S-duality N = 4 supersymmetric Yang–Mills theory

the coupling constants




θ

i




{\displaystyle \theta _{i}}

,



g


{\displaystyle g}

naturally pair in form:

τ
=


θ

2
π



+



4
π
i


g

2




.


{\displaystyle \tau ={\frac {\theta }{2\pi }}+{\frac {4\pi i}{g^{2}}}.}

the theory has symmetries shift



τ


{\displaystyle \tau }

integers. s-duality conjecture says there symmetry sends :



τ
↦



−
1



n

g


τ





{\displaystyle \tau \mapsto {\frac {-1}{n_{g}\tau }}}

switching group



g


{\displaystyle g}

langlands dual group.