Marek Lipert

M Smoluchowski

Nonlinear stability of the Taub-NUT soliton in 6+1 dimensions

Keywords: numbers: 04.20.Ex, 04.50.-h, 04.30.Nk, 04.70.Bw, 02.30.Jr

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Nonlinear stability of the Taub-NUT soliton in 6+1 dimensions

Introduction

It has been shown by Bizoń, Chmaj and Schmidt [1] that in odd number of spacetime dimensions one can perform a consistent cohomogeneity-two symmetry reduction of the vacuum Einstein equations which -in contrast to the spherically symmetric reductionadmits time dependent solutions. The key idea was to modify the standard spherically symmetric ansatz by replacing the round metric on the S n with the homogeneously squashed metric. By squashing we mean here the construction of a metric for S n that does not admit full symmetry group SO(n + 1). Instead, the metric is required to be invariant under transitive action of a specific subgroup of SO(n + 1). This idea, technically called the BCS ansatz, has been used for the stability analysis of the 4+1 dimensional Taub-NUT soliton, also known as the Kaluza-Klein (KK) monopole [2]. The monopole is important in M/String theory, as it is super-symmetric and can be used as a base for construction of certain BPS states [3,4]. The KK monopole was found to be stable within the ansatz (for small perturbations).

One might wonder why the KK monopole is stable. In order to address this issue we decided to change dimensionality of the model. This way we could figure out which features are dimension dependent. We chose to go from the 4+1 to the 6+1 spacetime dimensions. There is an additional advantage of this approach connected with the fact that the 6+1 dimensional Taub-NUT soliton (TN) is not super-symmetric. Therefore we can also see which aspects of relaxation rely on super-symmetry. We found that introduced changes do not destroy stability, as the TN soliton is nonlinearly stable against small finite-energy perturbations within the BCS ansatz. Our techniques used for stability investigation follow (and extend) the ones used in [2], therefore the reader is encouraged to read both papers in conjunction.

Further research revealed that strongly perturbed TN soliton loses stability and collapses, forming a black hole. This also occurred in the lower dimensional model. This time, however, the black hole solution is not known in closed form. We examined it in detail, proving its existence and computing its asymptotic expansion. This allowed us to compute its mass and compare it to the mass of its lower dimensional cousin.

Taub-NUT soliton

The Taub-NUT soliton in 6+1 dimensions is a static regular nontrivial solution of the vacuum Einstein equations. It can be explicitly found in [4]:

ds 2 6+1 = -dt 2 + (ρ + 2l) 2 2ρ(ρ + 4l) dρ 2 + ρ(ρ + 2l)dΣ 2 2 + 2l 2 ρ(ρ + 4l) (ρ + 2l) 2 σ 2 , ( 1
)
where l is a free positive parameter,

σ 2 = (dµ - 1 2 sin 2 ξσ 3 ) 2 , ( 2
)
dΣ 2 2 = dξ 2 + 1 4 sin 2 ξ (σ 2 1 + σ 2 2 + cos 2 ξ σ 2 3 ),
the latter being the Fubini-Study metric for CP 2 and σ 1 = sin ψ sin θdφ + cos ψdθ, σ 2 = cos ψ sin θdφsin ψdθ, σ 3 = dψ + cos θdφ are standard Maurer-Cartan forms for SU (2). Parameter ranges are as follows

0 ≤ ρ, 0 ≤ φ ≤ 2π, 0 ≤ θ ≤ π, 0 ≤ ψ ≤ 4π, 0 ≤ ξ ≤ π 2 , 0 ≤ µ ≤ 2π.
It is worth noticing that this solution lacks asymptotic flatness in a fashion similar to its 4+1 dimensional counterpart. It approaches R 5 times a circle of asymptotic radius √ 2l near infinity. Near ρ = 0, on the other hand, it is locally flat, which can be seen by introducing a new radial coordinate s = √ 2lρ. Then the solution is interpolating between two different vacua which justifies the name: soliton.

Preliminaries

The squashed S 5 can be constructed as a coset SU(3)/SU (2) with SU(3) ⊂ SO [START_REF] Kerin | [END_REF] being its symmetry group. The result is widely known and can be easily derived using the methods of [START_REF] Besse | Einstein Manifolds[END_REF][START_REF] Kerin | [END_REF][START_REF] Knapp | Lie Groups Beyond an Introduction[END_REF][START_REF] Bertini | [END_REF]. The BCS ansatz has the form

ds 2 = -Ae -2δ dt 2 + A -1 dr 2 + r 2 (e -8B σ 2 + e 2B dΣ 2 2 ), (3)
where σ 2 and dΣ 2 2 were defined in (2) and A, B, δ are functions of r and t. The function B, which measures squashing resulting from lower symmetry of our description of S 5 , serves as a dynamical degree of freedom and breaks Birkhoff's theorem for that case. It has been chosen in such a way that it does not change spheres' volume.

Substituting the metric (3) into the vacuum Einstein equations we get the following system of equations

A = - 4A r + 1 5r (24e -2B -4e -12B ) -4r(e 2δ A -1 Ḃ2 + AB 2 ), (4a)
Ȧ = -8rA ḂB , (4b)
δ = -4r(e 2δ A -2 Ḃ2 + B 2 ), (4c)
e δ A -1 r 5 Ḃ . -e -δ Ar 5 B + 6 5 e -δ r 3 (e -2B -e -12B ) = 0, ( 4d
)
where primes and dots denote derivatives with respect to r and t, respectively. Note that the equations are scale invariant. The Taub-NUT solution (1) expressed in terms of ansatz (3) takes the form

e -10B 0 = 2l 2 (ρ + 4l) (ρ + 2l) 3 , A 0 = 2(ρ + 4l)ρ (ρ + 2l) 2 ( dr dρ ) 2 , e 2δ 0 = A 0 , (5)
r 10 = 2l 2 ρ 5 (ρ + 2l) 2 (ρ + 4l). (6)
It can be proven that every static regular solution of (4a-4d) should exhibit Taylor series expansion near r = 0

B(r) = br 2 + O(r 4 ), A(r) = 1 -8b 2 r 4 + O(r 8 ), δ(r) = -4b 2 r 4 + O(r 6 )(7)
with √ b being the scale factor (a rigorous proof can be found in appendix A). It is easy to verify that (5) possesses the same expansion, with b = 1 16l 2 . Thus we identify l with the scaling freedom and conclude that up to scaling there exists the unique solution [START_REF] Besse | Einstein Manifolds[END_REF].

Linear stability

The linear stability analysis proceeds along the same lines as in the 4+1 dimensional case. Assuming

A(r, t) = A 0 (r) + A 1 (r, t), B(r, t) = B 0 (r) + B 1 (r, t), (8)
δ(r, t) = δ 0 (r) + δ 1 (r, t), and keeping only linear terms in perturbations, we get an equation for a single Fourier mode (ρ(r) is defined by equation ( 6)) , B 1 (ρ(r), t) = v(ρ)e iλt :

Lv(ρ) = λ 2 v(ρ), (9)
with

L = -2 ρ 2 (ρ + 2l) 2 d dρ ρ 3 (ρ + 4l) d dρ + V (ρ).
where V (ρ) is a long and complicated expression (given in the appendix B). This equation cannot be solved analytically for generic λ 2 . However, in order to demonstrate linear stability we only need to know the zero mode. This mode can be deduced from the scaling freedom. For b near 1 we have

B 0 (br) = B 0 (r) + r d dr B 0 (r)(b -1) + O((b -1) 2 ). ( 10
)
We define v 0 = r d dr B 0 (r). Changing r → ρ we get

v 0 (ρ) = (ρ + 5 l) ρ 4(ρ 2 + 5 ρ l + 5 l 2 ) , (11)
with 0 ≤ ρ, 0 < l. It can be easily verified that this solves (9) for λ 2 = 0. We see that the zero mode is manifestly positive for all ρ in the domain. This implies by the standard Sturm-Liouville theory that the operator L has no negative eigenvalues. Thus, the TN soliton is linearly stable within our ansatz. Note, that the zero mode is not a genuine eigenfunction because it is not square integrable. Rather, it is a resonance sitting at the bottom of the continuous spectrum.

Warped black holes

Black hole solutions play the key role in dynamics of a strongly perturbed Taub-NUT soliton. As far as we know, in contrast to the 4+1 dimensional case, these solutions are not known in closed form in 6+1 dimensions. Thus, from the mathematical point of view the very existence of such solutions is not obvious and requires a proof. Following Breitenhohner, Forgács and Maison [9] one can prove the local existence (see Appendix A). Global existence proof, as well as analysis of an asymptotic behaviour for large r is described below.

Let us begin by explaining what we mean by a black hole solution in our model. If we assume that for a given r H > 0 A(r H ) = 0 and substitute the formal power series expansion into the static equations, we obtain:

B(r) ∼ β + 6 4 e -2β -e -12β 6e -2β -e -12β r r H -1 , (12)
A(r) ∼ 4 5 (6e -2β -e -12β ) r r H -1 .
Theorems presented in the Appendix A guarantee that a local solution with the above behaviour for small r -r H exists and is analytic in β, r H and r -r H . By setting β = 0 one arrives at the ordinary Schwarzschild black hole with B = 0. Numerical simulations show that solution with β < 0 plays no role in TN soliton evolution. Below, the global solution with local behaviour (12) and β > 0 will be called the warped black hole (WBH).

Global properties

We now focus on establishing the global existence of WBH. Using logarithmic variable τ = ln r, static version of the equations (4a-4d) can be cast into the following form:

AB + 1 5 B 24e -2B -4e -12B -6 e -2B -e -12B = 0, (13a)
A = -4A 1 + B 2 + 1 5 24e -2B -4e -12B , (13b)
δ = -4B 2 . ( 13c
)
Here prime corresponds to differentiation with respect to τ . This is an autonomous system, which is a consequence of scaling invariance. Note that the first two equations are independent of the third one.

It can be easily verified that WBH solution obeys the following inequalities in a right-open range [τ H , τ 1 ) where τ H = ln r H :

0 < B < 1 4 , 0 < A < 1, 0 < B. (14)
Our argument proceeds in two steps. In the first step we show that if the local WBH solution preserving ( 14) is given in some right-open interval I = [τ 0 , τ + ), then the solution can be safely extended to Ī = [τ 0 , τ + ] preserving (14). In the second step we prove that the first step is equivalent to global existence of the WBH.

