Topological field theory of the initial singularity of spacetime*

We suggest a new solution of the initial spacetime singularity. In this approach the initial singularity of spacetime corresponds to a zero-size singular gravitational instanton characterized by a Riemannian metric configuration ( + + + + ) in dimension D = 4. Connected with some unexpected topological data corresponding to the zero scale of spacetime, the initial singularity is thus not considered in terms of divergences of physical fields but can be resolved within the framework of topological field theory. Then it is suggested that the `zero-scale singularity' can be understood in terms of topological invariants (in particular, the first Donaldson invariant ∑<sub>i</sub>(-1)<sup>n<sub>i</sub></sup>). With this perspective, here we introduce a new topological index, connected with zero scale, of the form {Z}<sub>β = 0</sub> = Tr (-1)<sup>s</sup>, which we call the `singularity invariant'. Interestingly, this invariant also corresponds to the invariant topological current yield by the hyperfinite II_∞ von Neumann algebra describing the zero scale of spacetime. Then we suggest that the (pre-)spacetime is in thermodynamical equilibrium at the Planck-scale and is therefore subject to the KMS condition. This might correspond to a unification phase between the `physical state' (Planck scale) and the `topological state' (zero scale). Then we conjecture that the transition from the topological phase of the spacetime (around the zero scale) to the physical phase observed beyond the Planck scale should be deeply connected to the supersymmetry breaking of the N = 2 supergravity.