N = 4 supersymmetric Yang–Mills (SYM) theory is a relativistic conformally invariant Lagrangian gauge theory describing fermions interacting via gauge field exchanges. In D=4 spacetime dimensions, N=4 is the maximal number of supersymmetries or supersymmetry charges.[1]
It is a toy theory based on Yang–Mills theory that does not model the real world, but is useful because it can act as a proving ground for approaches for attacking problems in more complex theories.[2] It describes a universe containing boson fields and fermion fields which are related by four supersymmetries (this means that transforming bosonic and fermionic fields in a certain way leaves the theory invariant). It is one of the simplest (in the sense that it has no free parameters except for the gauge group) and one of the few ultraviolet finite quantum field theories in 4 dimensions. It can be thought of as the most symmetric field theory that does not involve gravity.
Like all supersymmetric field theories, it may equivalently be formulated as a superfield theory on an extended superspace in which the spacetime variables are augmented by a number of anticommuting Grassmann variables which, for the case N=4, consist of 4 Dirac spinors, making a total of 16 independent anticommuting generators for the extended ring of superfunctions. The field equations are equivalent to the geometric condition that the supercurvature 2-form vanish identically on all super null lines.[3] [4] This is also known as the super-ambitwistor correspondence.
A similar super-ambitwistor characterization holds for D=10, N=1 dimensional super Yang–Mills theory,[5] [6] and the lower dimensional cases D=6, N=2 and D=4, N=4 may be derived from this via dimensional reduction.
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.
The Lagrangian for the theory is[1] [8]
L=\operatorname{tr}\left\{-
| 1 | |
| 2g2 |
F\mu\nuF\mu\nu+
| \thetaI | |
| 8\pi2 |
F\mu\nu\bar{F}\mu\nu-i\overline{λ}a\overline{\sigma}\muD\muλa-D\muXiD\muXi +g
| ab | |
| C | |
| i |
| i,λ | |
| λ | |
| b] |
+g\overline{C}iab\overline{λ}a[Xi,\overline{λ}
| ||||
[Xi,Xj]2\right\},
g
\thetaI
g
\thetaI
| k | |
| F | |
| \mu\nu |
=\partial\mu
| k | |
| A | |
| \nu-\partial |
\nu
| klm | |
| A | |
| \mu+f |
| l | |
| A | |
| \mu |
| m | |
| A | |
| \nu |
| k | |
| A | |
| \nu |
f
λa
\sigma\mu
D\mu
Xi
| ab | |
| C | |
| i |
The above Lagrangian can be found by beginning with the simpler ten-dimensional Lagrangian
L=\operatorname{tr}\left\{
| 1 | |
| g2 |
FIJFIJ-i\bar{λ}\GammaIDIλ\right\},
where I and J are now run from 0 through 9 and
\GammaI
(32=210/2)
\thetaI
The components
Ai
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.
See main article: Montonen–Olive duality.
The coupling constants
\thetaI
g
\tau:=
| \thetaI | + | |
| 2\pi |
| 4\pii | |
| g2 |
.
The theory has symmetries that shift
\tau
\tau\mapsto
| -1 | |
| nG\tau |
G
See main article: AdS/CFT correspondence.
This theory is also important[1] 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 (see below for what "planar" means in the present context).[9] As the number of colors (also denoted N) goes to infinity, the amplitudes scale like
N2-2g
1/N2-2g
g
Beisert et al. [12] give a review article demonstrating how in this situation local operators can be expressed via certain states in spin chains (in particular the Heisenberg spin chain), but based on a larger Lie superalgebra rather than
ak{su}(2)
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]
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
N → infty