Step I

Let us define an auxiliary function

f (B) = 6(e -2B -e -12B ) 24e -2B -4e -12B . ( 15
)
and observe, that 1 4 -> f(B) for r ∈ I and some finite > 0 dependent on the choice of I. Now, if we take B < 0 and look at (13a) we get

B > f(B), (16)
for r ∈ I. Therefore, we can safely assume that B < 0 for B > 1 4 -. This naturally leads to a conclusion that B (τ + ) < 1 4 . Next, we can put B = 0 in (13a) to obtain

B = 6 5A (e -2B -e -12B ) > 0, (17)
which means that B (τ + ) > 0. We can proceed with similar considerations for the function A. Setting A = 1 in (13b) we get

A = -4B 2 - 1 5 20 + 4e -12B -24e -2B < 0, (18)
which means that A(τ + ) < 1. Finally, setting A = 0 in (13b) we get

A = 1 5 (24e -2B -4e -12B ) > 0, (19)
which means that A(r + ) > 0. One should not be concerned with setting

A = 0 in -4A(1 + B 2 ) because B is finite in Ī. Finally, from B > 0 in Ī we get that B > 0 in Ī. Note, that B(τ + ) < ∞ because B (τ + ) < ∞.
Step II

Let us assume that a local WBH solution satisfying (14) in some [τ H , τ + ) cannot be extended to infinity preserving (14). Then, there exists τ s ≥ τ + where ( 14) is not satisfied. By our construction, the solution must be finite and preserve (14) for I = [τ + , τ s ). We can now use the results of Step I to get a contradiction -the solution is regular and preserves (14) in τ s .

We shall now prove that B ∼ τ for large τ . To see this, let us first note that B(τ ) → ∞ as τ → ∞. Indeed, we already know from ( 14) that B is monotonically increasing. Furthermore, if it was to reach a finite limit as τ grows, B would have to approach 0 and f (B(τ )) would attain a finite positive limit lim τ →∞ f (B(τ )) = ξ > 0. However, if we reverse a sign in (16) we immediately arrive at the contradiction because whenever B drops below ξ, B becomes positive repelling B from 0.

Next, let us observe that

d dτ f (B(τ )) = B f (B) > 0. ( 20
)
In the neighbourhood of τ H = ln r H we have B = f (B) which follows from (12). As we depart towards greater τ , B must drop below f (B) because from (16) we know that initially B = 0 < d dτ f (B(τ )). It becomes evident, by the same argument, that once B drops below f it can never cross it again. Therefore B is monotonically increasing (from reversed (16)) and bounded from above (because f is bounded from above and B < f). We get lim τ →∞ B (τ ) = ζ > 0, which implies

B(τ ) ∼ ζτ + Σ (21)
for large τ . Σ is an integration constant that, as we shall see, corresponds to the scale invariance.

Exact asymptotic behaviour

We shall now focus on establishing the large τ series expansion for WBH solutions. First, notice that substituting (21) into (13a) and neglecting e -12B yields the consistency condition

(4ζ -1)e -2B = 0 ( 22
)
which fixes ζ at 1 4 . Let us now rewrite B(τ ) = 1 4 τ + B(τ ). From ( 19) and (13b) we expect that A → 0 as τ → ∞. We also have B → 0 as τ → ∞. We shall now solve the system (13a-13b) through successive approximations. From the above assumptions we get for (13b):

A + 17 4 A = 24 5 e -1 2 τ (1 -2 B + (2 B) 2 2! -. . .) - (23)
- 4 5 e -3τ (1 -12 B + (12 B) 2 2! -. . .) -2A B -4 B 2 .
We expect that A as well as B will be expressed as a series of e -µnτ -like expressions for subsequent µ n 's so that they eventually all cancel each other. The first exponent to be cancelled is e -1 2 τ -therefore, neglecting higher order terms, we get

A + 17 4 A = 24 5 e -1 2 τ (24)
with a solution

A(τ ) = 32 25 e -1 2 τ + αe -17 4 τ , (25)
where α comes as an integration constant for homogeneous equation. We can now rewrite (23) using A(τ ) = 32 25 e -1 2 τ + αe -17 4 τ + Ã(τ ) to get:

à + 17 4 à = 24 5 e -1 2 τ (-2 B + (2 B) 2 2! -. . .) - 4 5 e -3τ (1 -12 B + (12 B) 2 2! -. . .) (26) -2αe -17 4 τ B - 64 25 e -1 2 τ B -2 à B -4 B 2 .
Note, that B can not have a e -1 2 τ term because there would be no possibility to cancel 24 5 e -1 2 τ (-2 B). Instead, this factor must combine to e -3τ in order to cancel -

Unfortunately, this is a wrong solution. One can see this by substituting it into (13a) and trying to find next order of expansion -unbalanced exponents appear. This leads to a conclusion that A must also contribute to e -3τ equation. It can be consistently done only by including e -3τ term in A as well as e -5 2 τ term in B:

B(τ ) = 1 4 τ + Ce -5 2 τ + γe -15 4 τ + B(τ ), A(τ ) = 32 25 e -1 2 τ + De -3τ + αe -17 4 τ + Ã(τ ),
where C and D are yet unknown coefficients, B has been redefined and à has been defined for the next step of the method. After substitution into (13b) we get an equation for terms proportional to e -3τ

(25

D + 64 C + 16) = 0. ( 29
)
Equation (13a) provides us with yet another equation for e -3τ term

1 -4C = 0. ( 30
)
We therefore get

C = 1 4 , D = -32 25 .
Everything is consistent and we can proceed to higher orders by guessing what terms should we include to cancel constant factors in equations for further exponents. It is interesting that we never arrive at any constraints for α or γ. This constants explicitly appear in higher coefficients of the expansion, but remain completely undetermined.

Our heuristic reasoning is justified by theorem from Appendix A which ensures that the above formal expansion defines well behaved three parameter family of local solutions near infinity. The theorem requires that we interpret the integration constant Σ from (21) as a scaling constant and that we parameterize solutions with {α, γ, Σ}. Below we present an expansion computed to one more order, translated into the original variable r = e τ :

A(r) ∼ 32 25 1 √ R - 32 25 1 R 3 + α R 17 4 - 12 25 1 R 11 2 + . . . , (31)
B(r) ∼ 1 4 ln R + 1 4 1 R 5 2 + γ R 15 4 + 3 8 1 R 5 + . . . ,
where R(r) = Σr. We should remark here that the TN soliton exhibits exactly the same expansion with γ = -

1 4 √ 2, α = 32 25 √ 2 and Σ = 1 √ 2l 2 .
This feature enables us to define a mass of a WBH in terms of asymptotic integrals.

Mass of the warped black hole

The definition of a mass for the WBH is not straightforward because commonly known mass definitions (e.g. Komar mass [10]) rely on asymptotic flatness of a given solution.

Yet, our spacetime is not asymptotically flat. It is locally asymptotically flat and we have to resort to other methods for mass computation. Abbott and Deser [11] have proposed a general definition of conserved charges. They replace asymptotically flat background with a spacetime of an arbitrary asymptotics which is supposed to be the vacuum of the system. Next, they use its isometries to generate conserved quantities. Below we present an application of this method to the case in hand.

We choose the TN soliton as a background solution and decompose the metric to factor-out the background:

g αβ = ḡαβ + h αβ (32)
One can define

H αβ = h αβ - 1 2 ḡαβ h, (33)
K αβγδ = 1 2 ḡαδ H γβ + ḡγβ H αδ -ḡαγ H βδ -ḡβδ H αγ . ( 34
)
All indices are raised and lowered using the background metric, ḡαβ . The TN soliton is static, therefore it possesses a Killing vector field ξα ∂ α = -∂ ∂t . The charge associated with this isometry can be interpreted as a mass and equals to

M AD = 1 2π 3 √ -ḡ ξα ∇β K tiαβ -K tjαi ∇j ξα dS i (35)
where dS i is the spatial surface area element at large radius, the index t denotes the time coordinate index. Greek indices run through all spacetime directions while Latin ones run through space directions. The integral should be performed at spatial infinity. That explains why this method is sometimes referred to as method of asymptotic integrals. We can, therefore, use the expansion (31) for computation of the above integral to obtain

M AD = 4 5 √ 2 - 5 4 α - 16 5 γ. ( 36
)
This result is valid only if the scaling parameter Σ of the background TN soliton is the same as for WBH. Otherwise the integral is divergent. Note that Σ does not appear in the above formula. It is interesting that the WBH solution is parameterized by different number of parameters near r H than near infinity. More precisely, near r H we have two parameters {β, r H } while near infinity -three {Σ, α, γ}. Therefore there must be a constraint involving Σ, α and γ which reduces the freedom at infinity. We determined this constraint numerically as follows. We chose r H , shot the solution to infinity and fine tuned β to get Σ = 1 from the fit. Then we were able to fit α and γ. The results are presented in figure 1 as a curve parameterized by r H . Combining this knowledge with the equation (36) we can compute the mass of the WBH basing solely on r H .

One might wonder what will happen if we perform shooting from infinity towards horizon and take α and γ lying outside the curve from Figure 1. Using numerical simulations we looked at this case and observed that A > 0 and becomes singular near r = 0. This is not a surprise, since the reasoning presented in the proof of existence of WBH (e.g. A can not cross 1 from below) is only valid when shooting forward, not backward.

Comparison to the 4+1 dimensional case

In the 4+1 dimensional model described in [1] the black hole solutions are explicitly known. It is instructive to look at the parameter dependence between near horizon and far from horizon power series expansions and compare it to 6+1 dimensional case where only numerical relations are known. Let us begin by recalling the lower dimensional ansatz:

ds 2 = -Ae -2δ dt 2 + A -1 dr 2 + 1 4 r 2 e 2B (dθ 2 + sin 2 θdφ 2 ) ( 3 7 )
+ 1 4
r 2 e -4B (dψ + cos θdφ) 2 .

The A and B functions have an analogous interpretation as in 6+1 dimensional model.

