Geometry of gerbes & D-branes

In the mathematical theory of electromagnetic fields, we know that the phases of moving a point particle around a closed path can in part be described in terms of a bundle with connection, a system of locally defined 1-forms glued together via the valued transition functions associated to the underlying bundle.

In string theory, a particle is described as a tiny /openclosed string the phases of moving a string around can be described in terms of a gerbe with connection and a curving. This curving is a system of locally defined 2-forms called the Kalb-Ramond field. The phase is sometimes called the holonomy along a closed surface (a closed trajectory of a string particle).

Strings can have various kinds of boundary conditions. Open strings can have different kinds of boundary conditions called Neumann and Dirichlet boundary conditions such that the endpoint of a string is fixed to move only on some submanifold, called the support of the D-brane. D-branes are actually dynamical objects which have fluctuations and can move around. All these require some deep understanding of geometry of gerbes. D-branes have found many interesting applications in theoretical physics and mathematics.

This project would lay down some of mathematical ground for string theory.