Emanuele Alesci

Carlo Rovelli

The complete LQG propagator: II. Asymptotic behavior of the vertex

In a previous article we have show that there are difficulties in obtaining the correct graviton propagator from the loop-quantum-gravity dynamics defined by the Barrett-Crane vertex amplitude. Here we show that a vertex amplitude that depends nontrivially on the intertwiners can yield the correct propagator. We give an explicit example of asymptotic behavior of a vertex amplitude that gives the correct full graviton propagator in the large distance limit.

Introduction

A technique for computing n-point functions in a background-independent context has been recently introduced [1,[START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF] and developed [START_REF] Livine | Group integral techniques for the spinfoam graviton propagator[END_REF]. Using this technique, we have found in a previous paper [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF] that the definition of the dynamics of loop quantum gravity (LQG) by means of the Barrett-Crane (BC) spinfoam vertex [START_REF] Barrett | Relativistic spin networks and quantum gravity[END_REF] fails to give the correct tensorial structure of the graviton propagator in the large-distance limit. The natural question is whether this is an intrinsic difficulty of the backgroundindependent loop and spinfoam formalism, or whether it is a specific difficulty of the BC vertex. Here we show that the answer is the second. We do so by explicitly exhibiting a vertex amplitude W that yields the correct propagator in the large distance limit. We have no claim that this vertex amplitude is physically correct. In fact, it is a rather artificial object, chosen by simply taking the asymptotic form of the BC vertex, and correcting the detail for which the BC vertex fails to work. Thus, W has at best an interest in the asymptotic region. But its existence shows that the background-independent loop and spinfoam formalism, can yield the full tensorial structure of the perturbative n-point functions.

Furthermore, the properties of W give some indications on the asymptotic that the dynamics can have, if it has to yield the correct low energy limit. The detail of the BC vertex that needs to be corrected turns out to be a phase in the intertwiner variables. A posteriori, the need for this phase appears pretty obvious on physical grounds, as we shall discuss in detail. This might provide a useful indication for selecting a definition of the dynamics alternative to the one provided by the BC vertex. While the BC vertex is defined by the SO(4) Wigner 10j symbol, an alternative vertex given by the square of an SU (2) Wigner 15j symbol has been introduced recently [START_REF] Engle | The loop-quantum-gravity vertex amplitude[END_REF]. This vertex can be derived also using coherent states techniques, and can be extended to the Lorentzian case and to arbitrary values of the Immirzi parameter [START_REF] Livine | A New spinfoam vertex for quantum gravity[END_REF]. It would be very interesting to see whether the asymptotics of this vertex exhibit the phase dependence that we find here to be required for the low energy limit.

In Section II we introduce the vertex W and we give a simple explanation of the reason why the additional phase is needed. In the rest of the paper we prove that W yields the correct full tensorial structure of the propagator. In developing this calculation we have stumbled upon an unexpected result that indicates that the state used in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF] is too symmetric. This does not affect the results of [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF], but forces us to reconsider the definition of the state. In section III, we discuss this issue in detail and give the appropriate boundary state. In Section IV we compute the propagator, and in Section V we compare it with the one computed in linearized quantum general relativity.

This paper is not self-contained. It is based on the paper [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF], where all relevant definitions are given. For an introduction to the formalism we use, see [START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF]; for a general introduction to background independent loop quantum gravity [START_REF] Rovelli | Loop space representation for quantum general relativity[END_REF], see [START_REF] Rovelli | Quantum Gravity[END_REF].

The vertex and its phase

Following [1,[START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF], the graviton propagator can be computed in a background independent context as the scalar product

G ij,kl q n,m = W | E (i) n • E (j) n -n (i) n • n (j) n E (k) m • E (l) m -n (k) m • n (l) m |Ψ q . ( 1
)
Here W | is the boundary functional, which can be intuitively understood as the path integral of the Einstein-Hilbert action on a finite spacetime region R, with given boundary configuration. The indices i, j, k, l, m, n, ... run over the values 1, ..., 5 and label the tetrahedra of a 4-simplex. The operator

E (i) n (denoted E (ni)
n in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF]) is the triad operator at the points n, contracted with (test) one-forms n (i) n (denoted n ni in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF]) at the same point. |Ψ q is a state on the boundary of R, picked on a given classical boundary (extinsic and extrinsic) geometry q.

Fixing such a boundary geometry is equivalent to fixing a background metric g in the interior, where g is the solution of the Einstein equations with boundary data q. The existence of such a background metric is part of the definition of the propagator, which is a measure of fluctuations around a given background. Criticisms to the approach of [1,[START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF] have been raised on the ground that a propagator makes no sense in a background independent context, because it is a quantity that depends on a background geometry. These criticisms follow from a misunderstanding of this point. The information about the background over which the propagator is defined is in the boundary state, via q.

We are interested in the value of (1) to first order in the GFT expansion parameter λ, and in the limit in which the boundary surface (whose size is determined by q) is large. On the physical interpretation of this limit, see [START_REF] Engle | The loop-quantum-gravity vertex amplitude[END_REF]. To first order, the leading contribution to W has support only on spin networks with a 4-simplex graph. If j = (j nm ) and i = (i n ) are, respectively, the ten spins and the five intertwiners that color this graph, then in this approximation (1) reads

G ij,kl q n,m = j,i W (j, i) E (i) n • E (j) n -n (i) n • n (j) n E (k) m • E (l) m -n (k) m • n (l) m Ψ(j, i). (2)
To this order, W is just determined by the amplitude of a single vertex. In [1,[START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF][START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF], (a suitable adjustment of) the BC vertex was chosen for W . The propagator depends only on the asymptotic behavior of the vertex. This has the structure [START_REF] Barrett | The asymptotics of an amplitude for the 4-simplex[END_REF] W

BC (j) ∼ e i 2 (δjGδj) e iΦ•δj + e -i 2 (δjGδj) e -iΦ•δj , ( 3
)
where G is the 10 × 10 matrix given by the second derivatives of the 4d Regge action around the symmetric state, δj is the difference between the ten spins j and their background value j 0 , and Φ is a 10d vector with all equal components, which were shown in [1,[START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF] to precisely match those determined by the background extrinsic curvature. The diagonal components of the propagator determined by

(1) turn out to be correct at first [1] and second [START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF] order, but the nondiagonal components fail to do so [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF].

Here we make a different choice for W . We choose a vertex W with an asymptotic form that includes a gaussian intertwiner-intertwiner and spin-intertwiner dependence, and -most crucially-a phase dependence on the intertwiner variables. To write this, introduce a 15d vector δI = (δj, δi), where δi is the difference between the five intertwiners i and their background value i 0 . Explicitly, δI α = (δj nm , δi n ) = (j nmj 0 , i ni 0 ), where α = (nm, n). And consider the state

W (j, i) = e i 2 (δIGδI) e iφ•δI + e -i 2 (δIGδI) e -iφ•δI . ( 4
)
Here G is now a 15 × 15 matrix and φ = (φ nm , φ n ) is a 15d vector. Its 10 spin components φ nm just reproduce the spin phase dependence of (3); while its five intertwiner components are equal and we fix them to have value

φ n = π 2 . ( 5
)
This phase dependence is the crucial detail that makes the calculation work.

Let us illustrate upfront the reason why this additional phase cures the problems that appeared with the BC vertex. The boundary state must have an intertwiner dependence, in order to have the correct semiclassical value of the mean values of the angles between the faces of the boundary tetrahedra. The mean value of an intertwiner variable i n -namely of the virtual link of the intertwiner in a given pairing-must have a certain value i 0 . For this, it is sufficient, say, that the state be a gaussian around i 0 . However, in quantum geometry the different angles of a tertrahedron do not commute [START_REF] Barbieri | Quantum tetrahedra and simplicial spin networks[END_REF]. Therefore a state with a behavior like exp{-(i ni 0 ) 2 } will be peaked on the virtual spin i n in one pairing, but it will not be peaked in the virtual spin in a different pairing. Therefore, the other angles of the tetrahedron will not be peaked on the correct semiclassical value. We can of course write a gaussian which is peaked on a variable as well as on another, non-commuting, variable. For instance, a standard Schrödinger wave packet ψ(x) = exp{-(x-x0) 2 2 σ + ip 0 x} is peaked on position as well as momentum. But in order to do so, we must have a phase dependence on the x. Similarly, the boundary state needs a phase dependence on the intertwiner variable i n , in order to be peaked on all angles. As shown in [START_REF] Rovelli | A Semiclassical tetrahedron[END_REF], the correct value for this is exp{i π 2 i n }. Now, the general mechanism through which the dynamical kernel reproduces the semiclassical dynamics in quantum mechanics is the cancellation of the phases between the propagation kernel and the boundary state. If this does not happens, the rapidly oscillating phases suppresses the amplitude. For instance, in the non-relativistic quantum mechanics of a free particle, the propagation kernel K(x, y) in a time t has a phase dependence from small fluctuations δx = xx 0 and δy = yy 0 of the form

K(x 0 + δx, y 0 + δy) = x 0 + δx|e -i h p 2 2m t |y 0 + δy ∼ C e -ip0δx e ip0δy . ( 6
)
where p 0 = m(y 0x 0 )/t. This phase precisely cancels the phase of an initial and final wave packets ψ i and ψ f centered on x 0 and y 0 , if these have the correct momentum. That is

ψ f |e -i h Ht |ψ i = dx dy e -(x-x 0 ) 2 2σ -i h p f x K(x, y) e -(y-y 0 ) 2 2σ + i h piy (7)
is suppressed by the oscillating phases unless p i = p f = p 0 . This is the standard mechanism through which quantum theory reproduces the (semi-)classical behavior. In quantum gravity, it is reasonable to expect the same to happen if we have to recover the Einstein equations in the semiclassical limit. That is, the propagation kernel W , must have a phase dependence that matches the one in a semiclassical boundary state. This is precisely the role of the phase exp{i π 2 i n } that we have included in (4). In the rest of the paper we show that a vertex amplitude that has the phase dependence as above can reproduce the tensorial structure of the graviton propagator. First, however, we must improve the definition of the vertex given above, and correct a problem with the definition of the state in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF].

