Matrix model formulation of four dimensional gravity

The problem of constructing a quantum theory of gravity has been tackled with very different strategies. An attractive possibility is that of encoding all possible space-times as specific Feynman diagrams of a suitable field theory as it happens for the matrix model formulation of two-dimensional quantum gravity (see for example [1] and references therein). In the perturbative approach to the matrix model the resulting Feynman diagrams have vertices which correspond to two-simplices, and propagators which correspond to edge-pairings, so a diagram leads to a surface obtained by glueing triangles. Indeed one is brought to the search for theories having Feynman diagrams in which vertices can be identified with n-simplices, and propagators with glueings of codimension-1 faces. If this happens, each Feynman diagram can be identified as n-dimensional simplicial complex. We will discuss how the Feynman diagrams of an n-tensor model can be interpreted in this way. Moreover, we will discuss, in dimension four, the condition that must be fulfilled in order that the resulting space is a four manifold [3].