N = 4 supersymmetric Yang–Mills theory

The Lagrangian for the theory is[2]

${\displaystyle L=\operatorname {tr} \left\{-{\frac {1}{2g^{2}}}F_{\mu \nu }F^{\mu \nu }+{\frac {\theta _{I}}{8\pi ^{2}}}F_{\mu \nu }{\bar {F}}^{\mu \nu }-i{\overline {\lambda }}^{a}{\overline {\sigma }}^{\mu }D_{\mu }\lambda _{a}-D_{\mu }X^{i}D^{\mu }X^{i}+gC_{i}^{ab}\lambda _{a}[X^{i},\lambda _{b}]+g{\overline {C}}_{iab}{\overline {\lambda }}^{a}[X^{i},{\overline {\lambda }}^{b}]+{\frac {g^{2}}{2}}[X^{i},X^{j}]^{2}\right\},}$

where ${\displaystyle F_{\mu \nu }^{k}=\partial _{\mu }A_{\nu }^{k}-\partial _{\nu }A_{\mu }^{k}+f^{klm}A_{\mu }^{l}A_{\nu }^{m}}$ and indices i,j = 1, ..., 6 as well as a, b = 1, ..., 4. ${\displaystyle f}$ represents the structure constants of the particular gauge group. ${\displaystyle C_{i}^{ab}}$ represents the structure constants of the R-symmetry group SU(4), which rotates the 4 supersymmetries. As a consequence of the nonrenormalization theorems, this supersymmetric field theory is in fact a superconformal field theory.

The above Lagrangian can be found by beginning with the simpler ten-dimensional Lagrangian

${\displaystyle L=\operatorname {tr} \left\{{\frac {1}{g^{2}}}F_{IJ}F^{IJ}-i{\bar {\lambda }}\Gamma ^{I}D_{I}\lambda \right\},}$

where I and J are now run from 0 through 9 and ${\displaystyle \Gamma ^{I}}$ are the 32 by 32 gamma matrices ${\displaystyle (32=2^{10/2})}$, followed by adding the term with ${\displaystyle \theta _{I}}$ which is a topological term.

The components ${\displaystyle A_{i}}$ of the gauge field for i = 4 to 9 become scalars upon eliminating the extra dimensions. This also gives an interpretation of the SO(6) R-symmetry as rotations in the extra compact dimensions.

By compactification on a T⁶, all the supercharges are preserved, giving N = 4 in the 4-dimensional theory.

A Type IIB string theory interpretation of the theory is the worldvolume theory of a stack of D3-branes.

The coupling constants ${\displaystyle \theta _{I}}$ and ${\displaystyle g}$ naturally pair together in the form:

${\displaystyle \tau ={\frac {\theta }{2\pi }}+{\frac {4\pi i}{g^{2}}}.}$

The theory has symmetries that shift ${\displaystyle \tau }$ by integers. The S-duality conjecture says there is also a symmetry which sends :${\displaystyle \tau \mapsto {\frac {-1}{n_{G}\tau }}}$ as well as switching the group ${\displaystyle G}$ to its Langlands dual group.

This theory is important also in the context of the holographic principle. There is a duality between Type IIB string theory on AdS₅ × S⁵ space (a product of 5-dimensional AdS space with a 5-dimensional sphere) and N = 4 super Yang–Mills on the 4-dimensional boundary of AdS₅. However, this particular realization of the AdS/CFT correspondence is not a realistic model of gravity, since gravity in our universe is 4-dimensional. Despite this, the AdS/CFT correspondence is the most successful realization of the holographic principle, a speculative idea about quantum gravity originally proposed by Gerard 't Hooft, who was expanding on work on black hole thermodynamics, and was improved and promoted in the context of string theory by Leonard Susskind.

There is evidence that N = 4 supersymmetric Yang–Mills theory has an integrable structure in the planar large N limit.[3] As the number of colors (also denoted N) goes to infinity, the amplitudes scale like ${\displaystyle N^{2-2g}}$, so that only the genus 0 (planar graph) contribution survives. Planar Yang–Mills theory is a theory with a very large (infinite) number of colors.

A planar limit is a limit in which scattering amplitudes are dominated by Feynman diagrams which can be given the structure of planar graphs.[4]

Beisert et al. give a review article demonstrating how in this situation local operators can be expressed via certain states in "spin" chains, but based on a larger Lie superalgebras rather than SU(2) for ordinary spin. These are amenable to Bethe ansatz techniques. They also construct an action of the associated Yangian on scattering amplitudes.[5]

Nima Arkani-Hamed et al. have also researched this subject. Using twistor theory, they find a description (the amplituhedron formalism) in terms of the positive Grassmannian.[6]

N = 4 super Yang–Mills can be derived from a simpler 10-dimensional theory, and yet supergravity and M-theory exist in 11 dimensions. The connection is that if the gauge group U(N) of SYM becomes infinite as ${\displaystyle N\rightarrow \infty }$ it becomes equivalent to an 11-dimensional theory known as matrix theory.

• 6D (2,0) superconformal field theory
• Extended supersymmetry