that constructive theory embodies a deeper point of view on quantum field theory than the textbook one. A modern constructive technique such as the cactus expansion used hereafter allows to resum perturbation theory by reorganizing it in a precise, explicit manner.

Power series and Borel sums
Renormalized perturbation theory writes field-theoretic observables O(λ), notably correlation functions, as formal power series in the coupling constant(s) λ: The coefficients O n are, in turn, given by a sum of Feynman amplitudes O G , where G spans the set of Feynman graphs with V (G) = n vertices: It was early realized by Dyson [39] that the construction of observables by power series in λ is bound to remain heuristic, because (7.1) cannot converge. Indeed, if it did, it would define an analytic function of λ in a disk around λ, and hence would be equally well-defined for λ ≥ 0 and λ ≤ 0. But this is physically absurd. Think of λ as the fine structure constant of quantum electrodynamics: λ ≤ 0 means that charge of the same sign attract each other. But the vacuum is clearly unstable in this regime: to decrease the energy of the field, create many electrons and as many positrons and let them collapse on each other. The same argument can be run for λφ 4 theory.
The consequence of the non-analyticity of O(λ) at λ = 0 is irrevocable: the perturbative series (7.1) is, at best, an asymptotic expansion 1 of O(λ) in the λ → 0 limit. But it is a classic theorem of Borel that there is an infinite number of smooth functions asymptotic to any power series: the sequence of coefficients (O n ) n≥0 does not uniquely defines the function O(λ), as it would if O(λ) were analytic in a disk centered on λ = 0.
But what if O(λ) is analytic in a disk tangent to λ = 0 in the Re λ > 0 halfspace, as in Fig. 7.1? This corresponds to the case mentioned above, where O(λ) has singularities on the λ ≤ 0 axis (a branch cut), where the vacuum is unstable.
Borel summability states that, under certain conditions formulated by Nevanlinna [88] and rediscovered by Sokal [129], the function O(λ) can be reconstructed from the coefficients (O n ) n≥0 . These conditions are that holds uniformly in r and λ.  Then the power series converges for |t| < 1 B , and admits an analytic continuation in the strip for some constant C. Moreover, O is represented in C R by the absolutely convergent integral B is then called the Borel transform of O, and the complex t plane is called the Borel plane.
There is a reciprocal to this theorem: consider the power series n≥0 O n λ n . If the power series n≥0 O n t n /n! converges in a disk {|t| < 1 B }, admits an analytic continuation B(t) in the strip S B and satisfies the bound (7.6) in this strip, then the function O defined by the integral representation (7.7) is analytic in C R , has ∞ n=0 O n λ n as Taylor series at the origin and satisfies the uniform remainder estimates (7.3). In this case we say that the series n≥0 O n λ n is Borel summable, and call the series n≥0 O n t n /n! its Borel transform and the function O its Borel sum. In other words, Borel summable series and Borel summable functions are in oneto-one correspondence just like are ordinary series and germs of analytic functions.

Proliferating and non-proliferating species
Even when the perturbative series (7.1) is actually Borel summable, it is very difficult to obtain the relevant information about its Borel transform, and in particular to show that (7.4) converges. The main reason for this is purely combinatorial: there are too many graphs at a fixed order n. Indeed, the typical bound for a Feynman amplitude is of the type [103] |O G | ≤ K V (G) (7.8) for some constant K. Hence, one usually has to deal with the series n≥0 a n K n λ n , (7.9) where a n is the number of Feynman graphs with n vertices. Its Borel summability is in turn subject to the convergence of n≥0 a n λ n n! , (7.10) which is known as the generating series of Feynman graphs. Unfortunately, it so happens that the latter has zero radius of convergence, because a n grows too fast (typically like n! 2 ). Rivasseau has called proliferating [106] a combinatorial species whose generating series has zero radius of convergence. Thus, to get started with the constructive program, one must abandon the Feynman expansion and try to replace it by a series labelled by a non-proliferating species, such as trees. 2 Each such tree should resum an infinite number of Feynman amplitudes [108]. This is precisely what the Magnen-Rivasseau cactus expansion does.

A constructive tool: the cactus expansion
Although it was initially designed to deal with matrix φ 4 theory [105], the cactus expansion applies to scalar models [79,110] and higher-order interactions [108] as well.
An important mathematical prerequisite for constructive field theory is the notion of degenerate Gaussian measure, which we now review.

