An Interacting Geometry Model and Induced Gravity

We propose the theory of quantum gravity with interactions introduced by topological principle. The fundamental property of such a theory is that its energymomentum tensor is an BRST anticommutator. Physical states are elements of BRST cohomology group. The model with only topological excitations, introduced recently by Witten is discussed from the point of view of induced gravity program. We find that the mass scale is induced dynamically by gravitational instantons. The low energy effective theory has gravitons, which occur as the collective excitations of geometry, when the metric becomes dynamical. Applications of cobordism theory to QG are discussed. The idea that topology of spacetime must play the fundamental role in any reasonable theory of Quantum Gravity (QG) was first put forward by Wheeler1. This highly original physical proposal was subsequently taken seriously by a number of people 2,3,4, especially after the period of intensive studies of nonperturbative properties of gauge theories (instantons, monopoles). The concept of gravitational instanton was introduced by Hawking4. The classical role of instantons is to break classical symmetries of the action functional for a given physical system. A beautiful example of this phenomenon is the t‘Hooft solution12 of the chiral symmetry breaking problem in gauge theories (the famous U(1) problem). We will see that this concept will play an important role in our proposal of the QG model whose only degrees of freedom are the global, topological degrees of freedom. The model we propose in this essay shares many qualitative properties with the modern theory of strings6. One may ask how we would expect to recover local physics, and in particular classical General Relativity (GR), from a model whose only degrees of freedom are the global, topological characteristics of the four-manifolds? We do not have the definite answer to this question yet, but it is clear that the four-metric on a topological four-manifold becomes a dynamical degree of freedom at an energy scale MP , which is of the order of the Planck mass. Therefore, if our model displays the property of dynamical scale symmetry breaking like Quantum Chromodynamics (QCD), then the dimensional transmutation phenomenon operative in QCD should take place. The dimensionless unrenormalized coupling constant g0 which parametrizes our model gets replaced by the renormalization group invariant mass parameter MP (g, μ), which will play the role of the Planck mass in the low energy