N=4 SYM Field Content, Action, and Superconformal Data

Four-dimensional $\mathcal N=4$ super-Yang–Mills theory is the dimensional reduction of ten-dimensional $\mathcal N=1$ Yang–Mills. That origin fixes the relative gauge, scalar, and Yukawa interactions. This page gives one complete local theory card and translates between the two field normalizations most often used in duality calculations.

Required background. Extended supersymmetry gauge dynamics supplies the $\mathcal N=2$ and $\mathcal N=1$ decompositions. Superconformal algebras and the CFT handoff supplies the distinction between Poincaré and conformal supercharges.

Helpful background. Supersymmetric Yang–Mills actions develops the lower-$\mathcal N$ superspace construction.

The maximally supersymmetric theory card

Let $\mathfrak g$ be a compact semisimple Lie algebra and let $\operatorname{Tr}$ be normalized so that $T(F)=1/2$ for the fundamental of $SU(N)$. Take Hermitian connection and scalar fields,

$$F=d\mathcal A-i\mathcal A\wedge\mathcal A, \qquad D_\mu X^I=\partial_\mu X^I-i[\mathcal A_\mu,X^I].$$

In the overall-coupling convention, the bosonic Minkowski action is

$$\begin{aligned} S_{\rm bos} ={}&\frac{1}{g_{\rm YM}^{2}}\int d^4x\, \operatorname{Tr}\!\left[ -\frac14 F_{\mu\nu}F^{\mu\nu} +\frac12 D_\mu X^I D^\mu X^I +\frac14[X^I,X^J][X^I,X^J] \right]\\ &+\frac{\theta}{8\pi^2}\int\operatorname{Tr}(F\wedge F), \qquad I,J=1,\ldots,6 . \end{aligned}$$

Because $[X^I,X^J]$ is anti-Hermitian, the potential energy

$$V(X)=-\frac{1}{4g_{\rm YM}^{2}} \operatorname{Tr}[X^I,X^J][X^I,X^J]$$

is nonnegative. Four adjoint Weyl fermions complete the multiplet. Their Yukawa couplings are the reduction of the ten-dimensional term $\frac{i}{2g_{\rm YM}^{2}}\operatorname{Tr}\bar\Psi\Gamma^M D_M\Psi$; the six internal gamma matrices are the Clebsch–Gordan maps relating the $\mathbf4$, $\overline{\mathbf4}$, and $\mathbf6$ of $SU(4)_R$. Dimensional reduction fixes all relative coefficients Brink, Schwarz, and Scherk 1977, §§2–3.

The fields transform under $Spin(3,1)\times SU(4)_R$ as

field	Lorentz representation	$SU(4)_R$ representation
$\mathcal A_\mu$	vector	$\mathbf1$
$\lambda^A_\alpha$	left Weyl spinor	$\mathbf4$
$\bar\lambda_{\dot\alpha A}$	right Weyl spinor	$\overline{\mathbf4}$
$X^I$	scalar	$\mathbf6$

The local data are not the whole theory. One must additionally choose a compact global form $G$ with Lie algebra $\mathfrak g$, a mutually local set of genuine line operators, and any discrete theta datum. Those choices do not alter the displayed adjoint-field action, but they change bundles, one-form symmetries, and the duality target.

Translating field conventions

Some authors put $g_{\rm YM}$ inside covariant derivatives and canonically normalize quadratic kinetic terms. The relation is

$$\mathcal A_\mu=g_{\rm YM}A_\mu^{\rm can}, \qquad X^I=g_{\rm YM}X_{\rm can}^I.$$

Consequently,

$$F(\mathcal A)=g_{\rm YM} \left(dA^{\rm can}-ig_{\rm YM}A^{\rm can}\wedge A^{\rm can}\right).$$

The cubic and quartic interactions then carry $g_{\rm YM}$ and $g_{\rm YM}^2$, respectively. Two quantities provide reliable translation checks:

$$\operatorname{Tr}_{R}\,\mathcal P \exp\!\left(i\oint\mathcal A\right) = \operatorname{Tr}_{R}\,\mathcal P \exp\!\left(ig_{\rm YM}\oint A^{\rm can}\right),$$

and, for a root $\alpha$ at a commuting scalar expectation value,

$$M_{W,\alpha}^{2} =\sum_I\lvert\alpha(X^I)\rvert^2 =g_{\rm YM}^{2}\sum_I \lvert\alpha(X_{\rm can}^I)\rvert^2 .$$

Thus a formula for a scalar expectation value cannot be compared across papers until the kinetic normalization is known.

