The universal von Neumann algebra of smooth four-manifolds

Making use of its smooth structure only, out of a connected oriented smooth $4$-manifold a von Neumann algebra is constructed. As a special four dimensional phenomenon this von Neumann algebra is approximated by algebraic (i.e., formal) curvature tensors of the underlying $4$-manifold and the von Neumann algebra itself is a hyperfinite factor of $\mathrm{II}_1$ type hence is unique up to abstract isomorphisms of von Neumann algebras. Nevertheless over a fixed $4$-manifold this von Neumann algebra admits a representation on a Hilbert space such that its unitary equivalence class is preserved by orientation-preserving diffeomorphisms. Consequently the Murray–von Neumann coupling constant of this representation is well-defined and gives rise to a new and computable real-valued smooth $4$-manifold invariant.

Some consequences of this construction for quantum gravity are also discussed. Namely reversing the construction by starting not with a particular smooth $4$-manifold but with the unique hyperfinite $\mathrm{II}_1$ factor, a conceptually simple but manifestly four dimensional, covariant, non-perturbative and genuinely quantum theory is introduced whose classical limit is general relativity in an appropriate sense. Therefore it is reasonable to consider it as a sort of quantum theory of gravity. In this model, among other interesting things, the observed positive but small value of the cosmological constant acquires a natural explanation.

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Published 12 April 2022