nLab Wheeler superspace


This entry is about the concept in gravity/cosmology. For the concept in supergeometry see at superspace.



In describing gravity (general relativity) on globally hyperbolic spacetimes as evolution along a given foliation of spacetime by spacelike slices, the Einstein equations describe a 1-parameter evolution in a configuration space/moduli space of fields on each spatial slice.

For applications to cosmology this space may be drastically reduced to retain only some very large scale feature. The resulting configuration space had been called “superspaces” by John Wheeler.

On this Wheeler superspace, the Einstein equations describe the classical mechanics of a configuration point moving around in superspace. Many or most approaches to quantum cosmology proceed by taking this reduced mechanical system on Wheeler superspace at face value and applying quantization to it, hoping that the resulting quantum mechanics on Wheeler superspace is a sensible approximation to full (and unknown) quantum gravity, at least for purposes of studying cosmic structure formation, cosmic inflation and similar issues.


In 2d gravity on string worldsheets

The worldsheet-theory of strings in string theory is D=2 gravity coupled to “matter”-fields (the string’s sigma-model “embedding fields”). For plain closed bosonic strings the corresponding Wheeler superspace is just the smooth loop space of the string’s target spacetime: each loop in spacetime is the configuration of a closed string at a fixed value of its chosen worldsheet-“time” parameter.

For the type II superstring the Wheeler superspace is essentially the smooth loop space of the exterior bundle of target spacetime, regarded as a supermanifold.

(Beware the clash of “super”-terminology here: the Wheeler superspace of the superstring is now a superspace-Wheeler superspace, a “super-superspace”, where the first “super” is in the sense of supersymmetry/graded geometry, while the second “super” in the sense of “configuration space”.

The Wheeler-DeWitt quantum mechanics of the type II superstring on its Wheeler superspace, in this sense, is (Witten 85, from p. 92 (32 of 39) on) a supersymmetric quantum mechanics of the form which for finite-dimensional leads to the relation between supersymmetry and Morse theory (Witten 82): the Morse-deformed super-charge is the Dirac-Ramond operator on smooth loop space (more on this in Schreiber 04).


Original articles:

The Wheeler-DeWitt equation is (5.5) in DeWitt 1967


See also:

“Quantum Theory of Gravity. I. The Canonical Theory”

Last revised on March 2, 2024 at 09:13:54. See the history of this page for a list of all contributions to it.