Positive Mass, Singularities, and Supersymmetry

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AbstractAward: DMS-0504890Principal Investigator: Mark A. SternIn collaboration with Anda Degeratu, the principal investigatorwill continue their study of harmonic spinors on manifolds withdegenerating spin structures. Their primary goal is to use thesedegenerating spin structures to extend Witten's technique forproving the positive mass theorem to new classes of higherdimensional nonspin manifolds. They will also investigate theapplication of index theory to these degenerate spin structuresin order to find new relations for the A hat genus on nonspinmanifolds. The principal investigator will also continue hisstudy of a geometric model of B fields and the higher p-formpotentials arising in string theory, which is suitable forunderstanding how these fields affect the application ofgeometric analysis to string theory. He will then apply thisimproved understanding to various questions in string theory,including joint work with Savdeep Sethi on the analysis ofgravitational anomalies of field theories arising from stringtheories on singular spaces.At low speeds and masses, general relativity should reduce toNewton's theory of gravity. This correspondence is confused bythe fact that there is no definition of mass in generalrelativity which allows one to define the mass density at asingle point in space. One can only define the mass of largeregions. Physicists have introduced several notions of mass toobviate this difficulty, the best known of which is the ADMmass. The positive mass conjecture posits that this complicatedADM mass is never negative. This is important because a negativemass universe seems physically unreasonable. Moreover, the truthof the positive mass conjecture has strong implications for thevery stability of space. In lower dimensions (including 4) thisconjecture was proved by Schoen and Yau. Recent work in stringtheory has led to the study of gravity in more than 4 spacetimedimensions and thus interest in higher dimensional positive masstheorems under investigation in this project. Many relatedquestions arising in the gravity theories produced by stringtheory are not currently amenable to careful mathematicalanalysis because the correct mathematical translation of some ofthe important string theory concepts, such as higher p formpotentials, remains unclear. One of the goals of this project isto provide and apply such a translation.