Multi-instantons and supersymmetric SU(N) gauge theories

Abstract

In this thesis the proposed exact results for low energy effective N = 2 supersymmetric SU(N) Yang-Mills gauge theory coupled to Nf fundamental matter multiplets in four dimensions are considered. The proposed exact results are based upon the work of Seiberg and Witten for low energy effective four dimensional M = 2 supersymmetric SU[2) Yang-Mills gauge theory coupled to Nf fundamental matter multiplets. The testing and matching of the proposed exact results via supersymmetric instanton calculus are the motivation for two studies. Firstly, we study the ADHM construction of instantons for gauge groups U(N) and SU(2) and for topological charge two and three. The ADHM constraints which implicitly specify instanton gauge field configurations are solved for the explicit exact general form of instantons with topological charge two and gauge group U[N). This is the first explicit and general multi-instanton configuration for the unitary gauge groups. The U[N) ADHM two-instanton configuration may be used in further tests and matching of the proposed exact results in low energy effective M =2 supersymmetric SU(iV) Yang-Mills gauge theories by comparison with direct instanton calculations. Secondly, a one-instanton level test is performed for the reparameterization scheme proposed by Argyres and Pelland matching the conjectured exact low energy results and instanton predictions for the instanton contributions to the prepotential of low energy effective M = 2 supersymmetric SU [N) Yang-Mills gauge theory with Nf = 2N mass-less fundamental matter multiplets. The constants within the reparameterization scheme which ensure agreement between the exact results and the instanton predictions for general N > 1 are derived for the entire quantum moduli space. This constitutes a non-trivial test of the proposed reparameterization scheme, which eliminates the discrepancies arising when the two sets of results are compared.