N = 4 supersymmetric Yang–Mills theory
Meaning of ''N'' and numbers of fields
Meaning of *N* and numbers of fields
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In *N* supersymmetric Yang–Mills theory, *N* denotes the number of independent supersymmetric operations that transform the [spin](/wiki/Spin_%28physics%29 "Spin (physics)")\-1 gauge field into spin\-1/2 fermionic fields.["*N* \= 4: Maximal Particles for Maximal Fun", from *4 gravitons* blog (2013\)](https://4gravitons.com/n4-maximal-particles-for-maximal-fun/) 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. 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](/wiki/Tensor "Tensor") field such as that of the [graviton](/wiki/Graviton "Graviton"). This is the [*N* \= 8 supergravity](/wiki/N_%3D_8_supergravity "N = 8 supergravity") theory.


What is the meaning of N in supersymmetric Yang–Mills theory?