# 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.

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## 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.

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## 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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