Geometry and Mathematical Physics of D-Branes

Duke University, Durham NC

Investigators

Abstract

The derived category of coherent sheaves has become a powerful tool in recent years in algebraic geometry. Remarkably this category also appears naturally in the mathematical physics of superstring theory where objects are D-branes, i.e., boundary conditions on open strings, and morphisms are Hilbert spaces associated to open string states. The proposer will investigate several aspects of this connection. The mathematics of tilting collections, quiver representations and matrix factorizations will be applied to notions such as monodromy, stability and moduli spaces to further understand string theory. It is expected that this research will lead to insight into the mathematics of the derived category of Calabi-Yau threefolds which play a central role in this proposal. String theory aims at providing an understanding of the fundamental physics of our universe, particularly the geometry of spacetime at short distances. The mathematics of string theory requires the use of sophisticated techniques from modern geometry. Indeed, ideas from string theory have inspired many recent important developements in the pure mathematics of geometry. This proposal will explore the connection between D-branes, which in a sense are subspaces of spacetime, and algebraic geometry, in which geometry is expressed purely in terms of algebraic equations. It is expected that new results both in pure mathematics and mathematical physics will be obtained.