Approaches to Quantum Gravity

320 D. Oritibetter, by a single matrix model in which the matrix dimension has been turnedfrom a parameter into a dynamical variable. The Feynman amplitudes are given byZ(Ɣ) = ∑ j dim( j)2−2g(Ɣ) , so the GFT above gives a quantization of BF theory(with gauge group G) on a closed triangulated surface, dual to Ɣ, ofgenusg(Ɣ),augmented by a sum over all such surfaces [20]. A similar quantization of 2d gravitywould use G = U(1), a restriction on the representations, and additional dataencoding bundle information [21].The extension to higher dimensions can proceed in two ways. In [15] thefirst“tensor model”, for an N × N × N tensor φ was introduced:S[φ] = ∑ α i( 12 φ α 1 α 2 α 3φ α1 α 2 α 3+ λ 4! φ α 1 α 2 α 3φ α3 α 4 α 5φ α5 α 2 α 6φ α6 α 4 α 1),which generates both manifold- and pseudo-manifold-like 3d simplicial complexes[15; 17]. This is turned easily into a GFT by a straightforward generalization of the2d case. The following kinetic and vertex terms:∫K(g i , ˜g i ) = dg ∏ δ(g i ˜g −1ig),GiorV(g ij , g ji ) = ∏ i∫Gdg i∏i< jδ(g i g ij g −1jig −1j),where the integrals impose the gauge invariance under the action of G, givetheGFT quantization of BF theories, for gauge group G, in any dimension [22; 23].In particular, in three dimensions, the choice [22] G = SO(3) or G = SO(2, 1)provides a quantization of 3D gravity in the Euclidean and Minkowskian signatures,respectively, and the so-called Ponzano–Regge spin foam model, while thechoice of the quantum group SU(2) q gives the Turaev–Viro topological invariant.The action is then:S[φ] = ∏ ∫φ(g 1 , g 2 , g 3 )φ(g 1 , g 2 , g 3 )iG+ λ 6∏∫dg i φ(g 1 , g 2 , g 3 )φ(g 3 , g 4 , g 5 )φ(g 5 , g 2 , g 6 )φ(g 6 , g 4 , g 1 ).4!i=1G(17.4)Lots is known about the last model (see chapter 16 by Freidel). Here, we mentiononly one result that is of interest for the general issue of GFT renormalization. Thisis the proof [25] that a simple modification of the GFT above gives a model whose