Gromov-Witten Theory

Harvard University, Cambridge MA

Investigators

Abstract

DMS-03401275 Tom Coates The principal investigator will study a number of topics in Gromov-Witten theory. These topics are linked by a common theme: a new perspective on Gromov--Witten theory, introduced by Givental, which reveals surprising connections with symplectic linear algebra, geometric quantization and the theory of loop groups. One objective of this research is to understand the geometric origin of the structures revealed by this point of view; the second and main objective is to apply the new computational tools that this perspective provides: 1. To the calculation of Gromov--Witten invariants of symplectic quotients; 2. To the computation of higher-genus Gromov--Witten invariants; 3. To the structure theory of non-semisimple Frobenius manifolds; and 4. To various structural and computational questions in quantum K-theory and quantum extraordinary cohomology. Gromov--Witten invariants have been studied intensively for the past decade in an effort to understand some of the mathematical consequences ofstring theory. As a string moves through space, it sweeps out a surface: the Gromov--Witten invariants of a space describe the number of these surfaces in the space which satisfy certain conditions --- for example, one could insist that the surfaces pass through a number of different points, or that they intersect with a particular curve. Gromov-Witten invariants give important information about the shape of a space. They play a vital role in Mirror Symmetry, a circle of ideas which expresses in mathematical form the equivalence of two different string theories. Mirror symmetry suggests that we can count surfaces in a space X (i.e. compute Gromov--Witten invariants of X) by studying differential equations on a space X' called the mirror of X. This is interesting from the point of view of both mathematics and physics: mathematicians had studied surface-counting and solving differential equations for more than a century without realising that the subjects had anything to do with one other, and some of the most convincing evidence for string theory to date comes not from experiment but from the gradual verification of its mathematical predictions.