$S$-duality testing and exceptional bundles

In this paper we discuss algebraic-geometric problems connected with the mathematical testing of the $S$-duality conjecture. In particular, we give a complete description of the field configurations in classical gauge theories for which the coefficient of the Gell–Mann–Low beta-function in the one-loop approximation equals zero. Realizing one of these configurations geometrically on Del Pezzo surfaces, we demonstrate its relation to exceptional bundles: every exceptional bundle whose cohomology is zero and whose slope is negative but exceeds the slope of the canonical class gives a correlation function for $S$-duality testing.