Degenerate Gaussian measures
If A is a n × n positive-definite matrix, the Gaussian measure dµ A associated to A is usually defined by where is x ∈ R n and dx is the standard n-dimensional Lebesgue measure. The inverse matrix C = A −1 is the covariance of the measure: When A = C is the unit matrix, we call dµ A the standard Gaussian measure on R n .
The key property of Gaussian measures is expressed by Wick's theorem 3 : for any smooth, summable function f , we have This implies that dµ A is completely determined by its covariance C. Now, observe that unlike (7.11), (7.13) still makes sense if C is non-invertible, i.e. only positive-semidefinite. For any such matrix C, we can thus define the degenerate Gaussian measure dµ C with covariance C by (7.13). When C actually has zero modes, it is not be absolutely continuous with respect to the Lebesgue measure dx: it is a Dirac delta along each zero mode.
One important observation about (degenerate) Gaussian measures is that they have a well-defined functional counterpart -unlike the Lebesgue measure. The covariance C is then a positive semi-definite operator, and by Minlos' theorem dµ C is supported on a suitable space of distributions. (In this setup the zero modes of C are interpreted as gauge modes.)

An absolutely convergent expansion
We are now ready to discuss the cactus expansion. Consider a quartic field theory λφ 4 , with partition function Here, dµ is some Gaussian measure, and we leave the number of components of φ (scalar, vector, matrix...) unspecified. The cactus expansion constructs Z(λ) as an absolutely convergent series where F ranges over the set of forests, i.e. graphs without loops. The amplitudes Z F (λ) have the following properties: 1. They factorize over the connected components of F . It follows that the free energy F (λ) := log Z(λ) is given by the very same expansion (7.15), with the sum restricted to connected forests, viz. trees.
2. They are analytic functions of λ in a Nevanlinna-Sokal disk C R , hence so are Z(λ) and F (λ).
3. They are easily bounded, and it is straightforward to check that the Taylor remainders of (7.15) about λ = 0 satisfy the Nevanlinna-Sokal bound (7.3).
Thus, in addition to giving a rigorous definition of (single-scale, i.e. with a ultraviolet cutoff) φ 4 theory, the cactus expansion naturally shows that it is the Borel sum of its perturbative series.

Four ingredients for a cactus expansion
The cactus expansion is a recipe with four ingredients: (1) an intermediate field representation, (3) a replica trick, (3) a forest formula and (4) a resolvent bound.
(1) The intermediate field representation is a very useful trick to represent a quartic interaction φ 4 in terms of a cubic one φσφ. Schematically, it reads where σ is a field with Gaussian ultralocal 4 measure dν (which in zero dimension would simply be the standard Gaussian measure over R). Performing the Gaussian integration over φ, one can then express the original path integral Z(λ) in terms of σ only. For instance, in zero spacetime dimension, viz. for the numerical integral we have The new interaction vertex V λ (σ) := − 1 2 Log(1 + i √ λσ) was called a loop vertex in [105]. In more than zero dimension, i.e. for a genuine field theory, a propagator C 1/2 would typically sandwich the σ field on both sides, and the loop vertex would be given by the trace of operator Log(1 + i √ λC 1/2 σC 1/2 ). This trick can be generalized to any correlation functions and to more complicated models [109].
(2) The replica trick is a property of degenerate Gaussian measures. Let dν(σ) denote the standard Gaussian measure on R, and V ∈ L n (R). Now, 'replicate' n times the variable σ, and consider the degenerate Gaussian measure dν n on R n with covariance C ij = σ i σ j = 1. The replica trick is then the statement that Let us emphasize that this is not a form of Fubini's theorem, which expresses an ndimensional integral as a product of n integrals. Here, we replace one 1-dimensional integral by an n-dimensional one, with (n − 1) delta functions.
(3) Consider a smooth function H of n(n−1) 2 variables h = (h l ), living on the lines l of the complete graph over n vertices. The so-called Brydges-Kennedy Taylor forest formula [1,32] is a Taylor interpolation of H with integral remainders indexed by labeled forests over n vertices: In this expression, F n denotes the set of forests over n vertices, the products are over lines l of each forest F , and h F is the n(n−1) For higher values of n, on the other hand, the outcome of (7.21) is genuinely non-trivial, as the case n = 3 already demonstrates (Fig. 7.2): (7.23) (4) If λ is a complex number with positive real part, and σ a real number, one has Let Σ be a Hermitian matrix. Taking the supremum over the spectrum of Σ in (7.24) gives the resolvent bound where · denotes the operator norm, A := sup |x|=1 |Ax|.

