Lecture Notes on Three Supersymmetric/Topological Systems in Quantum Field Theory

Abstract: ((1+1)-dimensional ${\cal N}=1$ super-symmetric field theory and (3+1)-dimensional ${\cal N}=2$ super-symmetric gauge theory are discussed in a, more or less, unified way, designed to identify the quantum BPS states in both systems. Euclidean 4-dimensional gauge theory with ${\cal N}=2$ twisted super-symmetry is also analized. ${\bf C}^\infty$-topological invariants are identified as certain n-point correlation functions in this QFT framework. The twist of the effective dual Abelian gauge theory is briefly described, both from mathematical and physical viewpoints. The physical nature of the topological defects arising in these systems, kinks, BPS and Dirac monopoles, BPST instantons, Liouville and Abrikosov-Nielsen-Olesen selfdual vortices, etcetera, is analyzed, The thread of the story connecting the QFT systems treated respectively in Sections \S.3 and \S.4 is the process of TWIST that leads from a conventional extended Supersymetric Gauge Theory to the topological ${\cal N}=2$ SUSY Donaldson QFT. Within Section \S.3 the $SL(2,\mathbb{Z}$-dualities establish a link between the weak coupling regime of the original gauge theory and the Wilsonian (abelian) effective gauge theory arising at low energies. We shall also look after the reminiscences of these dualities between the twisted TQFT systems of Section \S.4.