Boundary state and symmetry

Following [1,[START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF], we consider a boundary state defined as a gaussian wave packet, centered on the values determined by the background geometry q. Here Φ

q (j, i) = C e -1 2j 0 (δIAδI)+iφ•δI . ( 8
)
Where A is a 15 × 15 matrix and the normalization factor C is determined by W |Φ q = 1. The spin phase coefficients are fixed by the background extrinsic geometry [1]. The intertwiner phase coefficients are fixed by requirement that the state remain peaked after a change of pairing to the value φ n = π/2. [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF][START_REF] Rovelli | A Semiclassical tetrahedron[END_REF] At each node n we have three possible pairings, that we denote as x n , y n and z n . For instance, at the node 5, let x 5 = {(12)(34)}, y 5 = {(13)(24)}, z 5 = {( 14)(23)}, and denote i x5 = i {( 12)(34)} the intertwiner in the pairing x 5 , and so on. The vertex ( 4) and the state ( 8) are written in terms of the intertwiner variable i n , which is the virtual link of the node n in one chosen pairing. Because of this, the definition of these states depend on the pairing chosen. It follows that the vertex and the state do not have the full symmetry of the 4-simplex. The corresponding propagator turn out not to be invariant under SO( 4), as it should in the euclidean theory. In [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF], a simple strategy was adopted in order to overcome this difficulty: sum over the three pairings at each of the five nodes. The state was defined as

|Ψ q = mn j,im n Φ q (j, i mn ) |j, i mn , (9)
where m n = x, y, z for each node n. This sum implements the full symmetry of the 4-simplex.

Summing over the three bases removes the basis dependence.

In developing the calculations presented in the present paper, at first we adopted this same strategy. To our surprise, nothing worked, and something quite funny happened: the dependence on the intertwiner variables i n misteriously cancelled out in all components of the propagator!

The solution of the puzzle was to realize that to sum over the three basis with a correlation matrix A does implement the symmetry of the 4-simplex, but not just this symmetry. It implements a larger symmetry, that has the effect of cancelling the intertwiner dependence. Geometrically, this additional symmetry can be viewed as an independent rotation of each of the five tetrahedra forming the boundary of the 4-simplex.

To understand what happens, consider for instance the correlation j 12 i x5 between the spin j 12 which is the quantum number of the area of a triangle, and the intertwiner i x5 , which is the quantum number of the angle θ 12 between the faces 2 and 3 of the tetrahedron 5. More precisely, i x5 is the eigenvalue of the quantity A 2 2 +A 2 3 +A 2 A 3 cos(θ 12 ), where A i is the area of the face i of the tetrahedron 5. Now, if the state is summed over pairings, then it does not distinguish pairings, hence

j 12 i x5 = 1 3 ( j 12 i x5 + j 12 i y5 + j 12 i z5 ) . (10)
That is

j 12 i x5 = 1 3 j 12 3A 2 1 + A 2 2 + A 2 3 + A 2 4 + A 1 A 2 cos(θ 12 ) + A 1 A 3 cos(θ 13 ) + A 1 A 4 cos(θ 14 ) . ( 11
)
But let n i , i = 1, ..., 4 be the normal to the face i of the tetrahedron 5, with length

|n i | = A i . The closure relation reads i=1,4 n i = 0. ( 12
)
Taking the scalar product with n 1 gives

A 2 1 + A 1 A 2 cos(θ 12 ) + A 1 A 3 cos(θ 13 ) + A 1 A 4 cos(θ 14 ) = 0. ( 13
)
It follows from this equation and ( 11) that

j 12 i x5 = 1 3 j 12 (2A 2 1 + A 2 2 + A 2 3 + A 2 4 ) = 1 3
(2 j 12 j 15 + j 12 j 25 + j 12 j 35 + j 12 j 45 ). ( 14)

That is, the spin-intertwiner correlations are just functions of the spin-spin correlations for a state with this symmetry! The intertwiner dependence drops out! This means that the propagator is completely unaffected from the correlations involving the intertwiners. It then turns out that the sole spin-spin correlations in the state are not sufficient to give the full tensorial structure of the propagator. The solution of the difficulty is just to choose a boundary state and a kernel W that do not have the extra symmetry. The simplest possibility is to choose an abitrary pairing, and then to symmetrize only under the symmetries of the four-simplex. These are generated by the 5! permutations σ of the five vertices of the four-simplex. A permutation σ : {1, 2, 3, 4, 5} → {σ(1), σ(2), σ(3), σ(4), σ(5)} acts naturally on the boundary states

σ|j nm , i xn = |j σ(n)σ(m) , i σ(xn) ( 15
)
where the action σ(x n ) of the permutation on a node is defined by [START_REF] Reisenberger | A lattice worldsheet gr[END_REF] and can therefore change the original pairing at the node. We therefore define the boundary state by replacing [START_REF] Rovelli | Quantum Gravity[END_REF] with

σ({(ab)(cd) n }) = {(σ(a)σ(b))(σ(c)σ(d)) σ(n) } (
|Ψ q = σ σ|Φ q = σ j,i Φ q (j, i) σ|j, i . ( 17
)
This modification of the boundary state does not affect the conclusions of the paper [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF]. Similarly, we pose

|W = σ j,i W (j, i) σ|j, i . (18)
Before beginning the actual calculation of the propagator, consider what happens by contracting the vertex amplitude with the boundary state. We have the double sum over permutations

W |Ψ = σσ ′ jij ′ i ′ W (j, i)Φ(j ′ , i ′ ) σ(j, i)|σ ′ (j ′ , i ′ ) . ( 19
)
The scalar product is j, i|j

′ , i ′ = δ j,j ′ n i n |i ′ n , (20)
where i n |i ′ n is δ in,i ′ n if the two intertwiners are written in the same basis, and is the matrix of the change of basis, namely a 6j-symbol, otherwise. Now, it was observed in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF] that if one of these 6j-symbols enters in a sum like (19) then the sum is suppressed in the large j 0 limit, because the 6j-symbol contains a rapidly oscillating factor which is not compensated. Hence, in this limit we can effectively rewrite (19) in the form

W |Ψ = σσ ′ jij ′ i ′ W (j, i)Φ(j ′ , i ′ ) δ σj,σ ′ j ′ δ σi,σ ′ i ′ , (21)
where the second delta vanishes unless the two intertwiners have the same value and are written in the same basis. Up to accidental symmetry factors that we absorb in the state, we can then rewrite the scalar product in the form

W |Ψ = σ ji W (j, i)Φ(j, i) = 5! ji W (j, i)Φ(j, i). ( 22
)
We shall see that a similar simplification happens in the calculation of the matrix elements of the propagator.

The propagator

Let us begin by recalling the action of the grasping operators. This was computed in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF], to which we refer for the notation. Consider the operators acting on a node n. The diagonal action is simply

E (i) n • E (i) n |Φ q = C ni |Φ q (23)
where C ni is the Casimir of the representation associated to the link ni. The non-diagonal action depends on the pairing at the node n. We have three cases, depending on the three possible pairings. These are as follows. Say the node n is in the pairing (ij), (ef ), with positive orientation at the two trivalent vertices (i n , i, j) and (i n , e, f ). Then we have the diagonal double grasping

E (i) n • E (j) n |Φ q = j,i D ij n Φ(j, i) |j, i . ( 24
)
while the two possible non-diagonal graspings give

E (i) n • E (k) n |Φ q = j,i Φ(j, i) X ik n |j, i -Y ik n |j, (i n -1), i ′ -Z ik n |j, (i n + 1), i ′ (25)
and

E (i) n • E (l) n |Φ q = j,i Φ(j, i) X il n |j, i + Y il n |j, (i n -1)i ′ + Z il n |j, (i n + 1)i ′ . ( 26
)
and so on cyclically. The quantities D ij n , X ij n , Y ij n and Z ij n are defined in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF]. Here i ′ indicates the four intertwiners different from i n .

Inserting the expressions ( 17) and ( 18) in the expression (1) for the propagator, gives the double sum over permutations

G ij,kl q n,m = σ ′ σ j,i W (σ ′ (j), σ ′ (i)) E (i) n • E (j) n -n (i) n • n (j) n E (k) m • E (l) m -n (k) m • n (l) m Φ(σ(j), σ(i)) . (27)
The E operators do not change the spin, and the argument at the end of the last section can be repeated. This time, however, the residual sum over permutations remains, because the operators are not invariant under it

G ij,kl q n,m = σ   j,i W (σ(j), σ(i)) E (i) n • E (j) n -n (i) n • n (j) n E (k) m • E (l) m -n (k) m • n (l) m Φ(σ(j), σ(i))   . ( 28
)
By changing variables, we can move the symmetrization to the operators, hence writing

G ij,kl q n,m = σ Gσ(i)σ(j),σ(k)σ(l) q σ(n),σ(m) (29)
where Gij,kl

q n,m = j,i W (j, i) E (i) n • E (j) n -n (i) n • n (j) n E (k) m • E (l) m -n (k) m • n (l) m Φ(j, i). (30)
In other words, we can first compute the propagator with unsymmetrized states and vertex, and then symmetrize the propagator.

We can now begin the actual calculation of the various terms of the propagator. It is usuefull to distinguish three cases: the diagonal-diagonal components Gii,kk q n,m ; the diagonal-non-diagonal components Gii,kl q n,m ; and the non-diagonal-non-diagonal components Gij,kl q n,m , where again different indices are distinct. Let us considered the three cases separately.

