D-Brane Monodromies, Derived Categories and Boundary Linear Sigma Models

An important subclass of D-branes on a Calabi-Yau manifold, X, are in 1-1 correspondence with objects in D(X), the derived category of coherent sheaves on X. We study the action of the monodromies in Kaehler moduli space on these D-branes. We refine and extend a conjecture of Kontsevich about the form of one of the generators of these monodromies (the monodromy about the "conifold" locus) and show that one can do quite explicit calculations of the monodromy action in many examples. As one application, we verify a prediction of Mayr about the action of the monodromy about the Landau-Ginsburg locus of the quintic. Prompted by the result of this calculation, we propose a modification of the derived category which implements the physical requirement that the shift-by-6 functor should be the identity. Boundary Linear sigma-Models prove to be a very nice physical model of many of these derived category ideas, and we explain the correspondence between these two approaches