Quantum Gravity as a Broken Symmetry Phase of a BF Theory

We explain how General Relativity with a cosmological constant arises as a broken symmetry phase of a BF theory. In particular we show how to treat de Sitter and anti-de Sitter cases simultaneously. This is then used to formulate a quantisation of General Relativity through a spin foam perturbation theory. We then briefly discuss how to calculate the effective action in this quantization procedure.


Introduction
Quantization of General Relativity (GR) is still one of the outstanding problems in theoretical physics. Although the string theory has made a significant progress on this problem [1], the string theory approach is essentially a perturbation theory around the flat spacetime, which then makes it difficult to study the quantum cosmology problems. On the other hand, the Loop Quantum Gravity (LQG) approach is nonperturbative and background metric independent [2]; however, obtaining the perturbative and semiclassical results is difficult. This is related to the fact that LQG is not manifestly covariant under four-dimensional diffeomorphisms. This problem has been overcome by introducing the spin foam formalism [3], where the basic object of LQG, the spin network, a colored graph which lives in the space, is generalised to a spin foam (SF), a colored two-complex which lives in the spacetime. The SF models are typically obtained from the path-integral quantization of a BF theory [3], and the reason for this is that the Einstein-Hilbert action in the Palatini formalism where F = R and the two-form B is constrained by the relation The BF theories are topological, and their path-integral quantization is well understood [3]. Let the BF theory group be a Lie group G and let Λ label the finite-dimensional irreducible representations (irreps) of G, then the partition function can be written as a sum over the irreps of the spin foam amplitudes associated to the dual 2-complex Γ of a triangulation of M where the Λ's label the faces f of Γ, the ι's are the corresponding intertwiners which label the edges l of Γ, A τ are the tetrahedron amplitudes and A σ are the four-simplex amplitudes. The A l is a function of the dimensions of the four irreps that meet at an edge l, while A v is given by an evaluation of the four-symplex spin network where the ten edges carry the irreps Λ f and the five vertices carry the intertwiners ι l [3]. The infinite sums in (2) are divergent and the typical regularization is to replace the Lie group G by the quantum group G q where q is a root of unity [4]. In this case the set of Λ f becomes finite so that (2) becomes finite. However, the quantization of constrained BF theories is not that well understood, and in the GR case there is a proposal by Barrett and Crane [5,6] to implement the constraint (1) in the SF formalism by restricting the irreducible representations of SO(3, 1) that color the faces of the spin-foam complex as where J (Λ) are the Lorentz group generators in the representation Λ. Furthermore, one can argue that the admissible irreps have to be unitary, so that (3) selects the so called simple unitary irreps of the Lorentz group Λ = (j, 0) or Λ = (0, ρ) where 2j ∈ Z + and ρ ∈ (0, ∞) [6]. Given that a triangulation of a four-dimensional pseudo-Riemannian manifold can be always chosen such that all the triangles are spacelike, one can work with only the (0, ρ) irreps. In this way one obtains a spin foam model where the GR path integral is defined as a multiple integral over the face irreps of the corresponding spin foam amplitudes and (2) becomes where θ is a function of the four edge irreps and A rel σ is the relativistic evaluation of a four-simplex spin network.
One can show that for each non-degenerate triangulation of M the corresponding BC spin foam state sum is finite, provided that the θ amplitudes are appropriately chosen [7]. In this way one obtains a finite theory of quantum gravity; however, there are problems with this theory. First, the choice of the θ amplitudes is not unique, since there are other choices which also lead to a finite state sum [8]. Second, coupling fermionic matter is difficult because fermion fields couple to individual tetrads, while in the BC model one can couple the fermions only to a specific quadratic combination of the tetrads (the B field). And the last, and the most difficult problem is to find the semiclassical limit of this quantum gravity theory.

