Some problems in algebraic geometry with connections to string theory

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

A series of problems are proposed in algebraic geometry, inspired by string theory. Some problems in string theory relating to algebraic geometry are also proposed. These include the study of algebro-geometric invariants related to topological string amplitudes: Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants, all conjecturally related to each other. Particular problems include a rigorous definition of Gopakumar-Vafa invariants and their generalization from Calabi-Yau threefolds to more general threefolds, their computation in examples, and verification of the GW-GV correspondence, beginning with toric varieties and contractible curves. These invariants will be extended to threefolds which are not Calabi-Yau. Computational methods for Gromov-Witten and Donaldson-Thomas invariants are proposed. Several connections to physics will be pursued, including black hole entropy techniques. The project is a continuation of the exciting process of cross-fertilization between string theory in physics and geometry in mathematics. In string theory, the fundamental particles of nature are small vibrating strings, and physical properties depend on the geometry of the space that strings propagate in, much as the shape of a drum affects the sound that it makes. Physically relevant properties can be determined by calculations in pure geometry, in this instance calculations of intrinsic interest in mathematics. When physical methods of duality provide more than one mathematical model, a consequence is a deep unexpected insight into geometry. In this project, the PI will develop tools to investigate the mathematical validity of some of these predictions and will expand the geometric investigation beyond the physically relevant context. The PI will apply also geometric techniques to advance understanding of string theory, especially as it describes the statistical mechanics of a black hole. The primary mathematical content is the counting of curves in curved spaces called Calabi-Yau threefolds which form the small extra dimensions in string theory.