In the diagonal-diagonal case, from the expression of the last section, we have Gii,kk

q n,m = ji W (j, i) (C ni -|n (i) n | 2 )(C nk -|n (k) m | 2 ) Φ(j, i) (31)
As we have seen in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF] the background geometry determines the background link j 0

|n (i) n | 2 = C 2 (j 0 ) = j 0 (j 0 + 1) (32)
and

C ni = C 2 (j ni ). ( 33
)
In the large j 0 limit we have at leading order

C ni -|n (i) n | 2 ≈ 2j 0 δj ni (34)
the propagator components are then Gii,kk

q n,m = 4j 2 0 j,i W (j, i) δj ni δj mk Φ(j, i) (35)
The sum over permutations is now trivial. It only gives a 5! factor that cancels with the same factor in the normalization. We can therefore drop the tilde from ( 35).

In the diagonal-non-diagonal case, from (30) we have

Gij,kk q n,m = j,i W (j, i) E (i) n • E (j) n -n (i) n • n (j) n E (k) m • E (k) m -|n (k) m | Φ(j, i) (36)
now the second operator is diagonal and gives (34) at leading order; the action of the first operator instead gives only one of the three terms ( 24), ( 25), ( 26) depending on how the two links ni and nj are paired at the node n. The possible results (at leading order) are Gij,kk q n,m = j,i

W (j, i)2j 0 δj (mk) D (ij) n + j 2 0 3 Φ(j, i) (37)
if the two links are paired. The second term in the parenthesis comes from the fact that the background normals are fixed by the background geometry. In the large j 0 limit

n (i) n • n (nj) n ≈ - 1 3 (j 0 ) 2 . (38) And Gij,kk q n,m = j,i W (j, i) X ij n + j 2 0 3 -W (j, i ′ , i n -1) Y ij n -W (j, i ′ , i n + 1) Z ij n 2j 0 δj mk Φ(j, i), (39) or Gij,kk q n,m = j,i W (j, i) X ij n + j 2 0 3 + W (j, i ′ , i n -1) Y ij n + W (j, i ′ , i n + 1) Z ij n 2j 0 δj mk Φ(j, i), (40)
according to orientation, if they are not paired. In ( 39) and ( 40) the term in Y and Z cancel at the leading order for the following reason. First, recall from [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF] that Y and Z are equal at leading order. The difference between the Y -term and the Z-term is then only given by the ±1 in the argument of W . But the dependence of W on i n is of the form e i π 2 in . Hence (up to subleading terms in the large j 0 limit)

W (j, i ′ , i n + 1) = -W (j, i ′ , i n -1) (41)
The different between the two terms is just a sign and they cancel. Thus we have Gij,kk

q n,m = j,i W (j, i) X ij n + j 2 0 3 2j 0 δj mk Φ(j, i), (42)
anytime ni and nj are not paired.

In the large distance limit we have ( 38) and

D ij n -n (i) n • n (j) n = C 2 (i n ) -C 2 (j (ni) ) -C 2 (j (nj) ) 2 + 1 3 (j 0 ) 2 . ( 43
)
Introduce the fluctuations variables δj nj = j njj 0 , and δi n = i ni 0 and expand around the background values j 0 and i 0 . In the large j 0 limit (which is also large i 0 ). The dominant term of the ( 43) is

D ij n -n (ni) • n (nj) = δi n i 0 -δj ni j 0 -δj nj j 0 . (44)
Similarly, using the results of [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF], the X terms are approximated substituting C 2 (j) ≈ j 2 and keeping the dominant terms

X ij n = - 1 4 (i 0 ) 2 + 2j 0 δj ni + 2j 0 δj nj -2j 0 δj nf -2j 0 δj ne + 2i 0 δi n (45)
where nf and ne indicate the other two links of the node n. Recalling that i 0 = 2 √ 3 j 0 , we have that the first term of the sum cancels the norm of the n, leaving

X ij n + j 0 3 = - 1 4 (2j 0 δj ni + 2j 0 δj nj -2j 0 δj nf -2j 0 δj ne + 2i 0 δi n ) (46)
In conclusion, we have for the paired case Gij,kk

q n,m = 2j 2 0 j,i W (j, i) 2 √ 3 δi n -δj ni -δj nj δj mk Φ(j, i), (47)
and for the unpaired one Gij,kk

q n,m = j 2 0 j,i W (j, i) -δj ni -δj nj + δj nf + δj ne - 2 √ 3 δi n δj mk Φ(j, i). (48) Finally, the non-diagonal-non-diagonal case is Gij,kl q n,m = W | E (i) n • E (j) n -n (i) n • n (j) n E (k) m • E (l) m -n (k) m • n (l) m |Φ . ( 49
)
The calculations are clearly the same as above.

The final result is Gij,kl

q n,m = j 2 0 j,i W (j, i)K ij n K kl m Φ(j, i), (50)
where

K ij n = 2 √ 3 δi n -δj ni -δj nj (51)
if ni and nj are paired at n and

K ij n = 1 2 -δj ni -δj nj + δj nf + δj ne - 2 √ 3 δi n (52)
if they are not; while

K ii n = 2δj ni . ( 53
)
Both the state coefficients Φ(j, i) and the vertex coefficients W (j, i) are given by a gaussian in δI α . The phases in the boundary state cancels with the phase of one of the two terms of W , while the other term is suppressed for large j 0 . Thus, (50) reads Gij,kl

q n,m = j 2 0 j,i e -1 2j 0 M αβ δIαδI β K ij n K kl m , (54)
where

M = A + ij 0 G.
As in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF], we approximate the sum by a Gaussian integral with quadratic insertions. The result of the integral is easily expressed in terms of the matrix M -1 obtained inverting the 15 × 15 covariance matrix M , in the 10 spin variables δj nm and the five intertwiner variables δi n .

The symmetries of the matrix M -1 are the same as the symmetries of M , and are dictated by the symmetries of the problem. Which ones are these symmetries? At first sight, one is tempted to say that M -1 must respect the symmetries of the 4-simplex, and therefore it must be invariant under any permutation of the five vertices n. Therefore therefore it can have only seven independent components:

M -1 (ij)(ij) = c 2 , M -1 (ij)(ik) = c 1 , M -1 (ij)(kl) = c 3 , M -1 ii = c 4 , M -1 ij = c 5 , M -1 (ij)i = c 6 M -1 (ij)k = c 7 . ( 55
)
where different indices are distinct. The ratio for this being for instance that M 11 must be equal to M 22 because of the symmetry under the exchange of the vertex 1 and the vertex 2. However, this argument is incorrect. The reason is that the vertex function and the state function are written as a function of intertwiner variables i n which are tied to a given choice of pairing at each node. Specifically, we have chosen the pairing i . This choice breaks the symmetry under the permutations of the vertices, although this is not immediately evident. To see this, consider for instance the two matrix elements M -1 (12)3 and M -1 (12)4 . According to (55), they should be equal (both be equal to c 7 by symmetry. But notice that 1 and 2 are paired at the node 3, while they are not paired at the node 4. Therefore the two are not equal under the symmetries of the paired 4-simplex. To see this more formally, let us indicate explicitly the pairing in which the intertwiner is written by writing i

(ij)(ef ) n instead of i n . Then we see that M -1 (12)3 is of the form M -1 (ij)i (ij)(kl) n while M -1 (12)4 is of the form M -1 (ij)i (ik)(jl) n
, which makes it obvious that a permutation ijklm → i ′ j ′ k ′ l ′ m ′ cannot transform one into the other, since it cannot undo the fact that the ij indices of the link are paired at the node. As a consequence, we must for instance replace the last entry of (55) by

M -1 (ij)i (ij)(kl) n = c 7 M -1 (ij)i (ik)(jl) n = c 8 . (56)
and so on. Thus, the matrix M -1 may in general have a more complicated structure than (55). Now, the details of this structure depend on the pairing chosen. In fact, there are five possible inequivalent ways of choosing the pairings at the nodes, which do not transform into one another under permutations. These are illustrated in Figure 1.

i (23)(45) 1 i (34)(51) 2 i (45)(12) 3 i (51)(23) 4 i (12)(34) 5 i (25)(34) 1 i (31)(45) 2 i (42)(51) 3 i (53)(12) 4 i (14)(23) 5 i (34)(51) 2 i (45)(12) 3 i (51)(23) 4 i (12)(34) 5 i (25)(34) 1 i (31)(45) 2 i (45)(12) 3 i (51)(23) 4 i (14)(23) 5 i (25)(34) 1 i (45)(12) 3 i (51)(23) 4 i (14)(23) 5 i (31)(45) 2 i (23)(45) 1 Figure 1:
The five classes of pairings: from the upper left: [START_REF] Barrett | The asymptotics of an amplitude for the 4-simplex[END_REF], [START_REF] Barrett | Relativistic spin networks and quantum gravity[END_REF][START_REF] Barrett | Relativistic spin networks and quantum gravity[END_REF], [START_REF] Livine | A New spinfoam vertex for quantum gravity[END_REF][START_REF] Livine | Group integral techniques for the spinfoam graviton propagator[END_REF], [START_REF] Engle | The loop-quantum-gravity vertex amplitude[END_REF][START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF] and [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF][START_REF] Livine | Group integral techniques for the spinfoam graviton propagator[END_REF][START_REF] Livine | Group integral techniques for the spinfoam graviton propagator[END_REF].

The fact that they cannot be transformed into one another by a permutation can be deduced from the following consideration. In each diagram Figure 1, consider the sequences of links that can be followed without ever crossing an intertwiner. Observe that in the first case all links are clustered in a single cluster of length 10. In the second, they are clustered in two diagrams of lenght [START_REF] Barrett | Relativistic spin networks and quantum gravity[END_REF][START_REF] Barrett | Relativistic spin networks and quantum gravity[END_REF], and so on as indicated. Clearly a permutation cannot change the structure of these clusterings, and therefore these pairing choices cannot be transformed into one another under permutations. The five cases illustrated correspond to the five different 15j Wigner symbols illustrated in [START_REF] Yutsin | Mathematical Apparatus of the Theory of Angular Momentum (Israel program for Scientific Translation[END_REF]. These five classes define therefore distinct possibilities for the definitionns of vertex and the state. As here we are not interested in generality, we have just picked one of these: the first case. Also, since we are not interested in the full generality of an arbitrary gaussian vertex and state, we just assume a particular form, compatible with the symmetries, for the matrix M -1 . Specifically, we assume that M -1 has the form (55) with the last entry replaced by (56). That is, we assume the state depends on (at least) eight independent parameters that determine c = (c 1 , ..., c 8 ). The symmetries of the 4-symplex equivalence class admit a greater number of free parameters, but we do not need the most general possible gaussian state for what follows. Assuming thus this form for M -1 , we can then proceed with the calculation of (54).