GR as a symmetry breaking of a BF theory
The first two problems of the BC model suggest that one should look for a spin foam model of GR which arises from the quantization of a BF theory which is not constrained and includes the tetrads in the BF theory connection one-form. This also means that GR will appear through a symmetry breaking mechanism, since the larger symmetry of a bigger connection has to be broken to a Lorentz group connection plus the tetrads. Interestingly, such a mechanism was found by MacDowell and Mansouri [9] in the context of OSp (n|4) supergravity theories. The bosonic spatial symmetry group was Sp(4), which is the covering group of the anti-de Sitter group SO (3,2). Their action was quadratic in the field strengths and gave EH action with a negative cosmological constant. A BF theory formulation for a positive cosmological constant EH action was found by Smolin and Starodubtsev [10], and corresponds to the de Sitter group case, i.e. SO(4, 1).
Extending the Smolin-Starodubtsev result to the anti-de Sitter case is easy, since one can treat the both cases simultaneously in the following way. The Lie algebras of both groups can be represented as where ± corresponds to anti-de Sitter and de Sitter cases respectively. The corresponding connection can be written as where ω is the spin-connection, e a are the tetrads and λ is a dimensionful parameter. The curvature 2-form F = dA + A ∧ A is then given by where T a = de a + ω b a ∧ e b is the torsion. Let us introduce the Lie algebra valued 2-form B = b a P a + B ab J ab , then consider the following action This is the BF theory action which is perturbed by a symmetry breaking term, since the quadratic in B term is invariant only under the Lorentz subgroup. This action can be written as so that the b and B equations of motion imply vanishing of the torsion and The remaining equations correspond to the action The plus sign gives the MacDowell-Mansouri action, while the minus sign corresponds to the theory studied in [10]. Either way, it is easy to see that S * can be written as where G N = αλ −2 is the Newton constant and Λ = λ 2 2 is the cosmological constant and S top is proportional to the Euler class of M , which does not affect the equations of motion. Hence one obtains the Einstein equations for positive/negative cosmological constant. Since α = 2G N Λ, this gives that α is an extremally small number, which then justifies the view that GR with a small cosmological constant is a small deformation of a BF theory.

Spin foam perturbation theory
The action (4) is well-suited for a perturbative quantization. The standard approach is to calculate perturbatively the generating functional where J and j are the sources for the A and the B fields and U (A, B) is the perturbative interaction. This can be done via the formula This generating functional can be calculated by using the spin foam technology [11], and the result can be written as a state sum where is the insertion at an edge l of Γ while A σ (λ, J) is the modified 4-symplex amplitude, with the λ-edges attachments at its vertices and D (Λ f ) (e J f ) insertions at its edges. When the sources vanish, the state sum (5) reduces to the state sum (2). Notice that the state sum (5) is not of the same type as (2), because one has to label both the edges and the faces of Γ with the irreps of G. The expression (5) is an infinite sum, and has to be regularized. When G = SO(5), i.e. the Euclidian gravity case, the regularization consists of replacing the category of irreps of G by the category of irreps of the quantum group G q where q is a root of unity. When G = SO (3,2) or G = SO(4, 1), one has to use the category of unitary irreps, which are infinite-dimensional and typically have discrete and continuos series. This means that one will obtain both the state sums and the state integrals (like in the BC model case). Convergence properties of these models have not been yet investigated, and the hope is that the corresponding categories of quantum group irreps will yield convergent sums. Alternatively, one could try to generalize the discrete gauge fixing procedure introduced by Freidel and Louapre for the three-dimensional spin foam models [12,13]. Given a regularized Z 0 [j, J] one can calculate the generating functional perturbatively as The semiclassical properties of the theory where Z 0 is given by (5) can be explored by analyzing the effective action. The effective action can be calculated via the Legendre transform of Z[J, j] This can be done perturbatively in α, and if we denote (Ā l ,B f ) as X I , then Γ(X) = m≥0 α m n≥0 1 n! I 1 ,...,In C mn (I 1 · · · I n )X I 1 · · · X In = m≥0 α m Γ m (X).
In this way one can explore the semiclassical limit of the theory, and the crucial test for the physical relevance of the theory is whether or not Γ 0 + αΓ 1 gives the discretized classical action (4)

Conclusions
The spin foam perturbation theory is a promising approach for defining a viable quantisation of GR. However, one must resolve first the technical questions related with the regularization of the SF generating functional. This is something which may be complicated, but it can be done. Then one can calculate the effective action by the method outlined here and verify the classical limit. If this limit is GR, one can then proceed to calculate the higher order in α corrections. The matter can be coupled by using again the same method of the generating functional [11]. On the mathematical side, one can explore the state sum (5) for the case of compact groups and study what happens with the topological invariance. In the non-compact group case, one will need first to study the unitary irreps of quantum de Sitter and anti-de Sitter groups. One can also study Z 0 (j, J) for three-dimensional spacetimes, in which case the relevant groups are SO(4), SO(3, 1), SO(2, 2) and their covering groups. In the case of two-dimensional spacetimes the relevant groups are SO(3), SO(2, 1) and their covering groups.