Complex actions in two-dimensional topology change

DOI10.1088/0264-9381/14/1/018zbMATH Open0868.53069arXivgr-qc/9511023OpenAlexW3104090822MaRDI QIDQ4332178

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Publication date: 19 August 1997

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Abstract: We investigate topology change in (1+1) dimensions by analyzing the scalar-curvature action $1/2intRdV$ 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 $1/2sqrt-gR$ approaches a delta-function whose strength is complex: for the yarmulke family the strength is $$, where $$ is the rapidity parameter of the associated Misner space; for the trousers family it is simply $+2pii$. 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.

Full work available at URL: https://arxiv.org/abs/gr-qc/9511023

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