Manifolds with Special Holonomy and Applications

University Of Rochester, Rochester NY

Investigators

Abstract

The unification of the four fundamental forces of nature--electromagnetism, gravity, the strong and weak nuclear forces--is one of the greatest unsolved mysteries of physics. Over the last few decades, M-theory, a "theory of everything",  has emerged as a candidate for such a unification of these forces.  This project is about manifolds with special holonomy, spaces whose infinitesimal symmetries allow them to play a crucial role in M-theory 'compactifications'---that is, they model the tiny 'curled up' dimensions lurking at every point of spacetime.   In this project, the principal investigator will focus in particular on 6-dimensional Calabi-Yau manifolds (which play the analogous role of the curled-up dimensions of superstring theory) and spaces of dimension 7 and 8 whose symmetries fill out the special holonomy groups known as G2 and Spin(7), respectively.  Despite extensive research on Calabi-Yau manifolds, the geometric properties of G2 and Spin(7) manifolds are not well understood, and the problem of the existence of calibrated (i.e., volume minimizing) submanifolds is still wide open.  One goal of this project is to develop techniques that are robust enough to handle these difficult existence questions. Another goal is to study the deformation spaces of calibrated submanifolds, as understanding these spaces will ultimately be useful for M-theory compactifications.  The PI also believes that manifolds with special holonomy is an excellent topic for graduate research, and intends to continue to supervise PhD students.  She plans to encourage women and members of other under-represented groups to take up graduate study and continue to research careers in differential geometry, through activities that include advising, organizing seminars, special sessions, conference and "Women in Math" workshops. In this project, the PI plans to continue her work on Ricci-flat manifolds, their calibrated geometries and the compactifications of moduli spaces. In recent joint work with F. Arikan and H. Cho, she showed that every G2 manifold is an almost contact manifold. Studying the relations between contact and G2 structures can be useful to find the existence conditions of a G2 metric  on 7-manifolds (similar to the existence conditions of the Calabi-Yau metric). In another joint work with Cho and A.J. Todd, she investigated the properties of G2 manifolds from a symplectic point of view. Using contact and symplectic structures, the PI plans to construct Lagrangian and Legendrian type submanifolds  of G2 and Spin(7) manifolds. Also, in joint work with C. Robles, she applied the Cartan-Kahler theory to associative and Cayley embeddings into G2 and Spin(7) manifolds, and she plans to use these techniques to construct new examples of G2 and Spin(7) manifolds and study their contact and symplectic structures. In other joint work with D. Joyce, the PI studied deformations of asymptotically cylindrical coassociative submanifolds and their topological quantum field theories, and with Todd she also proved similar results for asymptotically cylindrical special Lagrangian submanifolds. The PI plans to apply these techniques on special Lagrangian moduli spaces inside Calabi-Yau manifolds to obtain a framework for the Floer homology program. Understanding the moduli spaces of these submanifolds will provide a better understanding of the mirror symmetry phenomenon.