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{{string theory}}
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The '''Green–Schwarz mechanism''' (sometimes called the '''Green–Schwarz anomaly cancellation mechanism''') is the main discovery that started the [[first superstring revolution]] in [[superstring theory]].<ref name="anomaly">{{cite doi|10.1016/0370-2693(84)91565-X}}</ref><ref>Clifford V. Johnson, "D-branes," section 7.1.4. Cambridge University Press, 2003</ref>
 
==Discovery==
In 1984, [[Michael Green (physicist)|Michael Green]] and [[John H. Schwarz]] realized that the [[anomaly (physics)|anomaly]] in [[type I string theory]] with the [[gauge group]] [[SO(32)]] cancels because of an extra "classical" contribution from a [[2-form]] field. They realized that one of the necessary conditions for a superstring theory to make sense is that the [[dimension]] of the [[gauge group]] of [[type I string theory]] must be [[496 (number)|496]] and then demonstrated this to be so.
 
In the original calculation, gauge anomalies, mixed anomalies, and gravitational anomalies were expected to arise<ref>{{cite doi|10.1103/PhysRevLett.50.1343
}}</ref> from a [[hexagon]] [[Feynman diagram]]. For the special choice of the [[gauge group]] SO(32) or [[E8 (mathematics)|E8 x E8]], however, the anomaly factorizes and may be cancelled by a tree diagram. In [[string theory]], this indeed occurs. The tree diagram describes the exchange of a virtual quantum of the B-field. It is somewhat counterintuitive to see that a tree diagram cancels a [[one-loop diagram]], but in reality, both of these diagrams arise as one-loop diagrams in [[superstring theory]] in which the anomaly cancellation is more transparent.
 
As recounted in ''[[The Elegant Universe]]'''s TV version, in the second episode, "The String's the Thing", section "Wrestling with String Theory", Green describes the night of the finding 496 on each side of the equals sign to a stormy night filled with lightning, and fondly recalls joking that "the gods are trying to prevent us from completing this calculation". Green soon entitled some of his subsequent lectures "[[Theory of Everything|The Theory of Everything]]".
 
==Details==
 
Anomalies in quantum theory arise from one-loop diagrams, with a chiral fermion in the loop and gauge fields, Ricci tensors, or global symmetry currents as the external legs. These diagrams  have the form of a triangle in 4 spacetime dimensions, which generalizes to a hexagon in D=10, thus involving 6 external lines.
The interesting anomaly in SUSY D=10 gauge theory is the hexagon which has a particular linear combination of the two-form gauge field strength and Ricci tensor, <math>F^6,\ F^4 R^2,\ F^2 R^4,\ R^6</math>, for the external lines.
Green and Schwarz realized that one can add a so-called Chern-Simons term to the classical action,
having the form <math>S_{GS} = \int B_{2}\wedge X_8</math>, where the integral is over the 10 dimensions, <math>B_{2}</math> is the rank-two Kalb-Ramond field, and <math>X_8</math> is
a gauge invariant combination of  
<math>F^4,\ F^2 R^2,\ R^4</math> (with space-time indices not contracted), which is precisely one of the factors appearing in the hexagon anomaly.
If the variation of <math>B_{2}</math> under the
transformations of gauge field for <math>F_{(2)}</math> and under general coordinate transformations is appropriately specified, then
the Green-Schwarz term <math>S_{GS}</math>, when combined with a trilinear vertex through exchange of a gauge boson, has precisely the right variation to cancel the hexagon anomaly.
 
==References==
{{Reflist}}
 
 
 
{{DEFAULTSORT:Green-Schwarz Mechanism}}
[[Category:Anomalies in physics]]
[[Category:String theory]]
[[Category:Quantum gravity]]

Latest revision as of 13:09, 5 May 2014

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