Dominant topologies in Euclidean quantum gravity

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according to the sign of the cosmological constant. For 3 > 0, saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For 3 < 0, on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the ‘density of topologies’ grows fast enough to overwhelm this suppression. The value 3 = 0 is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle–Hawking wavefunction. PACS numbers: 0460G, 9880H, 0420G, 0240K It has been 40 years since Wheeler first suggested that the topology of spacetime might be subject to quantum fluctuations [1]. We do not yet know whether the resulting picture of a ‘spacetime foam’ correctly describes the universe, but the potential implications are clearly important: for example, fluctuations of topology are a key element in Coleman’s proposed wormhole/baby universe solution to the cosmological constant problem [2]. If such fluctuations occur only at the Planck scale, a fully fledged quantum theory of gravity may be necessary to understand their effect. If they can occur at larger scales, however, it may be possible to treat the standard Einstein action as an effective field theory [3] from which we can draw useful conclusions. To understand the quantum mechanics of spacetime topology, one needs a formalism in which spacetime is treated as a unified entity. Canonical quantum gravity may allow us to investigate changes in spatial topology, but a path-integral approach seems more natural if we are interested in the topology of spacetime as a whole. In particular, much of the work on this subject (see, for example, [4–14]) has been based on path-integral techniques in Euclidean quantum gravity, that is, general relativity ‘Wick rotated’ to Riemannian (positivedefinite) metrics. If the Einstein action is treated as part of an effective field theory for distances larger than the Planck length, one should not worry too much about higher-loop corrections, which will be suppressed by powers of the Planck mass. It is thus sensible to treat the path integral in a saddle-point approximation. The purpose of this paper is to describe some features of saddle points and to discuss possible implications for spacetime foam. Some of the results presented here are old, but are not widely known among physicists; others are new. A brief report on results for 3 < 0 has appeared in [15]. † E-mail address: [email protected] 0264-9381/98/092629+10$19.50 c © 1998 IOP Publishing Ltd 2629