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| 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 [[Finland|Finnish]] [[Claus Montonen]] and [[United Kingdom|British]] [[David Olive]].
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| == Overview ==
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| In a four-dimensional [[Yang-Mills]] theory with [[extended supersymmetry|''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 monopole]]s. See also [[Seiberg duality]].
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| In fact, there exists a larger [[modular group|SL(2,'''Z''')]] symmetry where both ''g'' as well as [[theta-angle]] are transformed non-trivially.
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| == Mathematical formalism ==
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| The gauge coupling and [[theta-angle]] can be combined together to form one complex coupling
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| :<math> \tau = \frac{\theta}{2\pi}+\frac{4\pi i}{g^2}.</math>
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| Since the theta-angle is periodic, there is a symmetry
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| :<math> \tau \mapsto \tau + 1.</math>
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| The quantum mechanical theory with gauge group ''G'' (but not the classical theory, except in the case when the ''G'' is [[abelian group|abelian]]) is also invariant under the symmetry
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| :<math> \tau \mapsto \frac{-1}{n_G\tau}</math>
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| while the gauge group ''G'' is simultaneously replaced by its [[Langlands dual group]] <sup>''L''</sup>''G'' and <math>n_G</math> 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 ==
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| * [[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]]
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| {{DEFAULTSORT:Montonen-Olive duality}}
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| [[Category:Quantum field theory]]
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| [[Category:Duality theories]]
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| {{quantum-stub}}
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