Matching and Syzygies in N = 1 Gauge Theories

We investigate the connection between the moduli space of N = 1 su-persymmetric gauge theories and the set of polynomial gauge invariants constrained by classical/quantum relations called syzygies. We examine the existence of a superpotential reproducing these syzygies and the link with the 't Hooft anomaly matching between the fundamental fields at high energy and the gauge invariant degrees of freedom at low energy for the flavour symmetry group. We strengthen the connection by proving that the t'Hooft anomaly matching is equivalent to the existence of a polynomial superpotential describing the low energy dynamics. This corroborates a recent definition of confining theories. We illustrate our general result by considering two examples based on the SU (Nc) and Spin(7) gauge theories. We also examine the role of syzygies in the context of non-Abelian duality. We emphasize the relevance of non-perturbative effects in the dual magnetic theories in proving the equivalence of the electric and magnetic syzygies.