Topologicalfield theory of the initial singularity ofspacetime*

Abstract

We suggest a new solution of the initial spacetime singularity. Inthis approach the initial singularity of spacetime corresponds to azero-size singular gravitational instanton characterized by a Riemannianmetric configuration ( + + + + ) in dimension D = 4. Connectedwith someunexpected topological data corresponding to the zero scale ofspacetime, the initial singularity is thus not considered in terms ofdivergences of physical fields but can be resolved within the frameworkoftopological field theory. Then it is suggested that the`zero-scale singularity' can be understood in terms oftopological invariants (in particular, the first Donaldsoninvariant ∑i(-1)ni). With this perspective, herewe introduce a new topological index, connected with zero scale,of the form {Z}β = 0 = Tr (-1)s, which we call the`singularity invariant'. Interestingly, this invariant alsocorresponds to the invariant topological current yield by thehyperfinite II∞ von Neumann algebra describing thezero scale of spacetime. Then we suggest that the(pre-)spacetime is in thermodynamical equilibrium at thePlanck-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' (zeroscale). Then we conjecture that the transition from thetopological phase of the spacetime (around the zero scale) to thephysical phase observed beyond the Planck scale should bedeeply connected to the supersymmetry breaking of the N = 2supergravity.