Geometry in String Theory , Geometry in General Relativity

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Geometry plays an important role in any theory of quantum gravity, such as stringtheory. Recent advances in string theory have demonstrated the importance of newtypes of geometrical spaces that previously have not been much studied. For stringtheory constructions of phenomenologically realistic models of elementary particles viacompactifying extra dimensions,the presence of fluxes for fixing moduli, and hencedetermining coupling constants and masses, has now been deemed essential. Withfluxes, the generic internal geometry is no longer Calabi-Yau, but involves non-Kahlermanifolds with torsion. Yet much of the structure of non-Kahler geometries is not known.In a rather different direction, there has been considerable progress recently inunderstanding the gravity/gauge theory (AdS/CFT) correspondence in quite a rich classof geometrical backgrounds. In particular, our understanding of Sasaki-Einsteingeometry has improved greatly, and there are now a number of infinite families ofSasaki-Einstein AdS/CFT duals where both sides of the duality are known explicitly. Ingeneral, the AdS/CFT correspondence predicts a fascinating relationship betweengeometry and properties of conformal field theory. This project has two maincomponents. The PI will investigate the geometry of flux compactifications in heteroticstring theory. The objective is to understand the underlying mathematical structure ofnon-Kahler geometries of interest so that important physical questions like the numberof massless particles and their interactions can be answered. Besides geometricalanalysis, the application of string dualities and string worldsheet methods will be utilizedin the investigations. In addition, the PI will further develop Sasaki-Einstein geometry,with application to the AdS/CFT correspondence and conformal field theory. Goalsinclude formulating, as a precise mathematical statement, the map from geometry toconformal field theory, and addressing the important issue of existence and uniquenessof Sasaki-Einstein metrics. Another goal is to describe the geometrical structuresunderlying more general AdS/CFT backgrounds with fluxes, and also to study thegeometric description of renormlization group flows. The proposed project is at thefrontier of string theory research and is expected to make a substantial contribution tothe understanding of physics beyond the Standard Model, conformal field theory, andstring theory dualities. This will be achieved by addressing some of the foundationalmathematical problems that have arisen in the recent development of string theory. It isexpected that the results of the proposed program will also have a profound impact indifferent areas of mathematics, such as differential geometry and algebraic geometry.Broader Impact: A central aspect of this project is the training of graduate students andpostdoctoral fellows in applying the modern tools of mathematics to theoretical physicsand developing new mathematics to solve physical problems. The PI has a strongrecord in organizing workshops and conferences, including the annual Strings 2006conference in Beijing, for the string and mathematical physics community. He will alsocontinue to reach out to the media to publicize and educate the general public on theresearch of the math and physics communities.