For a real four manifold X, the S-duality conjecture of Vaffa-Witten in 1994 predicts that the S-transformation sends the gauge group SU(r) invariants counting instantons on X to the Langlands dual group SU(r)/Z_r invariants on X. Both of the invariants satisfy modularity properties. This is the mathematical meaning of the electro-magnetic duality in physics. In this talk I will present a mathematical method to attack the S-duality conjecture. The content of the proof may be related to the period-index conjecture for Brauer classes on smooth projective varieties. The talk is based on joint work with M. Kool, and with D. van Bree, A. Gholampour, and M. Kool.