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

Montonen, Claus; Olive, David (1977). "Magnetic monopoles as gauge particles?". Physics Letters B. 72 (1): 117–120. Bibcode:1977PhLB...72..117M. doi:10.1016/0370-2693(77)90076-4.
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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