Algebro-geometric aspects of \(S\)-duality. (Q1274036)

The author describes three series of algebro-geometric models for super Yang-Mills quantum field theories in dimension four. Then the \(S\)-duality conjecture is related to the modularity of generating functions of characteristic numbers of moduli spaces of stable bundles on algebraic surfaces. In this context, the finiteness condition for the number of of solutions of field equation is discussed; in one case, it turns out that the invariants of the moduli space should satisfy the Markov equation \(x^2+y^2+z^2=3xyz\).