From the above ansatz we get field equations. Below we present their static version

r(rB ) A + 2 3 rB 4 e -2 B -e -8 B - 4 3 e -2 B -e -8 B = 0, ( 38a
) rA = -2 A 1 + r 2 B 2 + 2 3 4e -2 B -e -8 B , (38b)
δ = -2rB 2 . ( 38c
)
where primes denote differentiation with respect to r. Next, we present KK solution

B 0 = 1 3 ln(1 + ρ m ), A 0 = (1 + 4ρ 3m ) 2 (1 + ρ m ) 8/3 , e 2δ 0 = A 0 , (39)
where

r = 2m 1/2 ρ 1/2 (1 + ρ m ) 1/6 (40)
and m > 0 is a free parameter which sets the scale (corresponding to l of ( 1)). We also recall the black hole solution [12]:

B = 1 3 ln Σ 2P , A = ρr 2 (3Σ + ∆) 2 36Σ 3 ∆ 2 , e 2δ = ∆ ρ A (41)
where

∆ = ρ + 3M - √ M 2 + 2P 2 , Σ = ρ + M + √ M 2 + 2P 2 , (42)
r = 4 2/3 P 1/3 Σ 1/6 ∆ 1/2 .
Here 0 < P ≤ 2M. To see that this is indeed a black hole we can find local asymptotics similar to ( 12)

B(r) ∼ β + 2(e -2β -e -8β ) 4e -2β -e -8β r r H -1 , (43)
A(r) ∼ 2 3 4e -2β -e -8β r r H -1 .
This time, though, we have explicit formulas connecting β and r H with P and M

β = 1 3 ln M + √ M 2 + 2P 2 2P , (44)
r 6 H = 4 4 P 2 (M + √ M 2 + 2P 2 )(3M - √ M 2 + 2P 2 ).
Next, careful asymptotic analysis applied to (38a-38b) gives us the following nearinfinity expansions:

A(r) ∼ 16 9 1 R + α R 5 2 + 8 9 -α γ + 16 3 γ 2 1 R 4 + . . . , (45)
B(r) ∼ 1 2 ln R + γ R 3 2 - 9 32 α γ + γ 2 + 1 8 1 R 3 .
. . , where α, γ are free parameters and R = Σr represents scaling freedom. In contrast with the 6+1 dimensional case, free parameters appear just after the leading order term. Note also, that the next order depends explicitly on these parameters. For KK monopole we get α = -16 9 , γ = 1 4 , Σ = 1 2m . For the black hole we have

α = - 8 9 M + √ M 2 + 2P 2 4 5/2 P 7/2 , γ = - M - √ M 2 + 2P 2 (4P ) 5/2 , Σ = 1 4P (46)
Therefore P is simply the scale factor. It is evident that two parameters {M, P } are translatable to {β, r H } on one hand and to {α, γ, Σ} on the other hand.

To compare these results with the 6+1 dimensional case we set P = 1 4 and get Σ = 1. One can now see that r H depends solely on M ≥ 1 8 and that for M = 1 8 we have r H = 0 -black hole degenerates to the KK monopole. β is also fixed by the choice of M so we can parametrize the black hole up to scaling by r H alone.

Next, elimination of M from (46) gives us a surprisingly simple relation between α and γ γ = -4 9

1 α . ( 47
)
It can be easily seen from Figure 1 that in the 6+1 dimensional case the corresponding relation is very different.

In the end we present mass of the 4+1 dimensional warped black hole computed using the same method as for 6+1 dimensional case:

M 4+1 = - 1 12 (8 + 32γ + 9α) . (48)
After substituting (46), we get

M 4+1 = 16M -2 3 .
(49)

Numerical results

In order to investigate nonlinear perturbations we have resorted to numerical simulations. Using second order finite difference code we have solved the equations (4a-4d) for several families of regular initial data representing perturbations of Taub-NUT soliton (1). The overall picture does not depend on the specific choice of a family.

The results shown below were generated for initial data of the form (using momentum variable P = e δ A -1 Ḃ):

B(0, r) = B 0 (ρ(r)), P (0, r) = p r r 0 4 e (r-r 0 ) 4 σ . ( 50
)
The scaling parameter of the initial soliton has been fixed to l 2 = 1 2 . Parameters σ i r 0 are fixed for given examples, whereas the amplitude p varies.

Small perturbations

It has been observed that for small perturbations, by which we mean small value of the control parameter p, the perturbation is being radiated to infinity and the solution settles down to the Taub-NUT soliton with the same parameter l as the perturbed one.

One might wonder why l is not altered by the evolution. If we recall that l is connected with the scaling constant by relation Σ = 1 2l 2 , we see that changes in l would change scaling. The mass formula (36) becomes then divergent -it would require an infinite energy to change scaling, whereas our perturbation is of finite energy.

An interesting aspect of the relaxation process is connected with the absence of r -4 term in expansion (31) of A. If one examines (4a) closely it can be seen that the compact-support perturbation imposed on the momentum variable instantly introduces r -4 term in the expansion of A for large r. If the perturbed soliton is really stable, this term should eventually vanish. More precisely, from (31) we know that the function

d(t, r) = r 4 (A(t, r) -A 0 (r)) (51)
represents the coefficient in front of r -4 term. Figure 4 presents the behaviour of this function.

Late time tail

In order to better understand the relaxation process we transformed the linear equation ( 9) into single mode wave equation For initial data (50) with a small amplitude (p = 0.02, r 0 = 3, σ = 1) we plot a series of snapshots of the function B(t,r) (where t is central proper time). The dashed line shows the unperturbed TN soliton. During the evolution the excess energy of the perturbation is clearly seen to be radiated away to infinity and the solution returns to equilibrium.

- d 2 dx 2 u(x) + V (x)u(x) = λ 2 u(x) ( 5 2 )
using transformation

v(ρ) = 4 ρ 3 (ρ + 4) 3 ρ 2 √ ρ + 2 (ρ + 4) u, (53) dx
= (ρ + 2) 2 2ρ(ρ + 4) dρ. (54)
The potential has the following asymptotic behaviour

V (x) = 2 x 2 + 2 √ 2 x 3 + 6 x 4 + O( 1 x 5 ). ( 55
)
According to work of Ching et al. [13] and Bizoń et al. [14] the solution should exhibit asymptotic power-law tail

u(x, t) ∼ 1 t 6 . (56)
We are aware of the possible existence of nonlinear tail, however our numerical simulations are in good agreement with (56). Furthermore, the amplitude of the tail scales linearly with initial data, which suggests that even if nonlinear tail is present it decays faster than the linear one.

Our simulations strongly suggest that the TN soliton is nonlinearly stable against small perturbations. However, just like its lower dimensional cousin, it is being destabilized by sufficiently strong perturbations. Warped black holes are important in this process.

Large perturbations

We have found that for sufficiently strong perturbations the inner part of the soliton collapses and horizon is being formed for r = r H > 0. The external part of the solution settles down to a WBH. The scaling parameter l is preserved.

Conclusions

W have shown that the TN soliton in 6+1 dimensions is classically stable just like the KK monopole. Therefore the stability is neither dimension specific nor supersymmetry dependent feature and its origin should be sought elsewhere. We have also shown that for sufficiently strong perturbations a collapse to the warped black hole occurs. The black hole properties have been studied in detail and the proof of global existence has been presented. Still, there are open questions concerning the model. The first one is connected with the critical behaviour on the threshold of black hole formation. In 4+1 case the discrete type II self similarity was observed and it might be interesting to compare those results with the observations concerning the case in hand. The other interesting feature that can be examined is the appearance of quasinormal modes in asymptotic pointwise convergence presented in the Figure 3.

Figure 1 . 4 √ 2

 142 Figure 1. The relation between asymptotic parameters α and γ for WBH is plotted as a parametric curve, horizon radius r H being the parameter. It can be seen that for r H → 0 the curve converges to {γ, α} of the soliton, γ = -1 4 √ 2 and α = 32 25 √ 2.

Figure 2 .

 2 Figure 2. Asymptotic stability of the Taub-NUT soliton in 6+1 spacetime dimensions.For initial data (50) with a small amplitude (p = 0.02, r 0 = 3, σ = 1) we plot a series of snapshots of the function B(t,r) (where t is central proper time). The dashed line shows the unperturbed TN soliton. During the evolution the excess energy of the perturbation is clearly seen to be radiated away to infinity and the solution returns to equilibrium.

Figure 3 .Figure 4 .

 34 Figure3. The convergence to the TN soliton. For the same initial data as in Figure2we plot ln |P (r 0 , t)| as a function of time for r 0 = 2.0. Asymptotic power-law tail can be seen.

Figure 5 .

 5 Figure 5. Instability of the Taub-NUT soliton for large perturbations. For initial data (50) with large amplitude (p = 0.1,r 0 = 3, σ = 1) we plot a series of snapshots of the function A(t,r) (where t is central proper time). During the evolution A(t,r) drops to zero near r = r H 2.174 which signals the formation of a horizon there. Outside the horizon the solutions settles down to the warped black hole (dashed line) with M = 0.612.

Acknowledgments

I want to express my gratitude to Piotr Bizoń for hours of enlightening discussions. This work was supported in part by the Polish Ministry of Science grant no. N N202 079235 (2008-2010).

Appendix A.

This appendix is devoted to the formal proof of the existence theorems for static equations. Let us begin with the theorem which has been proven in [9] (Proposition 12): Theorem 1. Suppose a system of differential equations for m+n functions u = (u 1 , ..., u m ) and v = (v 1 , ..., v n ),

with constants λ i with Re(λ i ) > 0 and integers µ i , ν i ≥ 1 and let C be an open subset of R m such that the functions f and g are analytic in a neighbourhood of t = 0, u = c, v = 0 for all c ∈ C. Then there exists a unique m-parameter family of solutions of the system (A.1) with boundary conditions (u, v) = (c, 0) at t = 0, defined for c ∈ C and |t| < t 0 (c) with some t 0 (c) > 0. These solutions satisfy

and are analytic in t and c. Now, we cast static equations into the form:

The first two equations are independent of the third. This allows us to consider only (A.2,A.3).

Appendix A.1. Existence near r = 0

We will try to prove the relations [START_REF] Knapp | Lie Groups Beyond an Introduction[END_REF]. In order to do this, we perform a transformation:

. According to our supposition on local behaviour we should get:

Now, equations (A.2,A.3) take the form:

A , r 2 . We need to perform another transformation:

)

), to finally get

where g, h are polynomials in u, v, 1 A , r 2 . This system has the form (A.1), thus from Theorem 1 we acquire Theorem 2. There exists an unique, one-parameter family of regular solutions for (A.2-A.3) near r = 0 analytic in b and r such that

We need a transformation which casts (A.2-A.3) to the form (A.1), around r H > 0 :

, (A.9)

We get

where f 1 and f 2 are polynomials in ω 1 , ω 2 , 1 ω 1 , 1 u 1 and

Next, transformations

give :

where

Those polynomials are regular around u 1 = r H , u 2 = β, v i = 0. According to Theorem 1 we get: Theorem 3. There exists an unique two-parameter family of regular solutions of the system (A.2-A.3) around r H : A(r H ) = 0, r H > 0, analytic in β, r H and r, so that:

Appendix A.3. Existence near infinity

Let us consider a coordinate change u = 1 r 1/4 . We then take ρ = Σu, Σ > 0:

Some more transformations are required:

Those transformations leave us with the set of equations:

where f, g are regular for ω → 0, A → α, ω → γ, S → Σ > 0, ρ → 0. Those equations are of the form (A.1). We now get from Theorem 1: Theorem 4. There exists an unique three-parameter family of regular solutions for the system (A.2-A.3) around infinity with asympotics B(r) ∼ ln r, analytical in Σ > 0, α and γ, so that:

), where R = Σr.