A toy example: φ 4 theory in 0 dimensions
To illustrate how these four ingredients can be combined to yield the cactus expansion, we now consider the φ 4 field theory in 0 dimension, 5 i.e. the integral (7.17). The functions z(λ) and f (λ) := log z(λ) it defines are analytic in the cut plane C \ R − , and hence admit, at best, a Borel expansion about λ = 0. Using the four ingredients presented above, we now show that this is indeed the case. We start from the intermediate field representation with loop vertex V λ (σ) := − 1 2 Log(1 + i √ λσ). First, expand the exponential in (7.26) in powers of V λ (σ), swap integration and summation 6 and apply the replica trick to the order-n term to obtain Next, consider the matrix C h , parametrized by an n(n−1)

2
-uple h and defined by C h ii := 1 and C h l := h l , where l = {ij} (i = j), and let dν h n (σ 1 , . . . , σ n ) be the Gaussian measure with covariance C h . Then apply the forest formula to the function This gives (7.29) or equivalently with F ranging over the set of all forests, and As announced, the summand factorizes along connected components of each forest. This implies that f (λ) = T z T (λ) (7.32) where now T ranges over the set of all trees. It is these trees T over loop vertices V λ (σ v ) which we coin cacti.
Since the dependence of the covariance of dν h T n (σ 1 , . . . , σ n ) in the h variables is linear, applying the derivative ∂/∂h l in (7.32) is easy using (7.13): it amounts to an additional insertions of ∂ 2 /∂σ s(l) ∂σ t(l) in the integral, where s(l) and t(l) are respectively the starting and ending vertices of the line l. Hence V λ (σ v ). (7.33) 6 Of course, it is precisely such an interchange between integration and summation that yields the divergent perturbative series. Note that here, however, the process is licit because dµ(σ) e 1/2|Log(1+i √ λσ)| < ∞, and so Lebesgue's dominated convergence theorem applies.
Consider the loop vertex V λ (σ v ), with coordination k v in the tree T . Thanks to the resolvent bound, we have 34) and thus, since there are n − 1 lines in a tree over n vertices, This bound goes through the normalized integrals over the σ's and the h's. Using Cayley's formula for the number of trees over n labeled vertices with fixed coordinations we find that z T (λ) ≤ 2 n−1 |λ| n−1 .

(7.37)
This shows that the cactus expansion (7.32) of f converges uniformly in a half-disk D R = {λ ∈ C, ℜλ ≥ 0, |λ| ≤ R}, with R < 1 2 , which obviously contains the Nevanlinna-Sokal disk C R/2 . Furthermore, the order-r Taylor-Lagrange remainder can easily be shown to satisfy the Nevanlinna-Sokal criterion (7.3). Indeed, consider a cactus amplitude z T (λ) with n loop vertices. By (7.33)-(7.34), each such amplitude is made of an explicit factor λ n−1 times an integral over dν h T n (σ 1 , . . . , σ n ) of a product of 2n − 2 resolvents where l v denote a half-line hooked to a vertex v. Hence for r ≤ n − 1, T r z T (λ) = z T (λ), and for r ≥ n But by (7.13) and since F = 2n−2 lv =1 R lv is solely a function of √ λσ we have Hence the Taylor-Lagrange formula applies to the exponential: (7.42) and the operator H k creates exactly k additional insertions of lines . The combinatorics of 2k derivations on a product of 2n − 2 resolvents costs a factor (2r−1))! (2n−1)! times a new Gaussian integral with 2n − 2 + 2k resolvents of the same type R l ′ v . The λ factor can be then transferred back to a √ λ factor in the resolvents: Hence, since k = r − n + 1 the convergent series T T r z T (λ) can be bounded exactly as before, except for two facts: each term contains a factor λ r and we have also to add to the bounds a factor This last factor is maximal for the trivial tree with n = 1, and certainly bounded by 2 r r!, from which it follows that for some constant K To summarize, the cactus expansion allows not only to trade the asymptotic perturbative series for the convergent expression (7.32), but also to check the Sokal-Nevanlinna criteria, proving Borel summability of (7.46). Of course, in this toy example, Borel summability of z is obvious and Borel summability of f = log z could be shown by more elementary methods; the power of the cactus expansion becomes manifest when it comes to the constructive analysis of the φ 4 field theory [79], and of the matrix φ 4 model [105]. And of Boulatov's group field theory.

