Montonen–Olive duality

In theoretical physics, Montonen–Olive duality is the oldest known example of S-duality or a strong-weak duality. It generalizes the electro-magnetic symmetry of Maxwell's equations. It is named after Finnish Claus Montonen and British David Olive.

Overview

In a four-dimensional Yang-Mills theory with N=4 supersymmetry, which is the case where the Montonen–Olive duality applies, one obtains a physically equivalent theory if one replaces the gauge coupling constant g by 1/g. This also involves an interchange of the electrically charged particles and magnetic monopoles. See also Seiberg duality.

In fact, there exists a larger SL(2,Z) symmetry where both g as well as theta-angle are transformed non-trivially.

Mathematical formalism

The gauge coupling and theta-angle can be combined together to form one complex coupling

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

Since the theta-angle is periodic, there is a symmetry

\tau \mapsto \tau + 1.

The quantum mechanical theory with gauge group G (but not the classical theory, except in the case when the G is abelian) is also invariant under the symmetry

\tau \mapsto \frac{-1}{n_G\tau}

while the gauge group G is simultaneously replaced by its Langlands dual group ^LG and $n_G$ is an integer depending on the choice of gauge group. In the case the theta-angle is 0, this reduces to the simple form of Montonen–Olive duality stated above.

References

Edward Witten, Notes from the 2006 Bowen Lectures, an overview of electric–magnetic duality in gauge theory and its relation to the Langlands program

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