Spacetime Geometry from Modular Theory

We prove that the spacetime metric of a globally hyperbolic Lorentzian manifold can beuniquely reconstructed from the modular data of a local quantum field theory. Starting fromTomita–Takesaki modular theory applied to wedge and double-cone algebras, we show that(i) modular flows of wedges generate the Poincaré group, (ii) modular flows of all regionsgenerate the full diffeomorphism group, and (iii) relative modular invariants together withspectral functionals determine the metric tensor. Under mild regularity assumptions on thealgebraic quantum field theory, we establish that the triple consisting of a type III1 factorwith prescribed symmetry (e.g., E8 ), its family of modular operators, and the net struc-ture uniquely determines the four-dimensional Lorentzian geometry up to diffeomorphism.We provide a concrete microscopic realization via M-theory compactification on an E8 -structured noncommutative eight-torus Tθ8 , where the deformation parameter is determinedby the E8 root lattice. The von Neumann completion of this noncommutative torus algebrayields the hyperfinite type III1 factor from which both the emergent spacetime geometry andE8 gauge bundle structure are reconstructed through modular theory. This establishes acomplete dictionary between microscopic noncommutative data and macroscopic geometricstructures, providing a rigorous realization of the principle that spacetime geometry emergesfrom quantum entanglement.