Borel summability and scaling behavior
Indeed, the fact that Boulatov's group field theory has a quartic interaction allows to use the cactus expansion to study its Borel summability. The great advantage of this method over Freidel and Louapre's [52] lies in the control it provides over the high-spin cutoff: by this token, we can unravel the constructive scaling limit of this group field theory.

A twofold regularization
We introduced Boulatov's model in the first chapter of this thesis. In the language of degenerate Gaussian measures, the Boulatov partition function is defined formally by where SU(2) 6 6 i=1 dg i φ(g 1 , g 2 , g 3 )φ(g 3 , g 4 , g 5 )φ(g 5 , g 2 , g 6 )φ(g 6 , g 4 , g 1 ), (7.48) and dµ[φ] is the Gaussian measure with covariance C φ defined by Here, c ranges over the cyclic permutations of three elements. This measure is clearly degenerate: the covariance C φ is the orthogonal projector onto the subspace of L 2 (SU(2) 3 ) invariant under SU(2) averaging and cyclic permutations. Of course, the partition function (7.47) is only formal and needs regularizations to become mathematically well-defined. The problem is twofold: • the Fourier space of the field φ is non-compact, although discrete, and hence 'ultraviolet' divergences arise, is not positive, hence unstable. This can be seen from its Fourier space formulation where the interaction term reduces to an oscillatory {6j} symbol.

Intermediate field representation
Let us now construct the cactus expansion of the BFL model. Let us then define the operator Σ coupling φ to σ Σφ(g 1 , g 2 , g 3 ) := dg 4 dg 5 σ(g 1 , g 2 , g 4 , g 5 )φ(g 3 , g 4 , g 5 ) (7.59) Pillow Tetrahedron in such a way that Sφ|σ 4 = φ|Σφ 3 , (7.60) where ·|· 3 is the standard inner product in L 2 (SU(2) 3 ). After integration over the original field φ, we obtain where the loop vertex is given by One easily checks that Σ := C φ ΣC φ is a Hermitian operator and therefore that the resolvent bound applies to the derivatives of this loop vertex just like in the toy example.

Cactus expansion
Following the same steps as in sec. 7.2.4 yields the cactus expansion of the BFL free energy: dh l dν h T n (σ 1 , . . . , σ n ) l∈T d 4 g s(l) d 4 g t(l) C σ (g s(l) ; g t(l) ) δ 2 δσ s(l) (g s(l) )δσ t(l) (g t(l) ) n v=1 V λ (σ v ). (7.63) At this stage, the only difference with the 0-dimensional case is the insertion of a covariance C σ (g s(l) ; g t(l) ) on each line l of the tree, and the integration with respect to the corresponding 4-uples of group elements g lv := (g lv i ) 4 i=1 , attached to the half-lines l v .

