The universal von Neumann algebra of smooth 4-manifolds with an application to gravity

Abstract: 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 ${\rm II}_1$ 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: In an appropriate sense this invariant along all simply connected closed $4$-manifolds is generated by its surely non-trivial value on ${\mathbb C}P^2$ (with its standard smooth structure) alone.

In the second half of the seminar (i.e. if time remains) 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 ${\rm II}_1$ 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.

Reference

1. G. Etesi: The universal von Neumann algebra of smooth four-manifolds, to appear in Adv. Theor. Math. Phys., arXiv: 1712.01828 [math-ph];

2. G. Etesi: Gravity as a four dimensional algebraic quantum field theory, Adv. Theor. Math. Phys. 20, 1049-1082 (2016), arXiv: 1402.5658 [hep-th].

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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