D-Branes on Landau-Ginsberg orbifolds and Calabi-Yau spaces.

Abstract

In this thesis the author had tried to identify the objects that correspond, at large volume to the fractional 2-branes in C^n / ZN orbifolds. The fractional two-branes have a fractional first Chern class and also have integer charges if we consider them as objects living in the ambient non-compact CY (the resolution of the orbifold), which is a line bundle over a projective base, rather than on the projective space itself. It is also noted, a strong parallel between the relation of the fractional two-branes (and more generally fractional 2p-branes) in the orbifold theory to the corresponding coherent sheaves at large volume and the notion of the quantum McKay correspondence due to Martinec and Moore. The fractional first Chern Class ensures that in the case of fractional 2-branes of the C^5 / Z5 orbifold, when the large volume analogue of these objects are restricted to the CY hypersurface, one of them precisely becomes a zero-brane on the CY hypersurface. It is to be noted that if we start with objects that have integer first Chern class on the ambient projective space, then we would always obtain d zero-branes on the CY hypersurface, where d is the degree of the polynomial equation describing the CY. The CFT description of states of the LG model are identified, that correspond to fractional two-brane and four-brane states in the ambient non-compact orbifold that are restricted to the CY hypersurface. It turns out to be a sub class of the B-type permutation branes of the Gepner Models.