Appendix B.

We present an explicit form of V (ρ) introduced in the proof of linear stability :

× 3l 2 ρ 12 + 60l 3 ρ 11 + 502l 4 ρ 10 + + 2250l 5 ρ 9 + 5665l 6 ρ 8 + 7500l 7 ρ 7 + 36ρ 6 P 6 Q 4 -3ρ 6 P Q 4 + 4000ρ 6 l 8 -42ρ 5 P Q 4 l + + 504ρ 5 P 6 Q 4 l -237ρ 4 P Q 4 l 2 + 2844ρ 4 P 6 Q 4 l 2 -690ρ 3 P Q 4 l 3 + 8280ρ 3 P 6 Q 4 l 3 + + 13140ρ 4 P 6 Q 4 l 2 -1095ρ 2 P Q 4 l 4 -900ρP Q 4 l 5 + 10800ρP 6 Q 4 l 5 -300P Q 4 l 6 + 3600P 6 Q 4 l 6 where P = l 2 (ρ + 4l) (ρ + 2l) 3


Alessandro Di Mare

Daniele Oriti

Emergent matter from 3d generalised group field theories

We identify classical solutions of a generalised group field theory model in 3 dimensions, and study the corresponding perturbations, deriving their effective dynamics. We discuss their interpretation as emergent matter fields. This allows us, on the one hand to test the proposed mechanism for emergence of matter as a phase of group field theory, and on the other hand to expose some limitations of the generalised group field theory formalism.

I. INTRODUCTION

In the last decades several approaches to quantum gravity have been developed, with important results [START_REF]Approaches to quantum gravity: Toward a new understanding of space, time and matter[END_REF]. Among them, Group Field Theories (GFT) [START_REF] Oriti | The group field theory approach to quantum gravity[END_REF][START_REF] Oriti | Quantum gravity as a quantum field theory of simplicial geometry[END_REF][START_REF] Freidel | Group field theory: an overview[END_REF] are, in our opinion, particularly promising. GFTs are quantum field theories defined over a group manifold, representing a "meta-space"of discrete spacetime geometries, and not spacetime itself, as a new algebraic and combinatorial realization of the "third quantization"idea [START_REF] Giddings | Baby universe, third quantization and the cosmological constant[END_REF]. Moreover, GFTs bring together most of the ingredients entering in other non perturbative and background independent approaches (such as loop quantum gravity, spin foam models and simplicial quantum gravity approaches) [START_REF] Oriti | The group field theory approach to quantum gravity[END_REF][START_REF] Oriti | Quantum gravity as a quantum field theory of simplicial geometry[END_REF]. Very little is known, still, about group field theory models for quantum gravity, in both 3 and 4 dimensions, and a lot of technical work should go into their analysis, for example their classical solutions, which will be one focus of the present work, and their purely field theoretic aspects, which are the subject of attention at present [START_REF] Freidel | Group field theory renormalization -the 3d case: Power counting of divergences[END_REF][START_REF] Magnen | Scaling behaviour of three-dimensional group field theory[END_REF][START_REF] Gurau | Colored group field theory[END_REF][START_REF] Ben-Geloun | Bosonic colored group field theory[END_REF].

An important issue in many (non-perturbative) quantum gravity formalisms is the inclusion and the correct description of matter fields. In the literature we can distiguish two different strategies. One approach is start from a model describing a quantum spacetime without matter and then add degrees of freedom describing matter, being fields, point particles or extended objects. This is the standard route both in loop quantum gravity [START_REF] Thiemann | Modern canonical quantum general relativity[END_REF][START_REF] Rovelli | Quantum Gravity[END_REF], spin foam models [START_REF] Freidel | Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles[END_REF][START_REF] Freidel | Ponzano-Regge model revisited III: Feynman diagrams and effective field theory[END_REF][START_REF] Oriti | Causality and matter propagation in 3-D spin foam quantum gravity[END_REF][START_REF] Oriti | A Spin foam model for pure gauge theory coupled to quantum gravity[END_REF][START_REF] Speziale | Coupling gauge theory to spinfoam 3d quantum gravity[END_REF] and simplicial quantum gravity [START_REF] Hamber | Discrete and continuum quantum gravity[END_REF][START_REF] Loll | Discrete approaches to quantum gravity in four dimensions[END_REF]. In a GFT context this means, for example, writing down a coupled GFT action [START_REF] Freidel | A group field theory for 3-D quantum gravity coupled to a scalar field[END_REF] for both gravity and matter fields that would produce, in perturbative expansion, a sum over simplicial complexes with dynamical geometry together with Feynman graphs for the matter fields living on the simplicial complexes. Or it means [START_REF] Dowdall | Wilson loops, geometric operators and fermions in 3d group field theory[END_REF] defining a GFT model whose Feynman amplitudes can be understood as state sum models for gravity with appropriate matter field observables. An alternative approach is to think of matter degrees of freedom as a subset of the same degrees of freedom defining the microstructure of quantum spacetime itself, and emerge as fluctuations around a background configuration of the same. This idea, pursued for example (in different ways) in [START_REF] Bilson-Thompson | Quantum gravity and the standard model[END_REF][START_REF] Kribs | Geometry from quantum particles[END_REF][START_REF] Fairbairn | 3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory[END_REF][START_REF] Girelli | 4d Deformed Special Relativity from Group Field Theories[END_REF], shares some similarities with analog gravity models in condensed matter systems, in which quasi-particles with an effective description as matter fields propagating on curved geometries emerge as fluctuations around stable vacua of the underlying many-body system (e.g. a Bose condensate), but are of course collective excitations of the same microscopic degrees of freedom just as well. A strategy for emergent matter within the GFT framework, in particular, first studied in [START_REF] Fairbairn | 3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory[END_REF][START_REF] Girelli | 4d Deformed Special Relativity from Group Field Theories[END_REF][START_REF] Oriti | Emergent non-commutative matter fields from Group Field Theory models of quantum spacetime[END_REF], is based on the following idea: starting from a fundamental GFT action, one looks for solutions of the corresponding equation of motion; then, one considers perturbations around these solutions, obtaining an effective dynamics for the perturbation field. The task is to identify a class of solutions and of perturbations, such that the effective dynamics for these perturbations takes the form of a matter field theory on an effective spacetime. It turns out [START_REF] Oriti | Emergent non-commutative matter fields from Group Field Theory models of quantum spacetime[END_REF] that the effective matter field theories emerging from GFTs following this procedure are noncommutative quantum field theories on non-commutative spacetimes with a Lie algebra structure, and with a curved momentum (group) manifold. This only makes this strategy all the more interesting. In fact, non-commutative geometry has been often advocated as the appropriate language to describe a quantum spacetime, i.e. a regime which, already far from the full non-perturbative quantum gravity dynamics at the (ultra-)Planckian scales, is still semi-classical in that it incorporates somehow the quantum gravity corrections to continuum physics at the effective level. This can take the form of an effective minimal (Planck) length scale in matter field theories, of a non trivial commutation relation between position operators in quantum mechanics (so that the notion of spacetime point, and of spacetime continuum, becomes meaningless), or of a generalised uncertainty principle, or of a form of co-gravity, i.e. curvature in momentum space [START_REF] Szabo | Quantum field theory on noncommutative spaces[END_REF][START_REF] Majid | Foundations of quantum group theory[END_REF]. It is actually this sort of effect, a curved momentum sector for particle (and field) kinematics, which shows up most naturally in a GFT context. Moreover, in recent years much work in the context of quantum gravity phenomenology [START_REF] Camelia | Quntum gravity phenomenology: status and prospectus[END_REF] was developed. On the one hand, this starts from the simple idea that several astrophysical systems could work as a magnifying lens to amplify quantum gravity effects even if Confidential: not for distribution. Submitted to IOP Publishing for peer review 12 April 2010 Planck scale suppressed, and make them accessible to experiments. On the other hand, much of this work relies on effective models of quantum gravity, rather than on the tentative fundamental formalisms currently available, and in particular on effective non-commutative models for matter kinematics and dynamics, one important example being so-called Deformed (or Doubly) Special Relativity [START_REF] Camelia | Doubly Special Relativity[END_REF]. This means that the recent results on the emergence of effective non-commutative matter field theories from GFT have the potential to help bridging the gap between the microscopic dynamics they define for quantum space and macroscopic physics and quantum gravity phenomenology.

In this paper we thus follow this strategy, and apply it to a generalised class of GFT models introduced in [START_REF] Oriti | A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity[END_REF][START_REF] Oriti | Group field theory and simplicial quantum gravity[END_REF][START_REF] Oriti | Encoding simplicial quantum geometry in group field theories[END_REF], focusing on the 3d case. This generalised GFT formalism has been introduced as an attempt to formulate GFT models, which on the one hand depended explicitly on metric-related variables, thus allowing a more manifest encoding of simplicial geometry at the level of the GFT action, and on the other hand possessed Feynman amplitudes with the explicit form of a simplicial gravity path integral, thus clarifying the relation between spin foam and simplicial gravity structures. Obviously, the viability and correctness of such a generalised GFT formalism is something to be tested. Therefore, we understand the results of this paper in two dual ways: 1) as a test of the strategy for the emergence matter from group field theories as perturbations around classical GFT solutions; in fact, we check here whether this strategy works nicely also for (much) less trivial models than those it has been applied to so far, and under which assumptions and conditions; 2) as a test of the generalised GFT formalism itself; in fact, we will see that some difficulties encountered in applying this procedure, and some unsatisfactory features of the resulting effective actions for perturbations, can naturally be understood as stemming from limitations of the generalised GFT formalism itself, rather than as failures of the "emergent matter strategy". Let us also stress that the same identification of classical solutions of the model and of the corresponding perturbative dynamics will be an interesting and highly non-trivial result, from a purely technical perspective.

