Flavour from partially resolved singularities

Abstract

In this letter we study topological open string field theory on D--branes in a IIB background given by non compact CY geometries $\mathcal{O}(n)\oplus \mathcal{O}(-2-n)$ on $\P 1$ with a singular point at which an extra fiber sits. We wrap $N$ D5-branes on $\P 1$ and $M$ effective D3-branes at singular points, which are actually D5--branes wrapped on a shrinking cycle. We calculate the holomorphic Chern-Simons partition function for the above models in a deformed complex structure and find that it reduces to multi--matrix models with flavour. These are the matrix models whose resolvents have been shown to satisfy the generalized Konishi anomaly equations with flavour. In the $n=0$ case, corresponding to a partial resolution of the $A_2$ singularity, the quantum superpotential in the $\mathcal{N}=1$ unitary SYM with one adjoint and $M$ fundamentals is obtained. The $n=1$ case is also studied and shown to give rise to two--matrix models which for a particular set of couplings can be exactly solved. We explicitly show how to solve such a class of models by a quantum equation of motion technique.

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