Complex actions in two-dimensional topology change

TL;DR

This paper analyzes topology change in 1+1 dimensions using scalar-curvature action at metric-degeneration points, revealing how different elementary cobordisms contribute to the path integral in quantum gravity.

Contribution

It introduces a Morse-theory inspired approach to regularize and evaluate the scalar curvature at topology-changing singularities, providing insights into their quantum amplitudes.

Findings

Crotch singularities are exponentially suppressed in the path integral.

Yarmulke singularities are exponentially enhanced in the path integral.

Topology change effects depend on the type of elementary cobordism considered.

Abstract

We investigate topology change in (1+1) dimensions by analyzing the scalar-curvature action $1/2 \int R dV$ 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 $1/2 \sqrt{-g} R$ approaches a delta-function whose strength is complex: for the yarmulke family the strength is $\beta -2\pi i$, where $\beta$ is the rapidity parameter of the associated Misner space; for the trousers family it is simply $+2\pi i$. 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.