Proposal: DMS 9870161 Principal Investigator: Mark Stern
In collaboration with S. Sethi, the PI will analyze the supersymmetric matrix model quantum mechanics which has been conjectured to provide a definition of M theory. Crucial to the conjecture is the existence of normalizable ground states in quantum mechanical gauge theories with 16 supersymmetries. These ground states correspond to gravitons in the model. We seek to show existence and uniqueness of a groundstate wavefunction, by computing an L2 index for a large family of associated non Fredholm, (generalized) Dirac operators. We will analyze the structure of the wavefunction in order to determine, for example, the "size" of the graviton. In further work related to the matrix model, in collaboration with S. Paban and Savdeep Sethi, the PI will study constraints on the effective action in Yang Mills theories with 16 supersymmetries. Mathematically, this may be viewed as a moduli problem for deformations of systems with 16 supersymmetries. In collaboration with W. Pardon, the PI proposes to establish a harmonic theory for the L2 cohomology of singular projective varieties with the metric induced by a projective embedding, extending their harmonic theory for varieties with isolated singularities. The goal of such an investigation is to bring to the study of the L2 cohomology (and conjecturally the intersection cohomology) of these varieties the same array of tools available for studying complete Kahler manifolds. This includes Hodge and Lefschetz decompositions and potentially Bochner type vanishing theorems.
The primary goal of this project is to explore a recently proposed model for a theory of quantum gravity known as the matrix model. Mathematically this model has many advantages, but it is not yet complete and is not yet clear that it gives a physically reasonable theory. We will perform various (mathematical) tests to determine if the model does have the conjectured desired physical properties. For example, we seek to determine whether it predicts the existence of a stable particle, "the graviton". Then we will try to determine the structure of this particle and see if the model gives a framework for computing predicted outcomes to simple processes such as scattering of gravitons.
