N=4 Super Yang–Mills Theory

Four-dimensional $\mathcal N=4$ super Yang–Mills theory combines a gauge field, six real scalars, and four Weyl fermions in the adjoint representation. Sixteen Poincaré supercharges fix their relative interactions. In its conformal vacuum it is an interacting theory with no running gauge coupling, making it central to studies of duality, amplitudes, and holography.

Multiplet and defining interactions

Take gauge group $SU(N)$, $N\ge2$, theta angle zero, and Hermitian adjoint fields. With $\operatorname{Tr}(T^aT^b)=\delta^{ab}/2$, absorb the gauge coupling into the connection and write

$$\begin{aligned} F_{\mu\nu}&=\partial_\mu A_\nu-\partial_\nu A_\mu-i[A_\mu,A_\nu],\\ D_\mu X^I&=\partial_\mu X^I-i[A_\mu,X^I], \qquad I=1,\ldots,6. \end{aligned}$$

The bosonic Lagrangian is

$$\mathcal L_{\mathrm{bos}} =\frac1{g_{\mathrm{YM}}^2}\operatorname{Tr}\left[ -\frac12F_{\mu\nu}F^{\mu\nu} +D_\mu X^I D^\mu X^I +\frac12[X^I,X^J][X^I,X^J]\right].$$

The four adjoint Weyl fermions have kinetic and Yukawa terms fixed by dimensional reduction of ten-dimensional $\mathcal N=1$ Yang–Mills theory; this construction specifies the fermionic completion, rather than permitting arbitrary Yukawa couplings. See Brink, Schwarz, and Scherk 1977, §§2–3.

The commutator of Hermitian matrices is anti-Hermitian, so its squared trace is nonpositive. Since $\mathcal L$ contains $-V$, the displayed scalar potential is nonnegative and vanishes for mutually commuting $X^I$.

Degrees of freedom and a beta-function check

Per gauge generator, two photon polarizations plus six scalars give eight bosonic on-shell degrees of freedom. Four Weyl fermions give eight fermionic ones.

The one-loop gauge coefficient provides another check:

$$b_0=C_A\left(\frac{11}{3} -\frac23\times4-\frac16\times6\right)=0.$$

The scalar coefficient counts real adjoint scalars and the fermion coefficient counts Weyl fields. Extended supersymmetry protects the vanishing beta function beyond this first check; that all-order statement is stronger than the arithmetic cancellation Weinberg 2000, §27.9.

Conformal vacuum and controlled limits

The conformal vacuum has vanishing scalar expectation values. Nonzero commuting scalar expectation values move onto a Coulomb branch, introduce mass scales, and break conformal symmetry spontaneously; this is a different vacuum of the same theory.

Gauge-invariant correlators, Wilson loops, and scaling dimensions are central observables. Some operators have protected dimensions; generic operators do not. Weak coupling controls a perturbative expansion. Taking $N$ large at fixed $\lambda=g_{\mathrm{YM}}^2N$ controls a different expansion, and large $N$ alone does not make $\lambda$ small. Strong-coupling holographic calculations require their own large-$N$ and large-$\lambda$ regimes.

The local action also does not choose every possible global line-operator spectrum if the global gauge group is changed.

Develop the calculation

N=4 SYM Field Content, Action, and Superconformal Data derives the dimensional reduction and protected data. Compare Yang–Mills Theory for the nonsupersymmetric running and Supersymmetric QCD for flavor-dependent infrared dynamics.

References

• Brink, Lars, John H. Schwarz, and Joël Scherk. “Supersymmetric Yang–Mills Theories.” Nuclear Physics B 121 (1977): 77–92. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge University Press, 2000, §27.9. DOI.