Each term of the normalized propagator is a sum of individual elements of the matrix M -1 . The overall dependence on j 0 is as in the diagonal case, and gives the expected inverse-square dependence. The normalization factor is

N -1 = j 2 0 d(δI α ) e -1 2j 0 M αβ δIαδI β (57)
The diagonal-diagonal term gives Gii,kk q n,m N j e

-1 2j 0 M αβ δIαδI β 2δj ni 2δj mk = 4 j 0 M -1 (ni) (mk) = 4 j 0 c 1 if i=k or i=m, 4 j 0 c 3 otherwise. ( 58
)
In this case G gives immediately G since the permutation does not mix c 1 and c 3 terms. Proceeding in the same way for the other cases, we get for the diagonal-non-diagonal term the two cases Gij,kk q n,m =

1 j 0 -2M -1 (mk) (ni) -2M -1 (mk) (nj) + 4 √ 3 M -1 (mk) n =      -4 j 0 c 1 -1 √ 3 c 7 if i=k and j=m, -4 j 0 c 3 -1 √ 3 c 7 if i = k and j = k,m, -2 j 0 c 1 + c 3 -2 √ 3 c 7 otherwise. (59) and Gij,kk q n,m = 1 j 0 -M -1 (mk) (ni) -M -1 (mk) (nj) + M -1 (mk) (np) + M -1 (mk) (nq) - 2 √ 3 M -1 (mk) (n) (60) =        2 j 0 (-c 1 + c 3 -1 √ 3 c 8 ) if i=k and j=m 2 j 0 (-c 3 + c 1 -1 √ 3 c 8 ) if i = k and j = k,m -2 √ 3j 0 c 8 otherwise (61)
depending on the pairing of the node n. For the non-diagonal-non-diagonal terms, we have the three possibilities: diagonal double grasping on the two nodes Gij,kl

q n,m = 1 j 0 4 3 M -1 m n - 2 √ 3 M -1 n mk + M -1 n ml + M -1 m ni + M -1 m nj + M -1 ni mk + M -1 ni ml + M -1 nj mk + M -1 nj ml ; (62)
diagonal double grasping on one node and non-diagonal on the other one

q n,m = 1 2j 0 - 4 3 M -1 n m + - 2 √ 3 M -1 n mk - 2 √ 3 M -1 n ml + 2 √ 3 M -1 n mp + 2 √ 3 M -1 n mq + + 2 √ 3 M -1 ni m + M -1 ni mk + M -1 ni ml -M -1 ni mp -M -1 ni mq + + 2 √ 3 M -1 nj m + M -1 nj mk + M -1 nj ml -M -1 nj mp -M -1 nj mq ; , (63)
and non-diagonal on both nodes Gij,kl q n,m = 1 4j 0

4 3 M -1 n m + + 2 √ 3 M -1 n mk + 2 √ 3 M -1 n ml - 2 √ 3 M -1 n mp - 2 √ 3 M -1 n mq + + 2 √ 3 M -1 ni m + M -1 ni mk + M -1 ni ml -M -1 ni mp -M -1 ni mq + + 2 √ 3 M -1 nj m + M -1 nj mk + M -1 nj ml -M -1 nj mp -M -1 nj mq + - 2 √ 3 M -1 ne,m -M -1 ne mk -M -1 ne ml + M -1 ne mp + M -1 ne mq + - 2 √ 3 M -1 nf,m -M -1 nf mk -M -1 nf ml + M -1 nf mp + M -1 nf mq (64)
whose expression in terms of the c And so on. Notice that the only the six parameters c 1 , c 2 , c 3 and c 5 , c 7 , c 8 enter the components of the propagator. The other two, namely c 4 and c 6 do not, because we are only looking at the propagator between points on different tetrahedra. The last step is to symmetrize the propagator under permutations. The only terms that change under permutations, at this point, are those due to the pairing. Hence, the only result of a sum over permutation is to combine the two coefficients c 7 and c 8 , which are the only pairing dependent ones. For instance, a straightforward calculation gives the diagonal-non-diagonal term (which has the peculiarity of not depending on the pairing class)

G ij,kk q n,m = σ Gσiσj,σkσk q σn,σm (65)
=      1 3j0 4(-c 1 + c 3 ) -8c 1 + 2 √ 3 (c 7 -c 8 ) if i=k and j=m, 1 3j0 4(-c 3 + c 1 ) -4c 3 + 2 √ 3 (c 7 -c 8 ) if i = k and j = k,m, 1 3j0 -2(c 1 + c 3 ) -1 √ 3 (c 7 -c 8 ) otherwise.
It is easy to see that the sum over permutation replaces all terms c 7 and c 8 with a term proportional to the linear combination (c 7c 8 ). In conclusion, the propagator depends on the five parameters c 1 , c 2 , c 3 , c 5 , (c 7c 8 ). Varying the parameters in the state we can span a five-dimensional space of tensors Gij,kl q n,m . In conclusion, G ii,kk q n,m turns out to be a matrix with the symmetries of the 4-simplex, freely dependending on five arbitrary parameters. Can this give the same propagator as the linearized theory?

Comparison with the linearized theory

The number of components of G ij,kl q n,m is large, and it may seem hard to believe that the five-parameters freedom in the state could be sufficient to recover the tensorial structure of the linearized propagator. However, there are two properties of the propagator that strongly constrain it. First, the symmetrization of the 4-simplex symmetries largely reduce the number of indepedent components. Second, as proven in [START_REF] Alesci | The complete LQG graviton propagator: I. Difficulties with the Barrett-Crane vertex[END_REF], the propagator satisfies the closure relation

i G ij,kl q n,m = 0. ( 66
)
Let us count the number of free parameters of an arbitrary tensor G ij,kl q n,m satisfying these requirements. Using (66), we can always express a term in which any of the four indices i, j, k, l is equal to either n or m as sum of terms not of this kind. This reduces the independent terms to, say G ij,kl q 1,2 where i, j, k, l = 3, 4, 5. A few pictures and a moment of reflection will convince the reader that the only independent ones of these are

G ii,ii q n,m , G ii,kk q n,m , G ij,kk q n,m , G ij,ij q n,m , G ij,ik q n,m . (67)
All the other terms can be obtained from these by a permutation of the indices. Therefore a tensor with these symmetries depends only on five parameters. This implies that adjusting the five parameters in the state, we can match any such tensor, and in particular the propagator. This can be checked by an explicit calculation of the propagator of the linearized theory in the harmonic gauge (on the compatibility of the radial and harmonic gauge, see [START_REF] Magliaro | Compatibility of radial, Lorenz and harmonic gauges[END_REF]). The quantity G ij,kl q n,m is the propagator projected in the directions normal to the faces of the tetrahedra. The 4d linearized graviton propagator is

G µνρσ = 1 2L 2 (δ µρ δ βγ + δ µσ δ βγ -δ µν δ ρσ ) (68)
and its projection on the four linear normals to reads

G ij,kl nm ≡ G µνρσ (n (i) n ) µ (n (j) n ) ν (n (k) m ) ρ (n (l) m ) σ (69)
We need the explicit expressions of the normals; to this aim, fix the coordinate of a four simplex giving the 5 vertices of a 4-simplex fixing the 4d-vectors e µ I where µ is the 4d space index and I(I = 1, .., 5) is the label of the vertex. The easiest way to deal with this 4d geometry is to introduce the bivectors

B µν IJ B µν IJ = e µ K ∧ e ν L + e µ L ∧ e ν M + e µ M ∧ e ν K ( 70
)
where the indices IJKLM form an even permutation of 1, 2, 3, 4, 5. If t 1 is the tetrahedron with vertexes e 2 , e 3 , e 4 , e 5 and so on cyclically, the bivector B µν nm will be the bivector normal to the triangle t nm shared by the tetrahedra t n and t m . The normal n m n to this triangle, in the 3 surface determined by the tetrahedron t n is (n m n ) ν = B µν nm (t n ) µ , where (t n ) µ , is the normal to the tetrahedron. Using this, it is a tedious but straightforward exercise to compute the components of the projected linearized propagator. Writing the bimatrix G ij,kl linearized 1,2 = (G kl ) ij , where ijkl = 3, 4, 5 we have which displays the equality of the various terms. The five different components have here values (-16, 6, -28, -7, 4)/512. A judicious choice of the parameters c 1 , c 2 , c 3 , c 5 , (c 7c 8 ) can match these values.

Conclusion and perspectives

We have shown that a vertex with an appropriate asymptotic expansion, combined with a suitable boundary state, can yield the full tensorial structure of the propagator.

In doing so, we have also learned several lessons. The main lesson is that the non-commutativity of the angles requires a semiclassical state to have an oscillatory behavior in the intertwiners. In order to match this behavior, and approximate the semiclassical dynamics, the vertex must have a similar oscillatory dependence on the intertwiners. (This should not affect with possible finitness properties of the model [START_REF] Perez | Finiteness of a spinfoam model for euclidean GR[END_REF].) The second lesson is that the symmetries of the boundary state must be considered with care, if we do not want to loose relevant dynamical information. Symmetrizing over the permutation of the vertices is a simple way of achieving a symmetric state without inserting additional unwanted symmetries. In doing so, however, one must take into account that a choice of pairing breaks the 4-simplex symmetry.

