In these talks I will describe mathematical models
of D-branes as sheaves and, more generally, derived categories.
I will begin with a gentle introduction to some relevant
mathematics (sheaves, Ext groups). After that, I will discuss
mathematical models of D-branes on large-radius Calabi-Yau manifolds
as sheaves, describing how sheaves
can be used to calculate open string spectra, and how such
sheaf-theoretic models can be extended in various directions
to take into account B field backgrounds, orbifold structures,
and nontrivial Higgs vevs.
Finally, I will give a short introduction to derived categories,
their physical realization as boundary states in the
B model topological field theory, and stability issues.
References:
Lecture Notes:
Basic references on sheaves & sheaf cohomology:
- R C Gunning, "Lectures on Riemann Surfaces",
Princeton University Press, 1966,
sections 2 and 3.
- P Griffiths and J Harris,
"Principles of Algebraic Geometry",
John Wiley, 1978,
section 0.3.
Basic references on Ext groups of sheaves:
- Griffiths and Harris, same text as above,
sections 5.3, 5.4
Basic reference on the B model topological field theory:
- Witten's "Mirror manifolds and topological field theory",
in "Mirror Symmetry I", ed. by S.-T. Yau.