COMPLEX ACTIONS IN 2-DIMENSIONAL TOPOLOGY CHANGE

We investigate topology change in (1 + 1) dimensions by analysing the scalar-curvature action 1/2 integral R dV at the points of metric-dege neration that (with minor exceptions) any non-trivial Lorentzian cobor dism necessarily possesses. in two dimensions any cobordism can be bui lt up as a combination of only two elementary types, the 'yarmulke' an d the 'trousers.' For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics that vanish smoo thly at a single point, resulting in a conical-type singularity there. In the yarmulke case, the distinguished point is analogous to a cosmo logical initial (or final) singularity, with the spacetime as a whole being obtained from one causal region of Misner space by adjoining a s ingle point. In the trousers case, the distinguished point is a 'crotc h singularity' that signals a change in the spacetime topology (this b eing also the fundamental vertex of string theory, if one makes that i nterpretation). We regularize the metrics by adding a small imaginary part, whose sign is fixed to be positive by the condition that it lead to a convergent scalar field path integral on the regularized spaceti me. As the regulator is removed, the scalar density 1/2 root -g R appr oaches a delta-function, whose strength is complex: for the yarmulke f amily the strength is beta - 2 pi i, where beta is the rapidity parame ter of the associated Misner space; for the trousers family it is simp ly +2 pi i. This implies that in the path integral over spacetime metr ics for Einstein gravity in three or more spacetime dimensions, topolo gy change via a crotch singularity is exponentially suppressed, wherea s appearance or disappearance of a universe via a yarmulke singularity is exponentially enhanced. We also contrast these results with the si tuation in a vielbein-cum-connection formulation of Einstein gravity.