The most interesting open question, in our opinion, is whether other vertex amplitudes considered (such as [START_REF] Reisenberger | A lattice worldsheet gr[END_REF]) and in particular the vertex amplitude recently studied in [START_REF] Engle | The loop-quantum-gravity vertex amplitude[END_REF][START_REF] Livine | A New spinfoam vertex for quantum gravity[END_REF] satisfies the requirements for yielding the correct full tensorial structure of the graviton propagator. In particular, whether there is an oscillation in the intertwiners. This issue can be addressed analytically, via a saddle point analysis of the asymptotic of the new vertex, or numerically, using the technology developed in [START_REF] Baez | Positivity of spin foam amplitudes[END_REF]. Some preliminary numerical indications appear to be optimistic [START_REF] Magliaro | Numerical indications on the semiclassical limit of the flipped vertex[END_REF]. Also, we think that the role of the five inequivalent structures illustrated in Figure 1 deserve to be better understood.


Merced Montesinos
email: merced@fis.cinvestav.mx

Alejandro Perez

Two-dimensional topological field theories coupled to four-dimensional BF theory

Keywords:

Four dimensional BF theory admits a natural coupling to extended sources supported on two dimensional surfaces or string world-sheets. Solutions of the theory are in one to one correspondence with solutions of Einstein equations with distributional matter (cosmic strings). We study new (topological field) theories that can be constructed by adding extra degrees of freedom to the two dimensional world-sheet. We show how two dimensional Yang-Mills degrees of freedom can be added on the world-sheet, producing in this way, an interactive (topological) theory of Yang-Mills fields with BF fields in four dimensions. We also show how a world-sheet tetrad can be naturally added. As in the previous case the set of solutions of these theories are contained in the set of solutions of Einstein's equations if one allows distributional matter supported on two dimensional surfaces. These theories are argued to be exactly quantizable. In the context of quantum gravity, one important motivation to study these models is to explore the possibility of constructing a background independent quantum field theory where local degrees of freedom at low energies arise from global topological (world-sheet) degrees of freedom at the fundamental level.

I. INTRODUCTION

Topological field theories are simple examples of background independent field theories for which quantization can be completely worked out. These theories are a natural play ground where conceptual as well as technical issues in background independent quantum theory can be addressed in detail. Three-dimensional vacuum general relativity is an important example of a topological field theory. Interestingly the topological nature of the theory can be maintained if matter is added in the form of topological defects representing massive and spinning point particles [START_REF] Carlip | Exact Quantum Scattering In (2+1)-Dimensional Gravity[END_REF]. Interest in the quantization of 2 + 1 gravity coupled to point particles has been revived in the context of the spin foam [START_REF] Perez | Spin foam models for quantum gravity[END_REF] and loop quantum gravity [START_REF] Thiemann | Modern Canonical Quantum General Relativity[END_REF] approaches to the nonperturbative and background-independent quantization of gravity. On the one hand this simple system provides a nontrivial example where the strict equivalence between the covariant and canonical approaches can be demonstrated [START_REF] Noui | Three dimensional loop quantum gravity: Coupling to point particles[END_REF]. On the other hand intriguing relationships with field theories with infinitely many degrees of freedom have been obtained [START_REF] Freidel | Ponzano-Regge model revisited. III: Feynman diagrams and effective field theory[END_REF][START_REF] Noui | Three dimensional loop quantum gravity: Towards a self-gravitating quantum Class[END_REF]. The generalization of these models to higher dimensions has been studied in [START_REF] Baez | Quantization of strings and branes coupled to BF theory[END_REF]. As it is shown there, membrane-like defects of dimension d -3 are a natural form of matter that couples to d-dimensional BF theory [START_REF] Horowitz | Exactly Soluble Diffeomorphism Invariant Theories[END_REF]. The resulting theory is in turn also a topological theory and can be completely quantized using the techniques of loop quantum gravity. Among these higher dimensional models the four-dimensional one (which couples to string-like defects) is of singular interest due to the special role played by 4-dimensional BF theory in the construction of spin foam models of four-dimensional quantum gravity.

At first look these strings are a rather dull form of matter: at their location there are conical singularities of the curvature tensor and the equations of motion imply that the string world sheet is locally flat [START_REF] Fairbairn | Quantization of string-like sources coupled to BF theory: physical scalar product and spinfoam models[END_REF] (vibrational modes of the strings are pure gauge). Nevertheless, as we will argue in this paper, the feature that makes these strings interesting is the fact that they are extended objects (this is also behind their exotic statistical properties [START_REF] Baez | Exotic statistics for strings in 4d BF theory[END_REF]). This will allow us to couple four-dimensional BF theory with more physically appealing degrees of freedom. As the set of possibilities is quite vast, we will restrict our attention to certain world-sheet theories that satisfy the following two properties: (a) they can be naturally (or minimally) coupled to BF theory in 4d, and (b) the coupled system defines a (topological) theory with no local degrees of freedom. Due to the close relationship between four-dimensional BF theory and gravity, requirement (a) is expected to produce physically interesting models, as they might provide natural candidates for the coupling of spin foam models of gravity with natural forms of matter. Requirement (b) implies that the models studied here are expected to be non-perturbatively quantizable.

We believe that the study of these simple topological models can be of more relevance than a simple exercise in the application of non-perturbative quantization techniques. We would like to explore the possibility that topological theories, containing low dimensional objects, could be used to construct a background independent quantum field theory with infinitely many ('quasi-local') degrees of freedom. This is in fact our motivation for imposing requirement (b) above.

The article is organized as follows: In Section II we briefly review the coupling of strings to four-dimensional BF theory. In Section III we show how Yang-Mills degrees of freedom can be added to the strings. We analyze the equations of motion of the coupled system and perform the canonical analysis to prove that the theory is topological. In Section IV we add a tetrad field on the world sheet and obtain an interesting model whose equations of motion resemble those of general relativity in a curious way. In Section V we study a purely two-dimensional model of background independent Yang-Mills theory which naturally follows from the results of the previous sections. In Section VII we present a speculative discussion about the possibility of using topological theories of the type introduced in this article in order to define a background independent quantum field theory with infinitely many degrees of freedom..

II. STRINGS COUPLED TO FOUR-DIMENSIONAL BF THEORY

The coupling of (d -3)-dimensional membranes to d-dimensional BF theory (defined for a large class of structure groups) was introduced in [START_REF] Baez | Quantization of strings and branes coupled to BF theory[END_REF]. Here we concentrate on the case of strings coupled to four-dimensional BF theory with structure group SO(3, 1) (see Refs. [START_REF] Mondragón | [END_REF] for its canonical analysis and Refs. [START_REF] Montesinos | Genuine covariant description of Hamiltonian dynamics[END_REF]13] for alternative action principles). If we denote M the four-dimensional space time manifold and W ⊂ M the two-dimensional world sheet of the string, the action defining the coupling is given by

S ST -BF = M B IJ ∧ F IJ (A) + τ W (B + d A q) IJ p IJ , (1)
where I, J = 1, .., 4, and if we denote T IJ ∈ so(3, 1) the generators of the Lie algebra then q = q IJ T IJ is a so(3, 1)valued 1-form on W and p = p IJ T IJ is a so(3, 1)-valued function on W . This action is invariant under the gauge transformations:

B → gBg -1 B → B + d A η A → gAg -1 + gdg -1 q → q -η q → gqg -1 p → gpg -1 , (2)
where g ∈ C ∞ (M , G) and η is any g-valued (d -3)-form. Varying the action with respect to the B field implies that the connection A is flat except at W :

F = -p δ W , (3)
where δ W is the distributional 2-form (current) associated to the string world-sheet. So, the string causes a conical singularity in the otherwise flat connection A. The strength of this singularity is determined by the field p, which plays the role of a 'momentum density' for the string. Note that while the connection A is singular in the directions transverse to W , it is smooth and indeed flat when restricted to W . Thus the equation of motion obtained from varying q makes sense:

d A p = 0. ( 4
)
This expresses conservation of momentum density and in fact implies that the field p remains in the same conjugacy class, hence it can be writen as p = τ λvλ -1 for v ∈ so(3, 1) a normalized vector and λ ∈ SO(3, 1). (the geometric meaning of s will be discussed below). Notice that the strength of the conical singularity at the location of the strings is in this sense non dynamical. This will change in the model of Section III.

Assuming the spacetime manifold is of the form M = Σ × R. We choose local coordinates (t, x a ) for which Σ is given as the hypersurface {t = 0}. By definition, x a with a = 1, 2, 3 are local coordinates on Σ. We also choose local coordinates (t, s) on the 2-dimensional world-sheet W , where s ∈ [0, 2π] is a coordinate along the onedimensional string formed by the intersection of W with Σ. Performing the standard Legendre transformation one obtains E a i = ǫ abc B ibc as the momentum canonically conjugate to A i b . Similarly, p IJ is the momentum canonically conjugate to q IJ 1 = q IJ a (∂ σ ) a . The phase space variables satisfy the following constraints:

L IJ := D a E a IJ -2δ S [q 1[I|M| p M J] ] ≈ 0 (5) K a IK := ǫ abc F IJ bc (x) + δ S [p IJ (∂ σ ) a ] ≈ 0, (6)
Here S ⊂ Σ denotes the one-dimensional curve representing the string, parametrized by x S (s), and for any field φ on S we define

δ S [φ] := S φ δ (3) (x -x S (s)).
The constraint ( 5) is the modified Gauss law of BF theory due to the presence of the string. The constraint ( 6) is the modified curvature constraint containing the dynamical information of the theory. This constraint implies that the connection A is flat away from the string S . Some care must be taken to correctly intepret the constraint for points on S . By analogy with the case of 3d gravity, the correct interpretation is that the holonomy of an infinitesimal loop circling the string at some point x ∈ S is exp(-p(x)) ∈ G, where p = τ λvλ -1 as before. This describes the conical singularity of the connection at the string world-sheet.

