AN OPERATOR ALGEBRAIC CHARACTERIZATION OF THE VACUUM EINSTEIN EQUATION IN FOUR DIMENSIONS

Speaker(s)

GÁBOR ETESI

Affiliation

Budapesti Műszaki és Gazdaságtudományi Egyetem, Hungary

Language of the talk

English

Date

March 19, 2025, 5:15 p.m.

Information about the event

IMPAN 405 & ZOOM

Title in Polish

AN OPERATOR ALGEBRAIC CHARACTERIZATION OF THE VACUUM EINSTEIN EQUATION IN FOUR DIMENSIONS

Seminar

North Atlantic Noncommutative Geometry Seminar

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In this talk, a new approach to the vacuum Einstein equation on four dimensional Riemannian manifolds (i.e. the Einstein condition on a four dimensional Riemannian metric) is offered in the framework of dynamical systems on hyperfinite II_1 factor von Neumann algebras. More precisely, we observe that out of a connected oriented smooth 4-manifold M, a von Neumann algebra can be constructed in a functorial way such that this algebra is a hyperfinite factor of type II_1. Hence, it is unique up to abstract isomorphisms of von Neumann algebras. On the other hand, every connected oriented smooth 4-manifold admits an embedding via projections into an abstractly given von Neumann algebra of this type. Moreover, this embedding induces a Riemannian metric g on M whose Riemannian curvature tensor, if appropriately bounded, belongs to the von Neumann algebra, and the metric induced Hodge operator acting on 2-forms gives rise to a unitary element of the algebra. Also, the Hodge operator generates a periodic inner *-automorphism of the von Neumann algebra rendering it a dynamical system. One notes that that (M,g) is Einstein if and only if the Riemannian curvature tensor belongs to the fixed-point set of the corresponding "Hodge dynamics". Since, by a classical 1974 result of Størmer, the fixed-point subalgebra of a periodic inner *-automorphism of a von Neumann algebra is a so-called normal subalgebra, some conjectural correspondence emerges between 4-dimensional Riemannian Einstein structures and normal subalgebras of hyperfinite II_1 factors.

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