The outline of the paper is as follows. In the remaining sections of the introduction, we recall how a non-commutative matter field theory for a scalar field emerges from the perturbations around classical GFT solutions in a simple 3d model, and then we introduce the 3d generalised GFT model. In section II, we consider a simplified version of the latter, identify a class of solutions of the classical GFT equations, and then study the corresponding perturbations; we will see that the basic elements of the complete analysis of perturbations to be done on the full model are already evident, together with the limitations of the same model in this respect. In section III, we move on to the complete model; we find some exact and approximate classical solutions of its equations of motion, and extract the effective dynamics for the corresponding perturbations, focusing on the approximate case. We conclude with a discussion of the results obtained and an outlook on further developments, as suggested also by our results.

A. Non-commutative matter from GFT

In this section we briefly review some results about the emergence of an effective noncommutative field theory in the 3d case [START_REF] Fairbairn | 3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory[END_REF]. We start from the following GFT action (Boulatov model), based on a group manifold being either SO(3) or SU (2):

S[φ] = 1 2 dg 1 dg 2 dg 3 φ(g 1 , g 2 , g 3 )φ(g 3 , g 2 , g 1 ) - λ 4! dg 1 .
. . dg 6 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 ), [START_REF]Approaches to quantum gravity: Toward a new understanding of space, time and matter[END_REF] where φ is a real field, that we suppose invariant under diagonal action of the group, φ(g 1 , g 2 , g 3 ) = φ(g 1 h, g 2 h, g 3 h).

The corresponding classical equation of motion is φ(g 3 , g 2 , g 1 ) = λ 3! dg 4 dg 5 dg 6 φ(g 3 , g 4 , g 5 )φ(g 5 , g 2 , g 6 )φ(g 6 , g 4 , g 1 ).

(

) 2
and a class of solutions of the same equation is parametrized by a function f : G → R, satisfying G dg f 2 (g) = 1, and given by:

φ f (g 1 , g 2 , g 3 ) = 3! λ G dh δ(g 1 h)f (g 2 h)δ(g 3 h). (3)
Since the Boulatov model defines a (3rd) quantization of BF theory, whose only configurations are locally flat geometries on arbitrary topology, such solutions can be interpreted as describing a quantum flat space. Now we want to study classical perturbations around these solutions, in particular those of the form: ψ(g 1 , g 2 , g 3 ) ≡ ψ(g 1 , g 3 ) = ψ(g 1 g -1

3 ). For them we obtain the following effective action:

S (f ) [ψ] ≡ S[φ f + ψ] -S[φ f ] = dgψ(g)K f (g)ψ(g -1 ) - µ 3! dg 1 dg 2 dg 3 δ(g 1 g 2 g 3 ) ψ(g 1 )ψ(g 2 )ψ(g 3 ) + - λ 4! dg 1 dg 2 dg 3 dg 4 δ(g 1 . . . g 4 ) ψ(g 1 )ψ(g 2 )ψ(g 3 )ψ(g 4 ). (4)
with kinetic term given by

K f (g) = 1 2 1 -2 dh f (h) 2 -dh f (h)f (hg) . (5)
The nature of this action as an effective scalar field theory on a noncommutative space is evident once we interpret the group manifold on which it is defined as momentum space, so that the constraints δ(g 1 g 2 g 3 ) and δ(g 1 . . . g 4 ) represent the momentum conservation. As a curved space, this momentum space is dual to a noncommutative configuration space with a Lie algebra structure (given by the su( 2)). This duality can be made explicit by introducing a Fourier transform mapping functions over the group to functions over the corresponding algebra [START_REF] Livine | Matrix Models as Non-commutative Field Theories on R 3[END_REF][START_REF] Freidel | Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity[END_REF], using which we can rewrite the above action in term of functions on R 3 endowed with a noncommutative -product [START_REF] Livine | Matrix Models as Non-commutative Field Theories on R 3[END_REF][START_REF] Freidel | Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity[END_REF], which can then be shown to be invariant under the quantum double of SU (2), a deformation of the euclidean 3d Poincaré group.

Similar results have been obtained also in the 4d case and in Lorentzian signature [START_REF] Girelli | 4d Deformed Special Relativity from Group Field Theories[END_REF], where it has been shown that, starting from a GFT model for SO(4, 1) BF theory, one can obtain an effective non-commutative field theory of the DSR type, i.e. for a scalar field living on the κ-Minkowski spacetime and with a De Sitter space of momenta. Obviously, this is a model of a more direct interest for quantum gravity phenomenology. We can conclude that perturbations around a non trivial quantum geometric background, defined by a solution of the GFT equations, follow the dynamics of an effective scalar field theory of a noncommutative type. So we can interpret matter as a phase of quantum geometry, therefore already contained in our quantum gravity models. Moreover, this type of analysis of GFT perturbations is useful to expose the deep link between the GFT formalism and noncommutative geometry.

B. Generalised GFT model

A generalised GFT formalism has been introduced in [START_REF] Oriti | A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity[END_REF] and further developed in both 3d and 4d in [START_REF] Oriti | Group field theory and simplicial quantum gravity[END_REF][START_REF] Oriti | Encoding simplicial quantum geometry in group field theories[END_REF], its purpose being encoding and controlling in a more explicit way simplicial geometry at the level of the GFT action, and obtaining Feynman amplitude with the manifest form of a simplicial gravity path integral. In turn, this should serve as a complete definition of a quantum dynamics of simplicial structures, to obtain a spin foam model with a clear underlying geometry and to strengthen the links between spin foam and simplicial gravity approaches.

In the 3d case, the fundamental variable is a real function of 3 group variables and 3 algebra variables: φ :

G 3 × g 3 → R, which satisfies φ(g 1 , g 2 , g 3 ; B 1 , .., B 3 ) = φ(g 1 h, g 2 h, g 3 h; hB 1 h -1 , hB 2 h -1 , hB 3 h -1
). The group and Lie algebra variables are interpreted as the discrete triad and connection variables of simplicial BF theory or, equivalently, of discrete 3d gravity. Its classical action is the following:

S = 1 2 1 (2π) 9 G 3 dg 1 dg 2 dg 3 g 3 d B 1 d B 2 d B 3 φ 123 Ô1 Ô2 Ô3 φ * 123 + - λ 4!(2π) 36 G 6 dg 1 ...dg 6 g 6 d B 1 ...d B 6 φ 123 φ 345 φ 526 φ 641 , (6)
where we have introduced the operator Ô = G + B 2 + m 2 8 , with G the Laplace-Beltrami operator on the group manifold G, and the notation

φ 123 = φ(g 1 , g 2 , g 3 ; B 1 , B 2 , B 3 ).
Notice that, even though the field is a function of Lie algebra triad variables, it is treated in this formalism as an ordinary function, and the Lie algebra itself is considered in its vector space aspects only, so that integrations are performed by means of the Lebesgue measure and no star product is introduced when multiplying more than one GFT field. This will have important consequences when studying the corresponding perturbations around classical solutions.

For the achievements and limitations of this formalism concerning its original goals, mentioned above, we refer to the literature. Here we will only try to see whether the results on perturbations obtained for the Boulatov model extend to this more complicated case, and at the same time use this analysis of perturbations as a test for the formalism, trying to expose further its limitations by making them explicit at the level of the resulting effective dynamics.

II. THE STATIC-ULTRALOCAL CASE

Let us begin from the analysis of the simpler "static-ultralocal"truncation of the model. It corresponds to the case in which the kinetic operator is "freezed" to the identity, imposing Ô = I in ( 6):

S = 1 2 1 (2π) 9 G 3 dg 1 dg 2 dg 3 g 3 d B 1 d B 2 d B 3 φ 123 φ 321 - λ 4!(2π) 36 G 6 dg 1 ...dg 6 g 6 d B 1 ...d B 6 φ 123 φ 345 φ 526 φ 641 . (7)
In the following we always assume that solutions depend on the moduli of the vectors B, and not from its direction, so that the invariance property of the field under the diagonal group action on the Lie algebra arguments hBh -1 is trivialized. Moreover, to semplify the notation, we define B = B . Anyway we use the notation d B, to distinguish integration over all algebra with integration over only the module dB.

A. Solutions of the equations of motion

The equation of motion corresponding to the above action is:

φ 123 = λ 3! 1 (2π) 27 G 3 dg 4 dg 5 dg 6 g 3 d B 4 d B 5 d B 6 φ 345 φ 526 φ 641 . ( 8
)
and it is easy verify that the following class of functions, a simple generalization of ( 3), define classical solutions:

ϕ 123 = 3!(2π) 27 λ G dh δ(g 1 h)ψ 1 (B 1 )f (g 2 h; B 2 )ψ 3 (B 3 )δ(g 3 h), (9)
where the functions f , ψ 1 and ψ 3 have to fulfil the following relations:

G dg g d B f 2 (g, B) = 1 g d B ψ 1 (B)ψ 3 (B) = 1. ( 10
)
We also require, for simplicity, the function f (g, B) to be separable in its variables:

f (g, B) = f 1 (g)f 2 (B).

B. Perturbations

Let us consider a generic perturbation (g 1 , g 2 , g 3 ; B 1 , B 2 , B 3 ), and compute the effective action:

S[ ] ≡ S[ϕ + ] -S[ϕ] = 1 2 [dg] [d B] 123 321 + - [dg]dh 1 [d B]ψ 1 (B 5 )f 1 (g 2 h 1 )f 2 (B 2 )f 1 (g 4 h 1 )f 2 (B 4 )ψ 3 (B 1 ) (h -1 1 , g 2 , g 3 ; B 1 , B 2 , B 3 ) (g 3 , g 4 , h -1 1 ; B 3 , B 4 , B 5 ) + - 1 2 [dg]dh 1 dh 2 [d B]ψ 1 (B 3 )f 1 (g 4 h 1 )f 2 (B 4 )ψ 3 (B 1 )ψ 3 (B 5 )ψ 1 (B 6 )f 1 (g 4 h 2 )f 2 (B 4 )ψ 3 (B 6 ) × × (h -1 2 , g 2 , h -1 1 ; B 1 , B 2 , B 3 ) (h -1 1 , g 2 , h -1 2 ; B 5 , B 2 , B 6 ) - λ 4!(2π) 27 [dg] [d B] 123 345 526 641 + - λ 3!(2π) 27 [dg]dh [d B] ψ 1 (B 6 )f 1 (g 4 h)f 2 (B 4 )ψ 3 (B 1 ) (h -1 , g 2 , g 3 ; B 1 , B 2 , B 3 ) 345 (g 5 , g 2 , h -1 ; B 5 , B 2 , B 6 ),
where we used the notation 123 = (g 1 , g 2 , g 3 ; B 1 , B 2 , B 3 ), where convenient. This is the action for a generic perturbation, only assumed to have the sam diagonal invariance of the fundamental GFT field, but it simplifies considerably, and can then be interpreted physically, by appropriate simplified choices on the perturbation field. In particular, having in mind the possible interpretation of the B and g variables as configuration and momentum variables, respectively, we try to reduce the dependence of the perturbation field to either of the two sectors. Case 1: Dynamical group sector Now we show how one can recover the same effective action in [START_REF] Fairbairn | 3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory[END_REF] with a particular choice of perturbation field. We assume a separable dependence on g and B variables (this is suggested also by the fact that neither the kinetic nor the interaction term couple the two sectors), and the dependence on a single group variable via the combination

g 1 g -1 3 : (g 1 , g 2 , g 3 ; B 1 , B 2 , B 3 ) = φ(g 1 g -1 3 )ξ(B 1 , B 2 , B 3
). Factoring out the integration over the B variables, we get the following effective action for the perturbation field φ(g):

