We consider topological sigma models with generalized Kähler target spaces. The mirror map is constructed explicitly for a special class of target spaces and the topological A and B model are shown to be mirror pairs in the sense that the observables, the instantons and the anomalies are mapped to each other. We also apply the construction to open topological models and show that A branes are mapped to B branes. Furthermore, we demonstrate a relation between the field strength on the brane and a two-vector on the mirror manifold.

We use free boson techniques to investigate A-D-E-quiver matrix models. Certain higher spin fields in the free boson formulation give rise to higher order loop equations valid at finite N . These fields form a special kind of W-algebra, called Casimir algebra. We compute explicitly the loop equations for A r and D r quiver models and check that at large N they are related to a deformation of the corresponding singular Calabi-Yau geometry.

Exact Renormalization Group techniques are applied to supersymmetric models in order to get some insights into the low energy effective actions of such theories. Starting from the ultra-violet finite mass deformed N = 4 supersymmetric Yang-Mills theory, one varies the regularising mass and compensates for it by introducing an effective Wilsonian action. (Polchinski's) renormalization group equation is modified in an essential way by the presence of rescaling (a.k.a. Konishi) anomaly, which is responsible for the beta-function. When supersymmetry is broken up to N = 1 the form of effective actions in terms of massless fields is quite reasonable, while in the case of the N = 2 model we appear to have problems related to instantons.f( M ) Ji Pi + h -c-+ I fd 2 6^Tg{M) Jf + h.c. + fd 4 e J v v]. J ,=i 2J i=i J -I

The existence of a new kind of branes for the open topological A-model is argued by using the generalized complex geometry of Hitchin and the SYZ picture of mirror symmetry. Mirror symmetry suggests to consider a bi-vector in the normal direction of the brane and a new definition of generalized complex submanifold. Using this definition, it is shown that there exists generalized complex submanifolds which are isotropic in a symplectic manifold. For certain target space manifolds this leads to isotropic Abranes, which should be considered in addition to Lagrangian and coisotropic A-branes.The Fukaya category should be enlarged with such branes, which might have interesting consequences for the homological mirror symmetry of Kontsevich. The stability condition for isotropic A-branes is studied using the worldsheet approach.