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 :$au\; =\; frac\{\; heta\}\{2pi\}+frac\{4pi\; i\}\{g^2\}$Since the theta-angle is periodic, there is a symmetry :$au\; mapsto\; au\; +\; 1$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:$au\; mapsto\; frac\{-1\}\{n\_G\; au\}$while the gauge group "G" is simultaneously replaced by its Langlands dual group ^"L""G" 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, [http://math.berkeley.edu/index.php?module=documents&JAS_DocumentManager_op=viewDocument&JAS_Document_id=116 "Notes from the 2006 Bowen Lectures"] , an overview of Electric-Magnetic duality in gauge theory and its relation to the Langlands program