S[φ] = dg φ(g)K(g)φ(g -1 ) - η 3! [dg] 3 δ(g 1 . . . g 3 )ψ(g 1 ) . . . ψ(g 3 ) - µ 4! [dg] δ(g 1 . . . g 4 )ψ(g 1 ) . . . ψ(g 4 ). (11)
Here, the kinetic term is given by

K f (g) = 1 2 α -2β dh f (h) 2 -γ dh f (h)f (hg) , (12)
which is the same of the action ( 4), with up to some appropriate redefinition of coupling constants:

α = d B 1 d B 2 d B 3 ξ(B 1 , B 2 , B 3 )ξ(B 3 , B 2 , B 1 ), β = d B 1 . . . d B 5 ψ 1 (B 5 )f 2 (B 2 )f 2 (B 4 )ξ(B 1 , B 2 , B 3 )ξ(B 3 , B 4 , B 5 ), γ = d B 1 . . . d B 6 ψ 1 (B 3 )f 2 (B 4 )ψ 3 (B 5 )ψ 1 (B 6 )f 2 (B 4 )ψ 3 (B 6 )ξ(B 1 , B 2 , B 3 )ξ(B 3 , B 4 , B 5 ), η = √ 6λ dgf 1 (g) d B 1 . . . d B 6 ψ 1 (B 6 )ψ 3 (B 1 )f 2 (B 4 )ξ(B 1 , B 2 , B 3 )ξ(B 3 , B 4 , B 5 )ξ(B 5 , B 2 , B 6 ) µ = d B 1 . . . d B 6 ξ(B 1 , B 2 , B 3 )ξ(B 3 , B 4 , B 5 )ξ(B 5 , B 2 , B 6 )ξ(B 6 , B 4 , B 1 ).
Of course the perturbation field must be chosen such that this new set of coupling constants does not diverge, with the functional dependence on the B sector to be considered as fixed, with no variation allowed in the classical theory and no fluctuations (functional integration) in the quantum theory. We see that we have recovered, even from the more complicated GFT formalism, the expected effective action for the perturbations, that can be interpreted as a noncommutative scalar field theory written in momentum space. Even in this drastic simplification of the formalism, we also see, however, that the way we could obtain it was to completely decouple the B and g sector and to leave only this last one as dynamical.

Case 2: Dynamical Lie algebra sector Now we focus instead on the B sector, and treat the group sector as non-dynamical. The idea is then to reproduce the same action as above, or a modification thereof, but written now in configuration space, according to the standard interpretation of the effective theory and of the variables appearing in the GFT, outlined above.

As a regularization, we introduce a cut-off B max in the B integration, and the constant V B = 4π

Bmax 0 dBB 2 , that is the (truncated) volume of R 3 . We then consider a perturbation of the form

123 = 1 √ V B δ( B 1 -B 2 ) ψ( B 2 + B 3 ) ( 1 3 )
with no dependence on group variables (notice that this does not lead to any divergence because of the compact nature of the group manifold). The dependence on the three variables B i has been chosen to be what would have followed from a Fourier transform from g to B variables of the function we used as perturbation in the previous case. Moreover we introduce a new condition on the solution imposing that dgf (g) = 0. With this condition the integration over the group variables kills the cubic interaction and some quadratic terms. The resulting effective action for this perturbation field is:

S[ψ] = 1 2 d B ψ 2 (B) + λ 4! d B ψ 4 (B) ( 1 4 )
that is an ordinary local λφ 4 theory with a trivial kinetic term (i.e. again a static-ultralocal field theory).

We can interpret this as a scalar field theory written in configuration space (given by R 3 ). The triviality of the kinetic term arises directly from the initial ultra-locality restriction, but could possibly be lifted by some more involved choice of classical solution or perturbation field, at the cost of introducing derivatives in the same. The point to notice, however, is that we find no sign of the expected underlying non-commutativity in this field theory. That is, while we find the expected local dependence on B variables that matches their interpretation as configuration variables in some sense dual to the group variables, and while we could try to modify our construction to obtain a more complicated (and interesting) kinetic term or interaction, it is evident that the conjugate nature of B and g sectors is lost and cannot be recovered. Due to the simplicity of the underlying model, it is also clear that this conjugate nature fails to be present in the effective action because it fails to be correctly implemented in the original generalised GFT formalism. We will come back to this point in the following.

III. FULL GENERALISED MODEL

In this section we consider the complete action ( 6) and the corresponding integro-differential equation of motion :

Ô1 Ô2 Ô3 φ 321 = λ 3!(2π) 27 G 3 dg 4 dg 5 dg 6 g 3 d B 4 d B 5 d B 6 φ 345 φ 526 φ 641 . (15)

A. Exact solution of the equation and perturbations

We first try to find an exact solution of this equation and to obtain an effective field theory for the perturbations. As it is apparent, the integro-differential equation 15 presents a very complicated structure and there is no standard or general procedure to solve it. For simplicity we look for solutions that are separable in the three set of variables, i.e. of the type:

φ 123 = dh ψ 1 (g 1 h, B 1 )f (g 2 h, B 2 )ψ(g 3 h, B 3 ), ( 16
)
where we have imposed the usual invariance under diagonal right action of the group, by explicit projection. It should be clear that this is a choice and that the equation could have solutions which do not fulfill the requirement of separability1 , and that more general type of functions can be considered. Unfortunately, we have not been able to identify any solution outside this simpler class of functions, despite intense efforts. Therefore, while we cannot exclude the existence of more general solutions, we are forced to work within this simpler class. Now, it is easy to show that any function φ 123 of the above form is an exact solution of the GFT equation of motion if the following conditions are satisfied:

1. the functions ψ 1 , f, ψ 3 are solutions of the equation

+ B 2 φ(g, B) = 0, (17)
so they are eigenfunctions of the operator Ô with eigenvalue -m 2 8 . 2. ψ 1 , ψ 3 and f satisfy

d B dg ψ 3 (gh 1 , B)ψ 1 (gh 2 , B) = δ(h -1 1 h 2 ) d B dg f 2 (g, B) = 1. ( 18
)
We now look for such functions, starting from condition 1. We now prove the following: Proposition: Let I be the set of functions defined on the Cartesian product G×g and square integrable with respect to the group variable (so that we can apply the Peter Weyl theorem to decompose them in irreducible representations j of G). The most general solution, in the set I, of the equation ( 17) is of the form

φ(g, B) = j,n,m φnm j (B)D j nm (g) with    φ j mn (B) arbitrary ∀B : B 2 = j(j + 1) φ j mn (B) = 0 elsewhere. ( 19
)
The proof is straightforward. Because of Peter-Weyl theorem, every solution of the equation, belonging the set I, can be written as φ(g, B) = j,n,m φnm j (B)D j nm (g), for some coefficient functions φ j mn (B). Being a solution, it has to satisfy:

∞ j=0 n,m
-j(j + 1) + B 2 φnm j (B) D j nm (g) = 0, which implies, since the D j mn (g) is a complete orthonormal basis in I, that: [-j(j + 1) + B 2 ] φnm j (B) = 0 ∀j, n, m. This equation can be satisfied only if the coefficient functions φ j mn (B) are null for all values such that B 2 = j(j + 1), while they can be arbitrary otherwise.

Therefore, the functions [START_REF] Freidel | A group field theory for 3-D quantum gravity coupled to a scalar field[END_REF], considered as function of B are non null only on spheres of radius j(j + 1). Thus the integral over the algebra of these eigenfunctions is obviously zero, because each sphere is a set of null measure. Therefore the conditions 2 and 3 unless the functions φnm j (B) are actually distributions on R 3 . The idea is then to define a 3 dimensional delta-like function, i.e. null everywhere, except on the origin and on the spheres of radius j(j + 1), with j ∈ N + , where it goes to infinity, and multiply it by a regular function of B to define completely the functions φnm j (B) . There are many possibilities to define this distribution with a limit procedure, and we do it in appendix, providing explicit examples. In the following we call that distribution α( B). Its only property we need is:

d B α 2 ( B)f (B) = ∞ j=0 (2j + 1) 2 f ( j(j + 1))), (20)
where both integral and sum have to converge. Let ψ 1 (g, B) and ψ 3 (g, B) solutions of [START_REF] Hamber | Discrete and continuum quantum gravity[END_REF]. We also introduce the following distributions:

ϕ 1 (g, B) = α( B)ψ 1 (g, B) ϕ 3 (g, B) = α( B)ψ 3 (g, B). (21)
These define solutions too, as they satisfy condition 1, as it can be easily checked. Now we consider condition 2, and compute the following integral:

dg d Bϕ 1 (gh 1 , B)ϕ 3 (gh 2 , B) = dg dBα 2 ( B)ψ 1 (gh 1 , B)ψ 3 (gh 2 , B) = = ∞ j=0 (2j + 1) 2 dg ψ 1 gh 1 , j(j + 1) ψ 3 gh 2 , j(j + 1) = δ(h 1 h -1 2 ).
Using the above theorem, we know that ψ g, j(j + 1) = nm ψ( j(j + 1))

nm j D j nm (g). ( 22
)
which results in the condition:

ψ1 nm1 j ψ3 nm3 j = δ m1m3 ∀j ∈ N , (23)
which characterizes functions satisfying condition 2.

