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 (including the conformal factor, fixed byspectral functional/area law scaling). Under regularity assumptions on the algebraic quan-tum field theory, most notably that the theory approaches a conformal fixed point in theUV, we establish that the triple consisting of a type III1 factor with prescribed symmetry(e.g., E8 ), its family of modular operators, and the net structure uniquely determines thefour-dimensional Lorentzian geometry up to diffeomorphism. We provide a concrete mi-croscopic realization via M-theory compactification on an E8 -structured noncommutativeeight-torus Tθ8 , where the deformation parameter is determined by the E8 root lattice. Thevon Neumann completion of this noncommutative torus algebra yields the hyperfinite typeIII1 factor from which both the emergent spacetime geometry and E8 gauge bundle structureare reconstructed through modular theory. Under conjectured structural principles specificto the M-theory construction (E8 rigidity implying nuclearity, anomaly cancellation ensur-ing local stress tensor), this establishes a dictionary between microscopic noncommutativedata and macroscopic geometric structures, realizing the principle that spacetime geometryemerges from quantum entanglement.