Complex actions in two-dimensional topology change

We investigate topology change in (1+1) dimensions by analyzing the scalarcurvature action 12 ∫ RdV at the points of metric-degeneration that (with minor exceptions) any nontrivial Lorentzian cobordism necessarily possesses. In two dimensions any cobordism can be built up as a combination of only two elementary types, the “yarmulke” and the “trousers.” For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics that vanish smoothly at a single point, resulting in a conical-type singularity there. In the yarmulke case, the distinguished point is analogous to a cosmological initial (or final) singularity, with the spacetime as a whole being obtained from one causal region of Misner space by adjoining a single point. In the trousers case, the distinguished point is a “crotch singularity” that signals a change in the spacetime topology (this being also the fundamental On leave of absence from Department of Physics, University of Helsinki. Electronic address: [email protected]