The BF phase space variables satisfy the standard commutation relations:

{E a i (x), A j b (y)} = δ a b δ j i δ (3) (x -y) {E a i (x), E b j (y)} = {A i a (x), A j b (y)} = 0. (7)
The phase space of the string is parametrized in terms of the momentum p IJ and the 'total angular momentum'

J IJ = 2q 1[I|M| p M J]
. The Poisson brackets of these variables are given by

{p IJ (s), J KL (s ′ )} = c ST IJKL p ST (s)δ (1) (s -s ′ ) {J IJ (s), J KL (s ′ )} = c ST IJKL J ST (s)δ (1) (s -s ′ ), (8)
where c ST IJKL are the structure constants of so(3, 1), and

{J IJ (s), λ(s ′ )} = -T IJ λ(s)δ (1) (s -s ′ ). (9)
The string variables are still subject to the following first class constraints:

tr[T IJ λzλ -1 ]J IJ = 0 tr[pλzλ -1 ] = τ tr[vz], (10)
where z ∈ g is such that [z, v] = 0. The last constraint is the generalization of the mass shell condition for point particles in 3d gravity. The Poisson bracket of the string variables with the BF variables is trivial, as well as the Poisson brackets among the p IJ .

A. Geometrical interpretation

Here we present a brief account of the analysis carried out in [START_REF] Fairbairn | Quantization of string-like sources coupled to BF theory: physical scalar product and spinfoam models[END_REF]. The full set of equations of motion of the theory is

F (A) = -p δ W d A B = -[q, p] δ W d A p| W = 0, φ * W (B + d A q) = 0 , (11)
where φ * W in the last equation denotes the pull back of the corresponding 2-forms to W . Therefore, the field configurations A = 0, B = 0, q = 0, p =constant gives a solution to the equations of motion in an open region U ⊂ M such that any open set containing points of W has points outside U . Since the theory is topological, all the solutions are equivalent to this one in U through a gauge transformation. Assume that we have a coordinate system in U with coordinate functions X I , (for I = 1, • • • , 4). In order to recover an interpretation of fields on a flat background we can make a gauge transformation of the type (2) with gauge parameter η IJ = X [I dX J] . In this gauge the solution is

B IJ ab = e I [a e J b] = δ I [a δ J b] q IJ a = X [I d a X J] . (12)
We see that in this gauge the B field defines a flat background geometry. There is still the residual gauge freedom that maintains this property of the B field given by gauge transformations of the form η 0 = df for some arbitrary f . We call this family of gauges flat gauges. The integrability conditions that follow from the equation dB = [q, p]δ W imply that d[p, q] = 0 or equivalently that [p, q] = dα for some potential α. If α = 0 it can be shown that [p, q] = 0 has non trivial solutions if s = p IJ p KL ǫ IJKL = 01 . In that case the string world-sheet X I (σ, t) is given by a plane in Minkowski spacetime passing through the origin defined by either the equation p IJ X I = 0 or ⋆p IJ X I = 0. We can translate the plane off the origin by choosing

α IJ = C [I X J] (this choice sends X I to X I + C I ). If s = 0 then equation [p, q] = 0 implies X I = 0.
One can establish a strict connection between these solutions and solutions of general relativity representing a cosmic string. In cylindrical coordinates {∂ t , ∂ r , ∂ ϕ , ∂ z } such that the string is lying along the z axis and goes through the origin the metric of a cosmic string solution of tension τ is:

ds 2 = g µν dx µ ⊗ dx ν = -dt 2 + dr 2 + (1 -a) 2 r 2 dϕ 2 + dz 2 , ( 13
)
where a = (1 -4Gτ ), G is the Newton constant. The dual co-frame for the above metric is written e 0 = dt e 1 = cos ϕdra r sin ϕdϕ e 2 = sin ϕdr + a r cos ϕdϕ e 3 = dz,

such that ds 2 = e I ⊗ e J η IJ . The spin connection (s.t.

d A e = 0) is A = A IJ µ J IJ dx µ = 4Gτ J 12 dϕ, (15)
where J IJ are the so(3, 1) generators. We can identify now the string momentum p above, namely p IJ J IJ = τ J 12 .

From the distributional identity ddϕ = 2πδ 2 (r)dxdy (x = r cos ϕ, y = r sin ϕ), it is immediate to compute the torsion T = T 0 e 0 and curvature F = F 12 σ 12 of the cosmic string induced metric:

T 0 = 0, F 12 = 8πGτ δ 2 (r) dxdy. ( 16
)
The above fields are clearly a solution of Einstein's equations with distributional matter

ǫ IJKL e J ∧ F KL = 8πGτ ǫ IJKL e J J IJ 12 δ W . ( 17
)
The previous solution is in one to one correspondence with the solution of (11)

B = (e ∧ e) * , A = 4Gτ J 12 dϕ, p = τ J 12 , q IJ = (zdt -tdz) δ [I 0 δ J] 3 = (zdt -tdz) J IJ 21 . (18)
One can construct a two string solution by 'superimposing' two solutions of the previous kind at different locations (notice that the equations are non linear so the new solution is not the sum of two solutions). It can be show that the torsion d A B is proportional to the distance separating the worldsheets in the flat-gauge where

B IJ ab = δ [I a δ J]
b . More strings can be added in a similar fashion.

III. MINIMAL COUPLING OF WORLD SHEET YANG-MILLS WITH 4D BF THEORY

Yang Mills theory in two dimensions can be written in a way that resembles BF theory if one is given a 2-form field ρ, namely

S Y M = W [E a F a (A) + ρE a E a ] , (19)
where a = 1, . . . , dim(g) are internal indices labeling the elements of a basis of the Lie algebra g of the gauge group of our choice G (we require G to be compact and g to have an invariant metric with which we raise and lower internal indices). The field A = A a µ dx µ ⊗J a is the g-valued connection 1-form, [J a , J b ] = f c ab J c where f c ab are the structure constants with respect to the basis {J a }. AUnder these assumptions the internal metric can be taken as k ab = cTr J a J b (assuming a matrix form for the generators J a and c is a constant that depends on the dimension of the representation of the J a ). The field E a is a collection of dim(g) many 0-forms. One can show that if ρ is non-degenerate (i.e., a volume form) the previous action is equivalent to the standard Yang Mills action

S Y M = W √ gg µν g ρρ F a µν F ρρ a ,
where the 2d metric g µν is such that ρ = √ gdx 1 ∧ dx2 . If one does the canonical analysis of the BF like action above one finds that the total Hamiltonian is not weakly vanishing due to the presence of the background structure provided by the (non-dynamical) ρ. It is also easy to check through the canonical analysis that the theory has no local degrees of freedom. Sometimes it is said that 2d YM is topological; however, this is not strictly the case because, even though the degrees of freedom are global (and certainly tied to the topology of W ), they are also related to the background structure ρ.

The simplest way of coupling two-dimensional Yang Mills theory with four-dimensional BF theory to produce a background independent field theory is to combine the B field and the world sheet variable p to produce a volume 2-form ρ = B IJ p IJ on the world sheet. The result is given by the following action:

S BF Y M = M B IJ ∧ F IJ (ω) + W B IJ E a E a -d ω q IJ p IJ + E a F a (A) (20)
The equations of motion of the new model are

F (ω) + δ W [E a E a p] = 0, d ω B + δ W [qp] = 0, φ * W (E a E a B -d ω q) = 0, (21)
and

2B • p E a + F a (A) = 0. (22)
We have not explicitly written the equations d ω p = 0, and d A E a = 0 as they implied by the integrability conditions arising from the Bianchi identities for the curvature of ω and A, respectively. Now we show that the new model is in fact a topological field theory (i.e. background independent with no local degrees of freedom). In order to do this we perform the 3+1 decomposition of the previous action and analyze its phase space structure. The unconstrained phase space is parametrized by the canonical variables (E µ IJ , A KL ν ) and (p IJ , q KL 1 ) (of the previous section) plus the Yang-Mills canonical pair (E a , A b 1 ). The constraints relating the bulk degrees of freedom with the ones on the world sheet are

L IJ := d Aµ E µ IJ + 2δ S [q [I|M| p M J] ] ≈ 0 ( 23
)
K µIJ := ǫ µνρ F IJ νρ (x) + δ S [E a E a p IJ ∂ µ σ ] ≈ 0. ( 24
)
Notice that L IJ is precisely the same as (5), while K IJ is a simple modification of (6). In fact there are new constraints

G a := d A E a ≈ 0, (25)
which is the standard Gauss law of Yang-Mills. These equations (together with Hamilton's equations of motion) imply that E a E a = constant. It is easy to see that the constraint algebra closes forming a first class system of 6 + 18 + dim(g) local constraints for the same number of configuration variables {q IJ 1 , A IJ µ , A a 1 }. The model has no local degrees of freedom 2 . The curvature constraint implies that the space-time connection is flat in the bulk and there is a conical singularity at the string. The strings on Σ can be viewed as flux lines of Yang-Mills electric field which back react with the environment producing a conical singularity whose strength is modulated by the Yang-Mills 'energy density'

ρ E = δ S [E a E a p IJ ].
As mentioned in the introduction the strength of the curvature singularity is now dynamical.

IV. ADDING A WORLD SHEET 'FRAME' FIELDS

The idea follows from the observation that the two-dimensional field theory defined by the following action has no local degrees of freedom

S = W dq IJ + * (e I ∧ e J ) p IJ + π I de I (26)
where * (e I ∧ e J ) = 1 2 ε IJ KL e K ∧ e L , W is a two-dimensional surface, q IJ = -q JI is a set of six 1-forms on W , e I is a set of four 1-forms on W , p IJ = -p JI is a set of six 0-forms (functions) on W , π I is a set of four 0-forms (functions) on W . In principle, there are other terms that can also be added to the action, for instance, (d * q IJ )p IJ = dq IJ * p IJ and (e I ∧ e J )p IJ .

