Geometry plays an important role in any theory of quantum gravity, such as string theory. Recent advances in string theory have demonstrated the importance of new types of geometrical spaces that previously have not been much studied. For string theory constructions of phenomenologically realistic models of elementary particles via compactifying extra dimensions,the presence of fluxes for fixing moduli, and hence determining coupling constants and masses, has now been deemed essential. With fluxes, the generic internal geometry is no longer Calabi-Yau, but involves non-Kahler manifolds 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 in understanding the gravity/gauge theory (AdS/CFT) correspondence in quite a rich class of geometrical backgrounds. In particular, our understanding of Sasaki-Einstein geometry has improved greatly, and there are now a number of infinite families of Sasaki-Einstein AdS/CFT duals where both sides of the duality are known explicitly. In general, the AdS/CFT correspondence predicts a fascinating relationship between geometry and properties of conformal field theory. This project has two main components. The PI will investigate the geometry of flux compactifications in heterotic string theory. The objective is to understand the underlying mathematical structure of non-Kahler geometries of interest so that important physical questions like the number of massless particles and their interactions can be answered. Besides geometrical analysis, the application of string dualities and string worldsheet methods will be utilized in the investigations. In addition, the PI will further develop Sasaki-Einstein geometry, with application to the AdS/CFT correspondence and conformal field theory. Goals include formulating, as a precise mathematical statement, the map from geometry to conformal field theory, and addressing the important issue of existence and uniqueness of Sasaki-Einstein metrics. Another goal is to describe the geometrical structures underlying more general AdS/CFT backgrounds with fluxes, and also to study the geometric description of renormlization group flows. The proposed project is at the frontier of string theory research and is expected to make a substantial contribution to the understanding of physics beyond the Standard Model, conformal field theory, and string theory dualities. This will be achieved by addressing some of the foundational mathematical problems that have arisen in the recent development of string theory. It is expected that the results of the proposed program will also have a profound impact in different areas of mathematics, such as differential geometry and algebraic geometry. Broader Impact: A central aspect of this project is the training of graduate students and postdoctoral fellows in applying the modern tools of mathematics to theoretical physics and developing new mathematics to solve physical problems. The PI has a strong record in organizing workshops and conferences, including the annual Strings 2006 conference in Beijing, for the string and mathematical physics community. He will also continue to reach out to the media to publicize and educate the general public on the research of the math and physics communities.