In $\mathcal N=1$ language the multiplet is one vector $V$ and three adjoint chirals $\Phi_i$. With canonically normalized chiral kinetic terms, a standard generator convention gives

$$W=\sqrt2\,g_{\rm YM}\, \operatorname{Tr}\bigl(\Phi_1[\Phi_2,\Phi_3]\bigr).$$

A rephasing of the chiral fields or anti-Hermitian generators changes the displayed sign or factor of $i$, but not the invariant statement: the superpotential coupling equals the gauge coupling up to the fixed normalization required by the hidden twelve supercharges.

Coupling and topological normalization

For a bundle whose instanton number

$$\nu=\frac{1}{8\pi^2}\int\operatorname{Tr}(F\wedge F)$$

is integral, the theta angle is $2\pi$ periodic and

$$\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g_{\rm YM}^{2}}.$$

Quotient gauge groups can admit fractional instanton number on general four-manifolds; the period of a continuous theta parameter and its correlation with discrete theta data must then be specified globally. The simple formula for $\tau$ remains the local coupling coordinate, but it does not by itself identify the same global theory after $\tau\mapsto\tau+1$.

Superconformal data

At the conformal point the algebra is $\mathfrak{psu}(2,2|4)$. Its bosonic subalgebra is $\mathfrak{so}(4,2)\oplus\mathfrak{su}(4)_R$, and it contains sixteen Poincaré plus sixteen conformal supercharges. The scalar bottom of the stress-tensor multiplet has dimension $2$ and transforms in the $\mathbf{20'}=[0,2,0]$ of $SU(4)_R$.

The Weyl-anomaly coefficients follow quickly in an $\mathcal N=1$ description. The gaugino has $R=1$; each of the three chiral multiplet fermions has $R=-1/3$. Hence

$$\operatorname{Tr}R =\dim\mathfrak g\,[1+3(-1/3)]=0,$$ $$\operatorname{Tr}R^3 =\dim\mathfrak g\,[1+3(-1/3)^3] =\frac89\dim\mathfrak g.$$

Using the exact anomaly relations

$$a=\frac{3}{32}\left(3\operatorname{Tr}R^3-\operatorname{Tr}R\right), \qquad c=\frac{1}{32}\left(9\operatorname{Tr}R^3-5\operatorname{Tr}R\right),$$

one obtains

$$a=c=\frac14\dim\mathfrak g.$$

The anomaly formula and its relation to superconformal central charges are established in Anselmi et al. 1998, §§2–3. The result depends on the Lie algebra, not on the global form, because it is a local anomaly. Partition functions and line sectors can still distinguish global theories with the same $a$ and $c$.

What this card does not fix

The Lagrangian is an electric-frame description. It does not make magnetic lines local, does not select a nonperturbative regulator preserving all desired structures, and does not prove S-duality. The superconformal algebra also does not protect every operator: long multiplets such as the Konishi multiplet have coupling-dependent dimensions.

The displayed theta normalization assumes a spin four-manifold when fermions are present and a trace convention with integral $\nu$ for the simply connected reference group. On non-spin manifolds or for quotient groups, quadratic refinements and discrete counterterms require a more detailed specification.

Exercises

1. Check the potential. Show that $V(X)\geq0$ for Hermitian $X^I$.

Solution

Write $C^{IJ}=i[X^I,X^J]$, which is Hermitian. Then $\operatorname{Tr}[X^I,X^J]^2=-\operatorname{Tr}(C^{IJ})^2\leq0$. The minus sign in $V$ therefore makes every summand nonnegative.

2. Reproduce the central charges. Insert the computed $\operatorname{Tr}R$ and $\operatorname{Tr}R^3$ into the anomaly relations.

Solution

For $a$, one finds $\frac{3}{32}\cdot3\cdot\frac89\dim\mathfrak g=\frac14\dim\mathfrak g$. For $c$, one finds $\frac{1}{32}\cdot9\cdot\frac89\dim\mathfrak g=\frac14\dim\mathfrak g$.

References

• Anselmi, Damiano, Daniel Z. Freedman, Marcus T. Grisaru, and Andreas A. Johansen. “Nonperturbative Formulas for Central Functions of Supersymmetric Gauge Theories.” Nuclear Physics B 526 (1998): 543–571. doi:10.1016/S0550-3213(98)00278-8.
• Brink, Lars, John H. Schwarz, and Joël Scherk. “Supersymmetric Yang–Mills Theories.” Nuclear Physics B 121 (1977): 77–92. doi:10.1016/0550-3213(77)90328-5.