Using the same distribution α( B), we can define a function satisfying condition 3 too. Let f be a function of the type 19; we define the distribution F (g, B) = α( B)f (g, B), and impose condition 3, to find

dg d B F 2 (g, B) = dg d Bα 2 ( B)f 2 (g, B) = ∞ j=0 (2j + 1) 2 dg f 2 (g, j(j + 1)). (24)
If this sum is convergent then we can normalize f so that the result is 1 and 3 is satisfied. Therefore, using functions satisfying the above conditons, we have found an exact solution of equation ( 15):

φ 123 = dh ϕ 1 (g 1 h, B 1 )F (g 2 h, B 2 )ϕ(g 3 h, B 3 ) = dh α( B 1 )ψ 1 (g 1 h, B 1 )α( B 2 )f (g 2 h, B 2 )α( B 3 )ψ 3 (g 3 h, B 3 ).
Now we write down the effective action for a generic perturbation around such solution:

S[ ] = 1 2 1 (2π) 9 [dg] [d B] 123 Ô1 Ô2 Ô3 321 - λ 4!(2π) 36 [dg] [d B] 123 345 526 641 + - 1 (2π) 9 [dg]dh 1 dh 2 [d B] ψ 1 (g 1 h 1 , B 1 )f (g 2 h 1 , B 2 )f (g 4 h 2 , B 4 )ψ 3 (g 5 h 2 , B 5 )Ψ(h -1 1 h 2 ) 526 641 - 1 2 1 (2π) 9 [dg]dh 1 dh 2 [d B] ψ 1 (g 1 h 1 , B 1 )ψ 3 (g 3 h 1 , B 3 )ψ 1 (g 5 h 2 , B 5 )ψ 3 (g 6 h 2 , B 6 )ξ(h -1 1 h 2 ) 345 641 - λ 3!(2π) 45 [dg] [d B]ψ 1 (g 1 h, B 1 )f (g 2 h, B 2 )ψ 3 (g 3 h, B 3 ) 345 526 641 (25)
where we have defined:

Ψ(h -1 1 h 2 ) ≡ dg d B ψ 3 (gh 1 , B)ψ 1 (gh 2 , B) ξ(h -1 1 h 3 ) ≡ dg d B f(gh 1 , B)f (gh 3 , B). ( 26
)
This is a very general result, holding for any solution and any perturbation. Now we would obtain, starting from 25, an action with a kinetic term of the type dx φ(x)K(x)φ(x), either in group or in Lie algebra variables, imposing some particular form of the perturbation field. This turns out to be very difficult, and we have not been able to identify any simple form of the perturbations that would give such simplification. Roughly speaking, it is due to the presence of the terms 526 641 and 345 641 , which are not equivalent, because we have not assumed any invariance of the GFT field, nor of the corresponding perturbation, under permutations of its arguments. There is no simple choice of perturbation field that gives a product of the type (x) (x), for example no field with a dependence on the B variables only through the linear combination aB 1 + bB 2 + cB 3 , or the analogue in the G variables. This is true, unless we assume that the fundamental GFT field as well as the classical solution chosen and the corresponding perturbation field are invariant under permutations of their group and Lie algebra arguments. Imposing this property however, would complicate the analysis considerably. Leaving aside the issue of perturbations, the problem can be seen as due to the non-separability of the solution chosen, in particular the functions f and Ψ. We cannot exclude that a different solution, or a more involved choice of perturbation would lead to a nicer effective action. However, we also note that, even if one solves somehow this first problem, another trouble comes from the distribution α( B), which appears in the effective action for the perturbations with different powers, leading to additional technical complications.

This suggests us to leave aside for the moment the issue of perturbations around exact solutions of the generalised model. We turn our attention, instead, to approximate solutions and their corresponding perturbations.

B. Approximate solutions and perturbations

The basic idea here is to look for eigenfunctions of the Laplace-Beltrami operator and, at the same time, good approximations of the delta function on the group. We note that every good approximation of the delta function on the group has to be sharply peaked on the identity. Therefore the Laplace-Beltrami operator applied to such a function will be close to zero everywhere but in a neighbourhood of the identity element. For this reason we study now the behaviour of the Laplace-Beltrami operator on the group SU (2) which is isomorphic to the 3-sphere S 3 , very close to the identity element, which we identify with the north pole. Now we fix a chart in a neighborhood of the identity element, and we use polar coordinates. The Laplace-Beltrami operator becomes:

= 1 √ g ∂ µ ( √ gg µν ∂ ν ) ≈ 1 θ 2 ∂ ∂θ θ 2 ∂ ∂θ + 1 θ 2 sin φ ∂ ∂φ sin φ ∂ ∂φ + 1 θ 2 sin 2 φ ∂ 2 ∂φ 2
If we apply this operator to functions f only dependent on the θ variable, we can consider only the first term in the previous formula, so we can use the operator:

A f = 1 θ 2 ∂ ∂θ θ 2 ∂f ∂θ = 1 θ ∂ 2 ∂θ 2 (θf ) , (27)
where the A indicates both the approximation in the operator and that we can apply it just on function of θ considerably different from zero only in a small neighbourhood of θ = 0. Now we consider the eigenvalue problem for this operator: A f (θ) = λf (θ). Among the solutions of this equation, one is particular interesting for our purposes f (θ) = e -θ a θ , where λ = 1 a 2 . Indeed we note that in the limit a → 0 + the function f (θ) goes to zero everywhere but in θ = 0 where it goes to infinity. Therefore, with appropriate normalization, this is a good representation of the Dirac delta function. Thus we define the function

δ a (θ) = e -θ a a 2 θ ( 28
)
that satisfies the following properties:

δ a (θ) ≈ A δ a (θ) = 1 a 2 δ a (θ) l i m a→0 + δ a (θ) = δ(θ), (29)
Of course we can rewrite the δ a function in term of the group variable, using the parametrization: g = cos θI+i sin θn• σ, with Tr(g) = 2 cos(θ) and n ∈ S 2 . Through this relation we can interpret the δ a (θ) as a δ a (g) on SU(2). Now we apply the operator Ô on δ a (g), with a ≈ 0 + , to find:

+ B 2 - m 2 8 δ a (g) ≈ 1 a 2 + B 2 - m 2 8 δ a (g) ≈ 1 a 2 δ a (g). (30)
Of course the approximation improves as a tends to zero. Indeed we have supposed that 1 a 2 >> B 2 -m 2 8 . In order for our solution of the equations of motion to satisfy this condition for all relevant values of B, we then assume that it is a function with compact support (i.e. identically zero outside of some bounded set) with respect to the B variables. Another important point is that we can handle δ a (g) like a true δ(g) function, as it satisfies dg δ a (gh -1 )f (g) ≈ f (h). The last step is to introduce another function ∆ a (θ) on the sphere S 3 . It is defined, for 0 ≤ θ < π/2 (i.e. for the the half-sphere corresponding to SO(3)), as

∆ a (θ) = 1 2 1 √ πa e -θ a θ = 1 2 √ π a 3 2 δ a (θ) ( 3 1 )
and it is extended to the whole S 3 ≈ SU (2) (i.e. to 0 ≤ θ < π), by requiring ∆ a (g) = -∆ a (-g), again for a ≈ 0. This function is almost vanishing everywhere but for g = I, -I and satisfies:

∆ a (θ) ≈ 1 a 2 ∆ a (θ) dg∆ a (g) = 0 dg ∆ 2 a (g) = 1. (32)
In the last two properties the integration is over SU (2). If we restricted the integration only over SO(3), the first integral gives 2 √ πa 3/2 and the second one gives 1/2. Finally using all the above, it is easy to verify that the following function is an approximate solution of ( 15):

φ 123 = 3!(2π) 27 λa 6 dh δ a (g 1 h)ψ 1 (B 1 )∆ a (g 2 h)f (B 2 )δ a (g 3 h)ψ 3 (B 3 ), (33)
where ψ 1 , f, ψ 3 are functions with compact support, dependent only on the module of B, and satisfy the conditions

d B f 2 (B) = 1 d B ψ 1 (B)ψ 3 (B) = 1. (34)
Next, we study perturbations around this new approximate solutions. The effective action for a generic perturbation field (g 1 , g 2 , g 3 ; B 1 , B 2 , B 3 ) is:

S[ ] ∝ 1 2 [dg] [d B] 123 Ô1 Ô2 Ô3 321 - λ 4!(2π) 27 [dg] [d B] 123 345 526 641 + - 1 a 6 [dg]dh [d B] ψ 1 (B 5 )∆ a (g 2 h)f (B 2 )∆ a (g 4 h)f (B 4 )ψ 3 (B 1 ) (h -1 , g 2 , g 3 ; B 1 , B 2 , B 3 ) (g 3 , g 4 , h -1 ; B 3 , B 4 , B 5 ) + - 1 2 1 a 6 [dg]dh 1 dh 2 [d B] ψ 1 (B 3 )∆ a (g 4 h 1 )ψ 3 (B 5 )ψ 1 (B 6 )∆ a (g 4 h 2 )ψ 3 (B 1 ) × × (h -1 2 , g 2 , h -1 1 ; B 1 , B 2 , B 3 ) (h -1 1 , g 2 , h -1 2 ; B 5 , B 2 , B 6 ) + - λ 3!(2π) 27 a 6 [dg]dh [d B] ψ 1 (B 6 )∆ a (g 4 h)f (B 4 )ψ 3 (B 1 ) (h -1 , g 2 , g 3 ; B 1 , B 2 , B 3 ) 345 (g 5 , g 2 , h -1 ; B 5 , B 2 , B 6 ) (35)
Here there is a first difference with respect to the previous case: due to the approximation, the linear terms, which would cancel exactly when perturbing around exact extrema of the action, do not cancel exactly but are nevertheless suppressed with a and we have not written them for this reason. In order to get a "good" effective theory we need again to impose some special conditions on the perturbation field. As in the static-ultralocal theory, we have two sets of variables, group and algebra, therefore we have more room for manouvering with respect to previous models. The goal is again to obtain a field theory with a kinetic term written in the usual form dx φ(x)K(x)φ(x), and thus with a field depending on a single (group or algebra) argument. Moreover we recall that the group and algebra variables have the interpretetion of momentum and configuration variables in the usual matter field theory of noncommutative type.

The first assumption is the separability of the perturbation field:

(g 1 , g 2 , g 3 ; B 1 , B 2 , B 3 ) = ϕ(g 1 , g 2 , g 3 )φ(B 1 , B 2 , B 3 ) ( 3 6 )
Next, having again in mind the possible interpretation of the B and g variables as configuration and momentum variables, respectively, we try to reduce the dependence of the perturbation field to either of the two sectors, by treating the other sector as non-dynamical, as in the static-ultralocal case.