In order to count the number of degrees of freedom let us perform the canonical analysis of this model. Let (y a ) = (y 0 , y 1 ) = (τ, σ) be local coordinates on W which is assumed to have the form W = S × R; the coordinate time τ labels the points along R and the space coordinate σ labels the points on S which is assumed to have the topology of S 1 . Therefore, using q IJ = q IJ a dy a = q IJ 0 dτ + q IJ 1 dσ, e I = e I 0 dτ + e I 1 dσ,

The action becomes

S = R dτ S dσ qIJ 1 p IJ + ėI 1 π I -λ IJ D IJ -λ I G I , (28)
where λ IJ := -q IJ 0 and λ I := -e I 0 are Lagrange multipliers imposing the constraints

D IJ = ∂ σ p IJ ≈ 0 (29) C I = ∂ σ π I + ε KL IJ e J 1 p KL ≈ 0. (30)
There are no more constraints. Smearing the constraints with test fields

D(N ) = S dσN IJ G IJ , C (
with [a, b] IJ := a I b Ja J b I . Thus all the 10 constraints are first class for the 10 configuration variables (q IJ 1 , e I 1 ). Therefore, the system has no local degrees of freedom, it is a topological field theory.

In the spirit of what was done in the previous section now we couple this world sheet action to the four-dimensional BF theory in such a way to maintain the topological character of the model. There is a natural choice of coupling leading to the new model introduced in this section, namely:

S BF Y M GR = M B IJ ∧ F IJ (ω) + W B IJ E a E a -d ω q IJ + * (e I ∧ e J ) p IJ + π I d ω e I + E a F a (A) . ( 33
)
We call this model BF Y M GR (where GR stands for general relativity) due to the suggestive similarity of the equations of motion with those of general relativity in the first order formalism. In order to make this statement more explicit let us analyze the equations of motion of the model. The observation is that on the world sheet variations with respect to p imply that B = E -2 ( * (e ∧ e)dq), hence the B field is simple up to a gauge transformation. Therefore, the simplicity constraints that reduce BF theory to general relativity are satisfied on the world sheet. The conclusion is more transparent is we study the remaining equations of motion. For instance we have

F IJ = -p IJ E 2 δ W → F IJ µν = -p IJ E 2 and ǫ IJKL e J p KL = d A π I , (34)
where F IJ µν is the smearing of the curvature tensor a two-dimensional surface dual to the world sheet along the coordinates µν, more precisely

F IJ µν := µ-ν F IJ .

Now we can appropriately combine the previous equations and obtain

ǫ µνρτ ǫ IJKL e J ν F KL ρτ = ǫ µν (d A π I ) ν E 2 , (35)
where ǫ µν := ǫ µνρτ (dt) ρ (dσ) τ , and we have assumed that E 2 is non vanishing in order to bring it to the right hand side. The previous equation has a suggestive similarity to Einstein's equation with source T µν = t I(µ e I ν) where t Iµ = (d A π I ) µ E 2 . This is why we call this topological model BF Y M GR.

We have emphasized the similarity of this model with Einstein's theory of gravity in order to motivate the introduction of this model. Now let us stress why this is quite different in fact. The main reason is that, in contrast with general relativity, this model is a topological theory with no local excitations. This conclusion becomes transparent in the Hamiltonian analysis which yields the following set of constraints for the canonical variables (E µ IJ , A KL ν ), (p IJ , q KL 1 ), (E a , A b 1 ), and (π I , e J 1 )

L IJ := d Aµ E µ IJ + 2δ S [q [I|M| p M J] ] ≈ 0, K µIJ := ǫ µνρ F IJ νρ (x) + δ S [E a E a p IJ ∂ µ σ ] ≈ 0, G a := d A E a ≈ 0
which are just the same as ( 23), ( 6) and (25) in addition to the new world sheet constraints

C I := d ω π I + 2e J * p IJ ≈ 0 (36)
It is easy to see using the results of the previous sections that the constraints form a first class set of 24 + dim(g) local constraints for the same number of configuration variables. The degrees of freedom are topological.

V. A TWO-DIMENSIONAL BACKGROUND INDEPENDENT YANG-MILLS THEORY

Using what we have learnt we can also define a 2-dimensional background independent Yang Mills theory by making the 2-form ρ appearing in eq. ( 19) dynamical in an world sheet intrinsic way: namely ρ = (e I ∧ e J )p IJ . The resulting action is

S T Y M = W dq IJ + * (e I ∧ e J ) + e I ∧ e J E a E a p IJ + E a F a (A) + π I de I . ( 37
)
The canonical analysis performed along the lines of the one corresponding to the previous model leads to the following constraints

G a = d A E a ≈ 0 (38) D IJ = ∂ σ p IJ ≈ 0 (39) C I = ∂ σ π I + 2e J 1 p IJ E a E a + 2e J 1 * p IJ ≈ 0 (40)
The first one is the familiar Gauss law of Yang Mills theory while the remaining ones correspond to the appropriate modification of the ones obtained above. The constraint algebra gives

{G(α), G(β)} = G([α, β] g ), {D(N ), D(M )} = 0, {D(N ), C(a)} = 0, {C(a), C(b)} = D( * [a, b] + E 2 [a, b]) + G(2[a, b] IJ p IJ E) (41)
with [a, b] IJ := a I b Ja J b I and [α, β] g is the commutator in the Lie algebra g. The constraint algebra closes and gives a first class system. As before we have 10 + dim(g) local constraints for the same number of configuration variables; hence the system is a topological field theory. We end this section with a remark. Notice that the constraint algebra has field dependent structure constants. This is characteristic of the constraint algebra of general relativity, although here the field dependence is much simpler since the quantity E 2 is constant on the world sheet due to the Gauss constraint. These are genuine field dependent structure constants.

VI. QUANTIZATION

We have shown how the coupling of four-dimensional BF theory to strings introduced in [START_REF] Baez | Quantization of strings and branes coupled to BF theory[END_REF] allows for the definition of a large class of topological field theories with physically interesting kinematical degrees of freedom. The set possibilities is indeed very large so we have concentrated here on two cases of special interest: world sheet Yang-Mills theories defined in terms of structure groups G possessing an ad G invariant metric in their Lie algebra g, and a world sheet tetrad (with intriguing resemblance with general relativity).

The fact that these models are topological indicates that their non-perturbative quantization should be well defined. Indeed the quantization of the model of Section III follows straightforwardly from the results of [START_REF] Baez | Quantization of strings and branes coupled to BF theory[END_REF] and [START_REF] Fairbairn | Quantization of string-like sources coupled to BF theory: physical scalar product and spinfoam models[END_REF]. This should be clear from the fact that the phase space structure presented in Section III is quite similar to the one of the theory briefly reviewed in Section II whose loop quantization is set up in [START_REF] Baez | Quantization of strings and branes coupled to BF theory[END_REF] and completely worked out in [START_REF] Fairbairn | Quantization of string-like sources coupled to BF theory: physical scalar product and spinfoam models[END_REF]. The only new ingredient are the Yang-Mills unconstrained degrees of freedom which are specially well suited for the application of loop variables techniques.

More precisely a basis of the kinematical Hilbert space-space of solutions of all quantum constraints with the sole exception of the curvature constraint (24)-of the model ( 20) is given by: (1) A bulk S0(3, 1) spin network functional of the S0(3, 1)-connection A based on a graph γ ∈ Σ with open ends at n points on the string S , (2) an n-point spin functional of λ (recall that the variable p = λvλ -1 for vg normalized and λ ∈ G), (3) a functional of the G-connection A given by the trace of the Wilson loop of A around the string S in an unitary irreducible representation of G (Figure 1). If G is compact we can always think of the latter quantum number as n ∈ N, where n labels the n-th eigenvalue ǫ n of the square of the electric field E a E a . The physical Hilbert space is obtained by imposing the quantum version of the constraint (24). This amounts for requiring the holonomy of loops around the string carrying Yang-Mills quantum flux number n ∈ N to be in the conjugacy class of exp (-ǫ n v). The techniques developed in [START_REF] Fairbairn | Quantization of string-like sources coupled to BF theory: physical scalar product and spinfoam models[END_REF] can be simply extended to treat this case. n m FIG. 1: The elements of a natural basis of the kinematical Hilbert can be written as the product of: 1) A functional of the Lorentz connection labelled by a graph in space and the assignment of unitary irreducible representations of the Lorentz group, i.e., a SO(3, 1) spin-network state (represented by the thin-lines graph), 2) An n-point spin function (represented here by the endpoints of the thin-lines-graph on the strings; see [START_REF] Baez | Quantization of strings and branes coupled to BF theory[END_REF] for the precise definition), 3) A functional of the Yang-Mills connection given by the product of Wilson loops on a unitary representation of the structure group G along the each string component.

Another important remark concerns the relationship of this model with 4-dimensional Yang-Mills coupled to general relativity. There is a close relationship between SO(3, 1) BF theory and general relativity [START_REF] Plebanski | On the separation of Einsteinian substructures[END_REF]15,16,17]. More precisely one can obtain the action of general relativity in the first order formulation by constraining the B field to be of the form B = (e ∧ e) * for a tetrad field e. This idea is in fact at the core of the definition of many spin foam models for four-dimensional quantum gravity [START_REF] Engle | The loop-quantum-gravity vertex-amplitude[END_REF]. Here we would like to point out that if such constraint is imposed on the B field appearing in the action (20) then the naive quantum amplitude for a world sheet configuration with quantized Yang-Mills electric field squared ǫ n is proportional to exp (i A p [W ]ǫ n ) where A p [W ] is the area of the world sheet computed with the area form (e ∧ e) * IJ p IJ . This is precisely the functional dependence of the Yang-Mills amplitude in any dimension [START_REF] Conrady | Analytic derivation of dual gluons and monopoles from SU(2) lattice Yang-Mills theory. I. BF Yang-Mills representation[END_REF]. We think that the model presented here might present a new perspective for the definition of a natural coupling of Yang-Mills fields with gravity in the context of spin foam models of quantum gravity.

