Remarks on theN=1SU(M+p)×SU(p)quiver gauge theory with flavor

Abstract

We study supersymmetric vacua of the $\mathcal{N}=1$ cascading $SU(M+p)\ifmmode\times\else\texttimes\fi{}SU(p)$ gauge theory with flavor---the theory on $p$ D3-branes and $M$ wrapped D5-branes at the tip of the conifold, and ${N}_{f}$ flavor D7-branes wrapping a holomorphic four-cycle inside the conifold. The Coulomb branch of the moduli space is inherited from the pure gauge theory without flavor and was thoroughly studied in the past. Besides, there is a Higgs branch where some D3- and/or D5-branes dissolve in the D7-branes forming the worldvolume gauge instantons. We study the Higgs branch both from the field theory and the bulk point of view. On the classical level the moduli space is closely related to the one of the $\mathcal{N}=2$ ${\mathbb{C}}^{2}/{\mathbb{Z}}_{2}$ orbifold theory, in particular, certain vacua of the $\mathcal{N}=1$ theory are related to noncommutative instantons on the resolved ${\mathbb{C}}^{2}/{\mathbb{Z}}_{2}$. On the quantum level the Higgs branch acquires corrections due to renormalization of the K\"ahler potential and nonperturbative effects in field theory. In the bulk this is encoded in the classical D7-brane geometry. We compute the vacuum expectation value (VEV)s of the protected operators and the field theory RG flow and find an agreement with the parallel computations in the bulk.