The Master Space of $N$=1 Gauge Theories

The full moduli space M of a class of N=1 supersymmetric gauge theories is studied. For gauge theories living on a stack of D3-branes at Calabi-Yau singularities X, M is a combination of the mesonic and baryonic branches, the former being the symmetric product of X. In consonance with the mathematical literature, the single brane moduli space is called the master space F. Illustrating with a host of explicit examples, we exhibit many algebro-geometric properties of the master space such as when F is toric Calabi-Yau, behaviour of its Hilbert series, its irreducible components and its symmetries. In conjunction with the plethystic programme, we investigate the counting of BPS gauge invariants, baryonic and mesonic, using the geometry of F and show how its refined Hilbert series not only engenders the generating functions for the counting but also beautifully encode ``hidden'' global symmetries of the gauge theory which manifest themselves as symmetries of the complete moduli space M for arbitrary number of branes.