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 constantin one theory to states and vacua with coupling constant in the dual theory. This has permitted the use of perturbation theory, normally useful only for "weakly coupled" theories with less than 1, to also describe the "strongly coupled" ( 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).
Montonen-Olive duality, Seiberg duality.
Many more examples come from string theory: S-duality relates
type IIB string theorywith the coupling constant to the same type IIB string theory with the coupling constant . Similarly, type I string theorywith the coupling is equivalent to the SO(32) heterotic stringtheory with the coupling constant . Perhaps most amazing are the S-dualities of type IIA string theoryand E8 heterotic string theorywith coupling constant to the higher dimensional M-theory with a compact dimension of size .
S-duality has been rigorously shown to hold in some lattice models. It depends on the
In particular, in 2 dimensions, if the vertices can take on values in a
locally compact Abelian groupG and the action/ energyonly depends on the edges (e.g. the Ising modelfor Z2, the Potts modelfor Zn, the XY modelfor U(1) ), then its dual via the Kramers-Wannier dualityto 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 modelover the dual group G'.
In 4 dimensions, a lattice gauge model with G as the
gauge groupwould be dual to a lattice gauge model with G' as the gauge group (with the electric and magnetic fields interchanged).
S-duality typically exchanges local charges with
S-duality (homotopy theory)
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