S-duality
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In theoretical physics, S-duality (also a strong-weak duality) is an equivalence of two quantum field theories, string theories, or M-theory. An S-duality transformation maps the states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1 / g in the dual theory. This has permitted the use of perturbation theory, normally useful only for "weakly coupled" theories with g less than 1, to also describe the "strongly coupled" (g greater than 1) regimes of string theory, by mapping them onto dual, weakly coupled regimes.
In the case of four-dimensional quantum field theories, S-duality was understood by Ashoke Sen, Nathan Seiberg, and others. In this context, it usually exchanges the electric and magnetic fields (and the electrically charged particles with magnetic monopoles).
See Montonen-Olive duality, Seiberg duality.
Many more examples come from string theory: S-duality relates type IIB string theory with the coupling constant g to the same type IIB string theory with the coupling constant 1 / g. Similarly, type I string theory with the coupling g is equivalent to the SO(32) heterotic string theory with the coupling constant 1 / g. Perhaps most amazing are the S-dualities of type IIA string theory and E8 heterotic string theory with coupling constant g to the higher dimensional M-theory with a compact dimension of size g.
S-duality has been rigorously shown to hold in some lattice models. It depends on the Pontryagin dual group.
In particular, in 2 dimensions, if the vertices can take on values in a locally compact Abelian group G and the action/energy only depends on the edges (e.g. the Ising model for Z2, the Potts model for Zn, the XY model for U(1) ), then its dual via the Kramers-Wannier duality to a model where the vertices take on values in the dual group G'.
In 3 dimensions, such a model would be dual to a lattice gauge model over the dual group G'.
In 4 dimensions, a lattice gauge model with G as the gauge group would be dual to a lattice gauge model with G' as the gauge group (with the electric and magnetic fields interchanged).
