Research ArticleNo Access

Quantum fluctuations of the metric, three-geometries and conformal embeddings in four-manifolds

Research Foundation of Southern California, 8861 Villa La Jolla Drive #13595, La Jolla, CA 92037, United States

https://doi.org/10.1142/S0219887821502340Cited by:0 (Source: Crossref)

Abstract

In this paper, connections between the path integrals for four-dimensional quantum gravity and string theory are emphasized. It is shown that there is a natural relation between these two path integrals based on the theorems on embeddings of two-dimensional surfaces in four dimensions and four-dimensional manifolds in ten dimensions. The isometry groups of the three-geometries that are spatial hypersurfaces confomally embedded in the four-manifolds are required to be subgroups of $G_{2}$ , which is the invariance group of the Pfaffian differential system satisfied by one form in the cotangent bundles on the four-manifolds. Based on this and other physical conditions, the three-geometries are restricted to be $S^{3}$ , $S^{2} \times S^{1}$ and $𝔼^{3}$ with a boundary, which may be included in the quantum gravitational path integral over four-manifolds which are closed at initial times followed by an exponential expansion compatible with supersymmetry.

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