17 research outputs found

    Uniformization, Unipotent Flows and the Riemann Hypothesis

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    We prove equidistribution of certain multidimensional unipotent flows in the moduli space of genus gg principally polarized abelian varieties (ppav). This is done by studying asymptotics of Γg∌Sp(2g,Z)\pmb{\Gamma}_{g} \sim Sp(2g,\mathbb{Z})-automorphic forms averaged along unipotent flows, toward the codimension-one component of the boundary of the ppav moduli space. We prove a link between the error estimate and the Riemann hypothesis. Further, we prove Γg−r\pmb{\Gamma}_{g - r} modularity of the function obtained by iterating the unipotent average process rr times. This shows uniformization of modular integrals of automorphic functions via unipotent flows

    Noncommutative deformation of four dimensional Einstein gravity

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    We construct a model for noncommutative gravity in four dimensions, which reduces to the Einstein-Hilbert action in the commutative limit. Our proposal is based on a gauge formulation of gravity with constraints. While the action is metric independent, the constraints insure that it is not topological. We find that the choice of the gauge group and of the constraints are crucial to recover a correct deformation of standard gravity. Using the Seiberg-Witten map the whole theory is described in terms of the vierbeins and of the Lorentz transformations of its commutative counterpart. We solve explicitly the constraints and exhibit the first order noncommutative corrections to the Einstein-Hilbert action.Comment: LaTex, 11 pages, comments added, to appear in Classical and Quantum Gravit

    Equidistribution Rates, Closed String Amplitudes, and the Riemann Hypothesis

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    We study asymptotic relations connecting unipotent averages of Sp(2g,Z)Sp(2g,\mathbb{Z}) automorphic forms to their integrals over the moduli space of principally polarized abelian varieties. We obtain reformulations of the Riemann hypothesis as a class of problems concerning the computation of the equidistribution convergence rate in those asymptotic relations. We discuss applications of our results to closed string amplitudes. Remarkably, the Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring in perturbative closed string theory.Comment: 15 page

    Eluding SUSY at every genus on stable closed string vacua

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    In closed string vacua, ergodicity of unipotent flows provide a key for relating vacuum stability to the UV behavior of spectra and interactions. Infrared finiteness at all genera in perturbation theory can be rephrased in terms of cancelations involving only tree-level closed strings scattering amplitudes. This provides quantitative results on the allowed deviations from supersymmetry on perturbative stable vacua. From a mathematical perspective, diagrammatic relations involving closed string amplitudes suggest a relevance of unipotent flows dynamics for the Schottky problem and for the construction of the superstring measure.Comment: v2, 17 pages, 8 figures, typos corrected, new figure added with 3 modular images of long horocycles,(obtained with Mathematica
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