Case 1: Dynamical group sector We recall that we have fixed the value of a very close to zero. Thus we notice immediatly that in the action [START_REF] Joung | Three Dimensional Quantum Geometry and Deformed Poincare Symmetry[END_REF] we have two dominant quadratic terms. We restrict our attention to them, i.e. to the free field theory sector. The dominant terms, comparing powers of 1/a, are

S[ ] = - 1 (2π) 9 1 a 6 dg 2 dg 3 dg 4 dh 1 d B 1 . . . d B 5 ψ 1 (B 5 )∆ a (g 2 h 1 )f (B 2 )∆ a (g 4 h 1 )f (B 4 )ψ 3 (B 1 ) × × (h -1 1 , g 2 , g 3 ; B 1 , B 2 , B 3 ) (g 3 , g 4 , h -1 1 ; B 3 , B 4 , B 5 ) + - 1 2 1 (2π) 9 1 a 6 dg 2 dg 4 dh 1 dh 2 d B 1 d B 2 d B 3 d B 5 d B 6 ψ 1 (B 3 )∆ a (g 4 h 1 )ψ 3 (B 5 )ψ 1 (B 6 )∆ a (g 4 h 2 )ψ 3 (B 6 ) × × (h -1 2 , g 2 , h -1 1 ; B 1 , B 2 , B 3 ) (h -1 1 , g 2 , h -1 2 ; B 5 , B 2 , B 6 )
Now we separate the variables like in [START_REF] Baratin | A non-commutative representation for group field theories and simplicial quantum gravity[END_REF] and we impose the usual condition ϕ(g 1 , g 2 , g 3 ) = ϕ(g 1 g -1 3 ). After integrating out B variables, treating the part of the perturbation field depending on them as non-dynamical, we get

S[ϕ] = 1 2 1 (2π) 9 C a 6 dg ϕ(g)K(g)ϕ(g -1 ), (37)
where the kinetic operator is

K(g) = dh ∆ a (h)∆ a (hg), (38)
and we have defined the constant

C = d B 1 B 2 B 3 B 5 B 6 ψ 1 (B 3 )ψ 3 (B 5 )ψ 1 (B 6 )ψ 3 (B 1 )φ 123 φ 526 .
We see that we obtain once more a nice field theory on a single group manifold, interpreted as (curved) momentum space, whose dynamics is again entirely determined by the classical GFT solution chosen, thus by the quantum spacetime we select when selecting the solution itself, and on which the effective scalar field propagates. Once more, we could Fourier transform from group to Lie algebra space, playing the role of effective spacetime, introducing the appropriate star product.

Case 2: Dynamical Lie algebra sector Here we restrict to the SU (2) case. Without this restriction our choice of perturbation field gives a non-local effective action. This case, from a technical point of view, is very similar to the static-ultralocal limit. Indeed we choose exactly the same type of perturbation field:

123 = 1 √ V B δ( B 1 -B 2 )ψ( B 2 + B 3 ) ( 3 9 )
where V B is again the (truncated) volume of R 3 . After a straightforward calculation, we get

S[ψ] = 1 128 d B ψ( B)K(B)ψ( B) + λ 4! d B ψ 4 ( B) ( 4 0 )
where

K(B) = (B 2 -m 2 2 ) 3
. This is a nice local effective action with a rather unusual kinematical term, with an algebraic dependence on the configuration space Lie algebra variable. Once more, the specific form of the kinetic term could possibly be modified by different choices of classical (approximate) solution or perturbation field, even though it is not straightforward to envisage how to do so. What can not be modified easily is again the fact that we find no sign of the expected underlying non-commutativity in this field theory, and that the conjugate nature of B and g sectors is lost. This confirms that something is dubious in the way the original GFT model, i.e. the generalised GFT formalism, because we can once more trace back the origin of this limitation in the effective field theory to the way the generalised GFT formalism deals with the dependence on the GFT field on the Lie algebra B variables.

IV. DISCUSSION AND CONCLUSIONS

Let us summarize the results we have obtained. We have studied the issue of classical solutions of the generalized 3d GFT model proposed in [START_REF] Oriti | A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity[END_REF][START_REF] Oriti | Group field theory and simplicial quantum gravity[END_REF][START_REF] Oriti | Encoding simplicial quantum geometry in group field theories[END_REF], and of the effective dynamics of the corresponding perturbations, trying to identify specific solutions and perturbations such that the dynamics of the latter can be understood as that of a scalar field on an effective (noncommutative) spactime. The motivations were the following. First of all, to improve our understanding of such models, at a more technical level. Second, to show that indeed effective matter field theories could be extracted as phases of such generalised models. This means on the one hand again solving the technical difficulties coming from the more involved structure of this model, as compared to the easier ones which have been dealt with up to now, but on the other hand showing that the idea of matter as emerging from the collective dynamics of the micro-costituents of quantum spacetime is more solid than could be originally thought, holding for a wider class of GFT models. Third, and conversely, to use this procedure for extracting effective matter theory from GFT as a test for the generalised GFT formalism itself. This means learning from the way this strategy for emergent matter is made to succeed or from the way it fails, in this context, what are the weak and dubious points of the formalism. Our results are indeed mixed. We have managed to identify exact solutions of the GFT equations of motion both in the simpler static-ultralocal truncation of the model and in the full, more complicated, one. In the truncated model we could extract rather straightforwardly an effective scalar field theory for the perturbations. When only the group sector of the model is treated as dynamical, the effective field theory takes indeed the expected form of a non-commutative field theory written in momentum space. One could then use a non-commutative Fourier transform to re-write the theory in configuration space, given by the su(2) Lie algebra. Because of the presence of both Lie algebra and group variables in the fundamental GFT model, one could also expect that it should be possible to obtain directly a non-commutative field theory for the perturbations in configuration space, by treating directly the only Lie algebra sector of the perturbations as dynamical. This turns out not to be the case. One obtains indeed an effective scalar field theory on the same Lie algebra, but without the wanted and expected non-commutative structure. This turn out to be the result both in the truncated model and in the full model, for which we have been able to extract a nice(r) effective dynamics for perturbations, i.e. one that could be interpreted as a scalar field theory on an emergent 3d spacetime, only within a certain approximation, that is only around approximate solutions of the equations of motion. Moreover, a closer look at the way the solutions of the equations of motions could be found and the effective dynamics for perturbations extracted shows that it has been necessary, also when dealing with the full model, to basically trivialize the kinetic term of the theory to the level of the corresponding equations, i.e. look for solutions for which the non-trivial part of the kinetic term gives a basically trivial contribution to the equations. While this last point, although somehow disappointing, could be argued to be solvable with more work and some more clever analysis of the same equations, the first point appears to us to be more fundamental, i.e. to suggest a fundamental limitation of the generalized GFT formalism itself. Moreover, we have worked with special types of classical solutions of the GFT equations, separable in the three pairs of group+algebra variables. Again, we cannot exclude that some of the difficulties we encountered in deriving satisfactory effective non-commutative field theories for matter are due to this restriction. While this is true, however, it also true that, again, the first point, i.e. the lack of non-commutative structure in the B variables, seems still more fundamental that this somewhat practical limitation. Let us be more precise. The generalised GFT formalism of [START_REF] Oriti | A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity[END_REF][START_REF] Oriti | Group field theory and simplicial quantum gravity[END_REF][START_REF] Oriti | Encoding simplicial quantum geometry in group field theories[END_REF] is based on the idea of enlarging the set of variables on which the GFT field depends from group elements alone to include Lie algebra elements as well, interpreted as discrete connection and B field of a BF-based description of simplicial geometry, which should become manifest at the level of the Feynman amplitudes. The two sets of variables are treated on equal footing and in parallel, while maintaining the GFT an ordinary, even if non-local with respect to the combinatorics of arguments, field theory. This implies two choices: first, treating su(2) as an ordinary vector space, thus neglecting its non-commutative nature, and, second, relaxing the conjugate nature (at the classical level) of B and g variables in the definition of the model (which characterizes BF theory), and more specifically in the choice of kinetic term. Some consequences of these choices from the point of view of simplicial geometry are discussed in [START_REF] Oriti | Group field theory and simplicial quantum gravity[END_REF][START_REF] Oriti | Encoding simplicial quantum geometry in group field theories[END_REF]. The results we have presented concerning the effective dynamics of GFT perturbations, in particular the "negative"ones, suggest that the way the Lie algebra B variables are introduced and treated in the generalised GFT formalism is not correct. The apparent need to trivialize the kinetic term in the choice of classical solution suggests that only when the conjugate nature of the B and g variables is implemented, and thus a BF kynematics is chosen, the corresponding classical solutions define an effective (quantum) flat spacetime as desired. The lack of noncommutative structure in the effective action in the Lie algebra sector suggests instead that the correct way of introducing the B variables in the formalism is not to enlarge the domain space of the GFT field, but to use from the start a noncommutative Fourier transform (in particular, the one introduced and studied in [START_REF] Freidel | Ponzano-Regge model revisited III: Feynman diagrams and effective field theory[END_REF][START_REF] Freidel | Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity[END_REF][START_REF] Joung | Three Dimensional Quantum Geometry and Deformed Poincare Symmetry[END_REF] to go from group space to Lie algebra space, turning the original formulation of GFTs into a noncommutative (and combinatorially non-local) one. All these suggestions are indeed taken into account in the new noncommutative formulation of group field theories developed in [START_REF] Baratin | A non-commutative representation for group field theories and simplicial quantum gravity[END_REF], which, on top of solving several difficulties we encountered here in the study of perturbations, brings many additional bonuses and further insights from the point of view of simplicial geometry and fundamental quantum gravity proper.

Appendix: the distribution α( B)

In this appendix we introduce in a geometrical way the distribution α( B). The geometric idea is shown in figure 1, where we have suppressed one dimension. Over each corona, we build a cylindrical one such that each one is built around a 2-sphere of radius j(j + 1), and has thickness 2b. Just like the Dirac delta function, we then send b to zero keeping the volume of each corona fixed. Let us define a set of functions:

Of course it is possible to define our distribution through smoother functions, e.g. gaussian-like functions centered in | B| = j(j + 1), and then follow the same procedure showed above, with the same result.

From a geometric point of view, the height of the j th cylinder, of volume (2j + 1) 2 , is multiplied by f ( j(j + 1)). Therefore the resulting volume is (with f having spherical symmetry):

d B α 2 ( B)f (B) = ∞ j=0
(2j + 1) 2 f ( j(j + 1)).

(45)

FIG. 1 :

 1 FIG. 1: Geometric idea of the distribution α( B)

  Now, suppose we need to calculate an integral of the typed B α 2 ( B)f (| B|).

Notice also that the requirement of right group invariance couples the arguments of the functions, away from separability.