N = 4 supersymmetric Yang–Mills theory

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N = 4 supersymmetric Yang–Mills (SYM) theory is a

In N supersymmetric Yang–Mills theory, N denotes the number of independent supersymmetric operations that transform the spin-1 gauge field into spin-1/2 fermionic fields.^[7] In an analogy with symmetries under rotations, N would be the number of independent rotations, N = 1 in a plane, N = 2 in 3D space, etc... That is, in a N = 4 SYM theory, the gauge boson can be "rotated" into N = 4 different supersymmetric fermion partners. In turns, each fermion can be rotated into four different bosons: one corresponds to the rotation back to the spin-1 gauge field, and the three others are spin-0 boson fields. Because in 3D space one may use different rotations to reach a same point (or here the same spin-0 boson), each spin-0 boson is superpartners of two different spin-1/2 fermions, not just one.^[7] So in total, one has only 6 spin-0 bosons, not 16.

Therefore, N = 4 SYM has 1 + 4 + 6 = 11 fields, namely: one vector field (the spin-1 gauge boson), four spinor fields (the spin-1/2 fermions) and six scalar fields (the spin-0 bosons). N = 4 is the maximum number of independent supersymmetries: starting from a spin-1 field and using more supersymmetries, e.g., N = 5, only rotates between the 11 fields. To have N > 4 independent supersymmetries, one needs to start from a gauge field of spin higher than 1, e.g., a spin-2 tensor field such as that of the graviton. This is the N = 8 supergravity theory.

Lagrangian

The Lagrangian for the theory is^[1][8]

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

where ${\displaystyle g}$ and ${\displaystyle \theta _{I}}$ are coupling constants (specifically ${\displaystyle g}$ is the gauge coupling and ${\displaystyle \theta _{I}}$ is the instanton angle), the

${\displaystyle F_{\mu \nu }^{k}=\partial _{\mu }A_{\nu }^{k}-\partial _{\nu }A_{\mu }^{k}+f^{klm}A_{\mu }^{l}A_{\nu }^{m}}$

${\displaystyle A_{\nu }^{k}}$

${\displaystyle f}$

${\displaystyle \lambda ^{a}}$

${\displaystyle \sigma ^{\mu }}$

${\displaystyle D_{\mu }}$

${\displaystyle X^{i}}$

${\displaystyle C_{i}^{ab}}$

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

A

The coupling constants ${\displaystyle \theta _{I}}$ and ${\displaystyle g}$ naturally pair together into a single coupling constant

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

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.

AdS/CFT correspondence

This theory is also important

As the number of colors (also denoted N) goes to infinity, the amplitudes scale like ${\displaystyle N^{2-2g}}$, so that only the

Feynman diagrams are graphs in which no propagator cross over another one, in contrast to non-planar Feynman graphs where one or more propagator goes over another one.^[10]

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 ${\displaystyle 1/N^{2-2g}}$ 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

large N limit

, the coupling ${\displaystyle g}$ vanishes and a perturbative formalism is therefore well-suited for large N calculations. Therefore, planar graphs are associated to the domain where perturbative calculations converge well.

Beisert et al.

Heisenberg spin chain), but based on a larger Lie superalgebra

rather than ${\displaystyle {\mathfrak {su}}(2)}$ for ordinary spin. These spin chains are integrable in the sense that they can be solved by the

scattering amplitudes

.

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.^[13]

Relation to 11-dimensional M-theory

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.^{[citation needed]}

See also

• 6D (2,0) superconformal field theory
• Extended supersymmetry
• N = 1 supersymmetric Yang–Mills theory
• N = 8 supergravity
• Seiberg–Witten theory

References

Citations

1. ^
   S2CID 119501374
   .
2. ^ Matt von Hippel (2013-05-21). "Earning a PhD by studying a theory that we know is wrong". Ars Technica.
3. doi:10.1016/0370-2693(78)90585-3
   .
4. doi:10.1016/0550-3213(85)90410-9
   .
5. doi:10.1016/0550-3213(86)90090-8
   .
6. S2CID 122622189
   .
7. ^ ^{a b} "N = 4: Maximal Particles for Maximal Fun", from 4 gravitons blog (2013)
8. ^ Luke Wassink (2009). "N = 4 Super Yang–Mills theory" (PDF). Archived from the original (PDF) on 2014-05-31. Retrieved 2013-05-22.
9. ISBN 9780511846373
   .
10. ^ "Planar vs. Non-Planar: A Colorful Story", from 4 gravitons blog (2013)
11. ^ planar limit in nLab
12. S2CID 254796664
    .
13. S2CID 119599921. {{cite journal}}: Cite journal requires |journal= (help
    )

Sources

• Kapustin, Anton; Witten, Edward (2007). "Electric-magnetic duality and the geometric Langlands program". Communications in Number Theory and Physics. 1 (1): 1–236.
  S2CID 30505126
  .