Cauchy-Schwarz inequalities
To get some insight into this cactus amplitude, it is handy to introduce a 'dual' representation of a tree T , as a planar partition of the disk. The boundary of the disk is obtained by turning around T , while the dotted lines partitioning it cross the boundary twice and each line of T exactly once, without crossing each other, see In such a picture, the resolvents (1 + i √ λ Σ) −1 are attached to the arcs on the boundary of the disk, while the covariances C σ are attached to the dotted lines. To bound (7.65), we can apply the Cauchy-Schwarz inequality along a line splitting the disk in two parts with the same number of consecutive resolvents. Indeed, the number of half-lines of a tree being even, it is always possible to pick two arcs with resolvents R 1 and R 2 , and express (7.65) as the inner product A|R 1 ⊗R 2 |B , where A and B contain the same number of arcs, and thus of resolvents (see Fig. 7.6). By the Cauchy-Schwarz inequality, we have | A|R 1 ⊗ R 2 |B | ≤ R 1 |R 2 A|A B|B ≤ 2 A|A B|B . (7.66) This process has two effects: thanks to the resolvent bound, it trades the original tree for two trees, with the same number of vertices, but with two resolvents replaced by the identity ( R 1 R 2 ≤ 2); each covariance C σ which is sandwiched in the inner product is replaced by a modified covariance C ′ σ := 1 + δT ′ , where T ′ identifies the two central arguments of σ instead of twisting them (Fig. 7.7): T ′ (g 1 , . . . , g 4 ; g ′ 1 , . . . , g ′ 4 ) := δ(g 1 g ′−1 1 )δ(g 4 g ′−1 4 )δ(g 2 g −1 3 )δ(g ′ 2 g ′−1 3 ). We can iterate the process (n − 1) times, until all resolvents are removed, and all covariances C σ replaced by C ′ σ . We are then left with a BFL Feynman graph, hence |F T (λ)| ≤ 2 n(T )−1 sup T ′ ∈Tn |A BFL (T ′ )|, (7.68) Now, from theorem (7.57) we have sup T ′ ∈Tn |A BFL (T ′ )| ≤ K n δ Λ 6+3n , hence |F T (λ)| ≤ 2 n(T )−1 K n δ Λ 6+3n(T ) . (7.69) Before we can conclude, we should check that this estimate also holds for the trivial tree with just one vertex (n = 1): since it contains no half-line, the loop vertex Log(1 + i √ λ Σ) is not acted upon by a σ-derivative, and therefore the resolvent bound does not apply. However, using standard convexity inequalities, we have for any z ∈ C |Log(1 + z)| ≤ 1 2 + 1 2 (|z| 2 + 4π 2 ),

A constructive scaling limit
The remarkable feature of this result lies in the control it provides over the summability radius: it shows that, as the cutoff Λ is sent to infinity, the coupling constant should shrink as Λ −3 . This immediately suggests the definition of a scaling limit for the BFL model, analogous to one discovered in matrix models by 't Hooft at the perturbative level [131], and confirmed by Rivasseau at the constructive level [105]. Define indeed a running coupling constant λ ′ (Λ) = λ Λ 3 . (7.75) Then, seen as a function of λ ′ , the BFL free energy per degree of freedom G (Λ) BFL (λ ′ ) has a summability radius which is independent of Λ. This fact suggests the existence of a limit lim Λ→∞ G (Λ) BFL (λ ′ ) (7.76) where λ ′ → 0. Three comments are in order.
• This scaling behavior was beyond the scope of the Freidel-Louapre analysis [52]. It is only thanks to our use of the (much less elementary) cactus expansion that we could establish this result.
• The constructive scaling (7.75) matches the perturbative scaling (7.57). In other words, the sum over all foams does not introduce further divergences in the highspin limit.
• The presence of the pillow term in the BFL action is key to this result. Let us emphasize however that the BFL scaling behavior does not coincide with the one of the original Boulatov model. A perturbative estimate established in [81] shows indeed that, in the pure Boulatov model, we have Let us conclude by remarking that, since we established in 2009 the perturbative (7.57) and constructive (7.75) scaling behavior of three-dimensional group field theory, Gurau and Rivasseau have made significant progress towards the identification of dominant contributions in the Boulatov [68] and Ooguri models [68,64], analogous to 't Hooft's planar diagrams [131].
Moreover, like a simplicial complex, K G possesses a dual CW complex ∆ G , whose k-cells f k α are in one-to-one correspondence with the (d − k)-cells e d−k α of K G (see [70] for details). Let E k α,q = β , e k β(q) = e k α (13) index the set of cofaces of e k α . Then ∆ G is such that i.e. the boundary operator ∂ k of ∆ G is the coboundary operator δ d−k of K G . Hence, as in Poincaré duality.
To relate the colored homology of the colored graph G to the cellular homology of the corresponding complex, it suffices to note that the B-bubbles of G are in one-toone correspondence with the cells of K G arising from simplices with vertices labelled in ∆ d \ B. of K G , and therefore a k-cell f k b of ∆ G . It is not hard to check that this correspondence is actually bijective. Moreover, inspection of (9) and (14) shows that where F k the mapping b → f k b . In other words, we have an isomorphism of chain complexes F = (F k ) d k=0 : C * (G) → C * (∆ G ), and thus In particular, the colored Betti numbers of G coincide with the cellular Betti numbers of ∆ G . Note moreover that, since the 0-th Betti number of a topological space counts the number of its connected components, we have b d (G) = b 0 (K G ) = 1 and b 0 (G) = b 0 (∆ G ) = 1.