N = 4 supersymmetric Yang–Mills theory
Integrability <!--'Planar limit', 'Planar Yang-Mills theory', and 'Planar Yang-Mills theory' redirect here-->
Integrability
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There is evidence that *[N](/wiki/Extended_supersymmetry "Extended supersymmetry")* \= 4 supersymmetric Yang–Mills theory has an integrable structure in the planar [large *N* limit](/wiki/Large_N_limit "Large N limit") (see below for what "planar" means in the present context). As the number of colors (also denoted *N*) goes to infinity, the amplitudes scale like N^{2\-2g}, so that only the [genus 0 (planar graph)](/wiki/Planar_graph "Planar graph") contribution survives. **Planar [Feynman diagrams](/wiki/Feynman_diagrams "Feynman diagrams")** are graphs in which no [propagator](/wiki/Propagator "Propagator") cross over another one, in contrast to **non\-planar Feynman graphs** where one or more propagator goes over another one.["Planar vs. Non\-Planar: A Colorful Story", from *4 gravitons* blog (2013\)](https://4gravitons.com/2013/09/27/planar-vs-non-planar-a-colorful-story/) A non\-planar graph has a smaller number of possible gauge loops compared to a similar planar graph. Non\-planar graphs are thus suppressed by factors 1/N^{2\-2g} compared to planar ones which therefore dominate in the large *N* limit. Consequently, a **planar Yang–Mills theory** denotes a theory in the large *N* limit, with *N* usually the number of [colors](/wiki/Color_charge "Color charge"). Likewise, a **planar limit** is a limit in which scattering amplitudes are dominated by [Feynman diagrams](/wiki/Feynman_diagrams "Feynman diagrams") which can be given the structure of planar graphs.[planar limit in nLab](https://ncatlab.org/nlab/show/planar+limit) In the [large *N* limit](/wiki/Large_N_limit "Large N limit"), the coupling  g  vanishes and a [perturbative formalism](/wiki/Perturbation_theory_%28quantum_mechanics%29 "Perturbation theory (quantum mechanics)") is therefore well\-suited for large *N* calculations. Therefore, planar graphs are associated to the domain where perturbative calculations converge well.

[Beisert](/wiki/Niklas_Beisert "Niklas Beisert") et al.  give a review article demonstrating how in this situation local operators can be expressed via certain states in [spin chains](/wiki/Spin_chain "Spin chain") (in particular the [Heisenberg spin chain](/wiki/Heisenberg_spin_chain "Heisenberg spin chain")), but based on a larger [Lie superalgebra](/wiki/Lie_superalgebra "Lie superalgebra") rather than \\mathfrak{su}(2\) for ordinary spin. These spin chains are integrable in the sense that they can be solved by the [Bethe ansatz](/wiki/Bethe_ansatz "Bethe ansatz") method. They also construct an [action](/wiki/Algebra_representation "Algebra representation") of the associated [Yangian](/wiki/Yangian "Yangian") on [scattering amplitudes](/wiki/Scattering_amplitudes "Scattering amplitudes").

[Nima Arkani\-Hamed](/wiki/Nima_Arkani-Hamed "Nima Arkani-Hamed") et al. have also researched this subject. Using [twistor theory](/wiki/Twistor_theory "Twistor theory"), they find a description (the [amplituhedron](/wiki/Amplituhedron "Amplituhedron") formalism) in terms of the positive [Grassmannian](/wiki/Grassmannian "Grassmannian").


Can integrability in supersymmetric Yang–Mills theory be applied to other theories?