NonFredholm Index Theory, Matrix Models, and Hodge Theory

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AbstractProposal: DMS 9870161Principal Investigator: Mark Stern In collaboration with S. Sethi, the PI will analyze the supersymmetricmatrix model quantum mechanics which has been conjectured to provide adefinition of M theory. Crucial to the conjecture is the existence ofnormalizable ground states in quantum mechanical gauge theories with16 supersymmetries. These ground states correspond to gravitons inthe model. We seek to show existence and uniqueness of a groundstatewavefunction, by computing an L2 index for a large family ofassociated non Fredholm, (generalized) Dirac operators. We willanalyze the structure of the wavefunction in order to determine, forexample, the "size" of the graviton. In further work related to thematrix model, in collaboration with S. Paban and Savdeep Sethi, the PIwill study constraints on the effective action in Yang Mills theorieswith 16 supersymmetries. Mathematically, this may be viewed as amoduli problem for deformations of systems with 16 supersymmetries.In collaboration with W. Pardon, the PI proposes to establish aharmonic theory for the L2 cohomology of singular projective varietieswith the metric induced by a projective embedding, extending theirharmonic theory for varieties with isolated singularities. The goalof such an investigation is to bring to the study of the L2 cohomology(and conjecturally the intersection cohomology) of these varieties thesame array of tools available for studying complete Kahler manifolds.This includes Hodge and Lefschetz decompositions and potentiallyBochner type vanishing theorems.The primary goal of this project is to explore a recently proposedmodel for a theory of quantum gravity known as the matrix model.Mathematically this model has many advantages, but it is not yetcomplete and is not yet clear that it gives a physically reasonabletheory. We will perform various (mathematical) tests to determine ifthe model does have the conjectured desired physical properties. Forexample, we seek to determine whether it predicts the existence of astable particle, "the graviton". Then we will try to determine thestructure of this particle and see if the model gives a framework forcomputing predicted outcomes to simple processes such as scattering ofgravitons.