On the algebraic geometry of $S$-duality

This paper deals with algebro-geometric questions arising in the verification of the $S$-duality conjecture for supersymmetric Yang–Mills quantum field theories in the four-dimensional case. We describe all the cases for the gauge groups of rank 1 and 2, where the Gell-Man–Law beta-function is either zero or negative, and point out some series of such cases for gauge groups of arbitrary rank. Realization of one of these series on the complex projective plane demonstrates a relationship with exceptional bundles.