It would be interesting to undertake the quantization of the model of Section IV. This would require the nonperturbative quantization of the tetrad field e I 1 and its conjugate momentum π I . We would like to study this question in detail in the future. Nevertheless, it seems clear that topological invariance should considerable simplify matters. Its seems that if this question can be resolved then one should be able to quantize the model of Section V. An interesting feature of these models (from loop quantum gravity perspective) is that their constraint algebras represent simpler models of that of general relativity, since as in the latter, they possess field dependent structure constants. Perhaps some technical issues concerning the quantization of such theories can be clarified in this simpler context. The model of Section IV is in addition interesting because of its additional resemblance to general relativity.

VII. SOME SPECULATIVE REMARKS

Let us finish with more speculative considerations which are however an important additional motivation for the study presented here. The most fundamental question of loop quantum gravity is whether one can construct a quantum field theory in the absence of a non-dynamical background metric. Several known results such as the quantization of Chern-Simons theory, 2+1 gravity, BF theory, etc., show that this is possible at least when dealing with topological field theories. The difficult question is whether one can construct an explicit non trivial example of background independent quantum field theory (with infinitely many degrees of freedom, i.e., infinitely many physical observables). One can argue that the entire framework of standard quantum field theory is based on the notion of particle, where a Fourier modes are the basic building block in the construction of standard quantum field theories. Similarly, we would like to explore the possibility that the finitely many degrees of freedom encoded in topological models, of the kind presented here, might be put together (be 'second quantized') in order to define a QFT with infinitely many degrees of freedom. Our ideas are at this stage rather heuristic with some aspects based in unproven assumptions motivated by properties of very simple models [START_REF] Livine | 2D manifold-independent spinfoam theory[END_REF]. The degree to which these assumptions can be made into factual statements will be explored elsewhere.

The basic idea goes as follows: In the model of [START_REF] Baez | Quantization of strings and branes coupled to BF theory[END_REF] as well as those presented here, the topology of the space time manifold M and the embedded world sheet W are held fixed. Under these conditions the transition amplitudes between kinematical states can be computed. When the topology of the world sheet is trivial (e.g. a cylinder W = S 1 × R or an ensemble of any arbitrary number of disconnected cylinders) these amplitudes can be used to define the so called physical inner product of the (canonically defined) quantum theory. Let us call H n with n ∈ N, the physical Hilbert space so defined for the quantum theory associated with classical configuration space containing n disconnected strings. One can construct a theory with infinitely many degrees of freedom defining the 'Fock' space F = ⊕ ∞ n=0 H n with the infinite set of quantum observables associated with the multi-string states (for the explicit construction in the particle case see [START_REF] Noui | Three dimensional loop quantum gravity: Towards a self-gravitating quantum Class[END_REF]). However, from our perspective3 such a theory seems rather trivial because there is no interaction between the H n 's for different values of n.

FIG. 2: Interacting string world sheets

When the world sheet topology is non trivial (e.g. it has branching components as in Figure 2 and/or non vanishing genus) the quantum amplitudes are still well defined (in the spin foam representation) but have no clear-cut physical interpretation 4 . It is tempting to interpret these amplitudes as providing the definition of physical interacting transition amplitudes in a theory where the kinematical Hilbert space is the Hilbert space F defined above. This interpretation would be consistent if: (1) the sum over world sheet topologies would be convergent, and (2) the transition amplitudes define a positive semidefinite inner product in F . This last requirement is highly non trivialit is the counterpart of unitarity in background dependent quantum field theory. If these conditions hold, this would provide a consistent way of rendering the world sheet topology dynamical achieving the goal of defining a non trivial (i.e. interacting) quantum field theory with infinitely many degrees of freedom: the latter given by the ensemble global degrees of freedom of all world sheet topologies.

Due to the fact that topology of two-dimensional orientable manifolds is characterized by a single integer (the genus g) condition (1) above can be satisfied if the amplitudes are suitably damped for high g. In fact the sum over two-dimensional topologies does converge in simple models such as 2d BF theory (see for instance [START_REF] Livine | 2D manifold-independent spinfoam theory[END_REF]). Some positive indication that property (2) could be realized for models of the kind presented here also comes from the study of this simple case. However, the model in [START_REF] Livine | 2D manifold-independent spinfoam theory[END_REF] is too simple an the sum over world sheets does not lead to a theory with infinitely many degrees of freedom. If the sum over world sheet topologies can be achieved in the models presented here, due to the the non trivial character of the degrees of freedom involved, be believe they might lead to non trivial examples of background independent field theories with infinitely many degrees of freedom. We would like to explore this possibility in the future.

VIII. DISCUSSION

We have shown how the extended nature of the conical defects that naturally couple to four dimensional BF theory allow for the introduction of physically interesting world-sheet fields while keeping the topological character of the theory. These models are expected to be non-perturbatively quantizable. In particular, the coupling of Yang-Mills theory with BF theory described in Section III can be quantized in a rather direct way by using the thecniques of Refs. [START_REF] Baez | Quantization of strings and branes coupled to BF theory[END_REF][START_REF] Fairbairn | Quantization of string-like sources coupled to BF theory: physical scalar product and spinfoam models[END_REF]. For this theory we get at a remarkably simple description of states in the kinematical Hilbert space where bulk-geometry spin network states are dual to Yang-Mills electric field flux lines (see Figure 1). The strength of the conical singularities at the location of flux lines is proportional to the electric field square.

The models are in close relationship with gravity in at least two independent ways. On the one hand, as we argued in Section II A, solutions of the topological models are in one to one correspondence with solutions of Einstein's equations. This correspondence between solutions has to be interpreted with due care as the gauge symmetries of our models is much larger than the one of general relativity. In particular local excitations such as gravitons are pure gauge in our model. Nevertheless the correspondence among solutions might be of relevance if some of the hopes described in the previous section could be realized. On the other hand, our model is linked to gravity along the well known relationship between four dimensional BF theory and general relativity explicitly exhibit in the Plebansky formulation of gravity. In particular, it would be interesting to compare our model with the coupling to Yang-Mills theories proposed in [START_REF] Oriti | A spin foam model for pure gauge theory coupled to quantum gravity[END_REF].

These models are simple but non trivial. In particular, the presence of geometric degrees of freedom as well as matter-like degrees of freedom make them potentially useful for the study of various conceptual difficulties in nonperturbative quantum gravity.

After the 1+1 decomposition, (x µ ) = (x 1 , x 2 ) = (τ, σ), each of the terms become: The action becomes (neglecting space boundary terms)

S = dτ ∧ dσ E a Ȧa 1 + π IJ ωIJ 1 + p I ėI 1 -λ a G a -λ I C I -λ IJ D IJ (

  Poisson brackets {D(N ), D(M )} = 0, {D(N ), C(a)} = 0, {C(a), C(b)} = D( * [a, b])

  43)withG a := d A E a (44) C I := d ω π I + βp * IJ e J 1 E a E a + γp IJ e J 1 E a E a D IJ := d ω p IJ + 1 2 (π I e J1π J e I1 ) .Smearing the constraints with test fieldsG(α) := S dσα a G a , C(λ) := S dσλ I C I , D(N ) := S dσN IJ D IJ(45)The constraint algebra gives{C(λ), C(Λ)} = S [λ, Λ] IJ 2γ φ 2 E a E a D IJ + 4γ φ 2 E a [λ, Λ] IJ p IJ G a , [λ, Λ] IJ := 1 2 λ I Λ Jλ J Λ I (46) {G(α), G(β)} = G([α, β]) (47) {G(α), C(M )} = {G(α), D(N )} = 0 {D(N ), D(M )} = D([N, M ]) {D(N ), C(β)} = C(N • β) {C(M ), C(N )} = E 2 D([N, M ]) + 2G([N, M ] • p E)where [N, M ] I and (N • β) I := N IJ β J is the commutator in the Lie algebra so(4). The constraints are all first class which leads to the conclusion that there are zero local degrees of freedom.

  The constant τ defines the string tension. Conjugacy classes of so(3, 1) are labelled by the two Lorentz Casimirs. So far we have fixed only one by choosing the string tension τ 2 = p IJ p IJ . The other Casimir defines an extra parameter s = p IJ p KL ǫ IJKL

If we allow for complex p IJ , then solutions exist also if p IJ is self-dual (or anti self-dual).

There is a subtlety concerning the constraints K IJ . In fact when we are away from the string the source term vanishes and the Bianchi identity implies that only

out of the 6 ones are independent. On the string the Bianchi identity implies d A p = 0 which is indeed and independent condition.

In[START_REF] Noui | Three dimensional loop quantum gravity: Towards a self-gravitating quantum Class[END_REF] the context in which F is introduce is quite different. There one uses it to setup a perturbation theory.

A field theoretic interpretation as Feynman diagrams in the context of perturbation theory of an associated effective field theory is proposed in[START_REF] Freidel | Ponzano-Regge model revisited. III: Feynman diagrams and effective field theory[END_REF].

Acknowledgements

This work was supported in part by: the Agence Nationale de la Recherche, Grant No. ANR-06-BLAN-0050, the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) through a visiting professor fellowship, and by CONACyT, México, Grant No. 56159-F.

IX. APPENDIX

Here we construct a truly background independent model which will lead to a genuinely topological theory. The discussion of the first part of this paper gives a clear way to defining a background independent analog. The action is

where we have replaced q by q = * (e I ∧ e J )p IJ in the previous action,

ν and-in order to make the e I and p IJ fields dynamical-added the natural term p IJ F IJ (ω)+ π I d ω e I which also requires the introduction of the connection ω IJ . Of course there are other additional fields which can be added to the action (42) but, for the moment, let us look just at this action. The parameters β, and γ are coupling constants.