We investigate topology change in (1+1) dimensions by analyzing the
scalar-curvature action 1/2∫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 vertex of string theory, if one makes that
interpretation). 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 spacetime. As the
regulator is removed, the scalar density 1/2−g​R approaches a
delta-function whose strength is complex: for the yarmulke family the strength
is β−2πi, where β is the rapidity parameter of the associated
Misner space; for the trousers family it is simply +2Ï€i. This implies that
in the path integral over spacetime metrics for Einstein gravity in three or
more spacetime dimensions, topology change via a crotch singularity is
exponentially suppressed, whereas appearance or disappearance of a universe via
a yarmulke singularity is exponentially enhanced.Comment: 34 pages, REVTeX v3.0. (Presentational reorganization; core results
unchanged.