Quantum Gravity as a Deformed Topological Quantum Field Theory

It is known that the Einstein-Hilbert action with a positive cosmological constant can be represented as a perturbation of the SO(4,1) BF theory by a symmetry-breaking term quadratic in the B field. Introducing fermionic matter generates additional terms in the action which are polynomial in the tetrads and the spin connection. We describe how to construct the generating functional in the spin foam formalism for a generic BF theory when the sources for the B and the gauge field are present. This functional can be used to obtain a path integral for General Relativity with matter as a perturbative series whose the lowest order term is a path integral for a topological gravity coupled to matter.Comment: 7 pages, talk presented at the QG05 conference, 12-16 September 2005, Cala Gonone, Ital

Wilson loops, geometric operators and fermions in 3d group field theory

Group field theories whose Feynman diagrams describe 3d gravity with a varying configuration of Wilson loop observables and 3d gravity with volume observables at each vertex are defined. The volume observables are created by the usual spin network grasping operators which require the introduction of vector fields on the group. We then use this to define group field theories that give a previously defined spin foam model for fermion fields coupled to gravity, and the simpler quenched approximation, by using tensor fields on the group. The group field theory naturally includes the sum over fermionic loops at each order of the perturbation theory.Comment: 13 pages, many figures, uses psfra

Hidden Quantum Gravity in 3d Feynman diagrams

In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results.Comment: 35 pages, 4 figures, some comments adde

Bubble divergences from cellular cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory, the Ponzano-Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined in general, because of a phenomenon called `bubble divergences'. A common expectation is that the degree of these divergences is given by the number of `bubbles' of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian -- in both cases, the divergence degree is given by the second Betti number of the 2-complex.Comment: 5 page

N=2 supersymmetric spin foams in three dimensions

We construct the spin foam model for N=2 supergravity in three dimensions. Classically, it is a BF theory with gauge algebra osp(2|2). This algebra has representations which are not completely reducible. This complicates the procedure when building a state sum. Fortunately, one can and should excise these representations. We show that the restricted subset of representations form a subcategory closed under tensor product. The resulting state-sum is once again a topological invariant. Furthermore, within this framework one can identify positively and negatively charged fermions propagating on the spin foam. These results on osp(2|2) representations and intertwiners apply more generally to spin network states for N=2 loop quantum supergravity (in 3+1 dimensions) where it allows to define a notion of BPS states.Comment: 12 page

From twistors to twisted geometries

In a previous paper we showed that the phase space of loop quantum gravity on a fixed graph can be parametrized in terms of twisted geometries, quantities describing the intrinsic and extrinsic discrete geometry of a cellular decomposition dual to the graph. Here we unravel the origin of the phase space from a geometric interpretation of twistors.Comment: 9 page

Emergent non-commutative matter fields from Group Field Theory models of quantum spacetime

We offer a perspective on some recent results obtained in the context of the group field theory approach to quantum gravity, on top of reviewing them briefly. These concern a natural mechanism for the emergence of non-commutative field theories for matter directly from the GFT action, in both 3 and 4 dimensions and in both Riemannian and Lorentzian signatures. As such they represent an important step, we argue, in bridging the gap between a quantum, discrete picture of a pre-geometric spacetime and the effective continuum geometric physics of gravity and matter, using ideas and tools from field theory and condensed matter analog gravity models, applied directly at the GFT level.Comment: 13 pages, no figures; uses JPConf style; contribution to the proceedings of the D.I.C.E. 2008 worksho

Spin foam model from canonical quantization

We suggest a modification of the Barrett-Crane spin foam model of 4-dimensional Lorentzian general relativity motivated by the canonical quantization. The starting point is Lorentz covariant loop quantum gravity. Its kinematical Hilbert space is found as a space of the so-called projected spin networks. These spin networks are identified with the boundary states of a spin foam model and provide a generalization of the unique Barrette-Crane intertwiner. We propose a way to modify the Barrett-Crane quantization procedure to arrive at this generalization: the B field (bi-vectors) should be promoted not to generators of the gauge algebra, but to their certain projection. The modification is also justified by the canonical analysis of Plebanski formulation. Finally, we compare our construction with other proposals to modify the Barret-Crane model.Comment: 26 pages; presentation improved, important changes concerning the closure constraint and the vertex amplitude; minor correctio

Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory

We study the no gravity limit G_{N}-> 0 of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the G_{N} expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagatorsComment: 46 pages, the wrong file was first submitte

Canonical analysis of the BCEA topological matter model coupled to gravitation in (2+1) dimensions

We consider a topological field theory derived from the Chern - Simons action in (2+1) dimensions with the I(ISO(2,1)) group,and we investigate in detail the canonical structure of this theory.Originally developed as a topological theory of Einstein gravity minimally coupled to topological matter fields in (2+1) dimensions, it admits a BTZ black-hole solutions, and can be generalized to arbitrary dimensions.In this paper, we further study the canonical structure of the theory in (2+1) dimensions, by identifying all the distinct gauge equivalence classes of solutions as they result from holonomy considerations. The equivalence classes are discussed in detail, and examples of solutions representative of each class are constructed or identified.Comment: 17 pages, no figure