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

Four-dimensional $\mathcal N=4$ super-Yang–Mills theory packages a gauge connection, six adjoint scalars, and four adjoint Weyl fermions into one multiplet. Sixteen supersymmetries fix every relative interaction, but they do not choose the global gauge group or genuine-line spectrum. This page freezes a local $\mathfrak{su}(N)$ convention, derives its action from ten dimensions, and records the superconformal data needed by later BPS and duality pages.

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 and between shortening data and dynamical CFT data.

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

Local fields, trace, and supersymmetry transformations

For the explicit card, take local gauge algebra $\mathfrak{su}(N)$ with Hermitian generators

$$\operatorname{Tr}(T^aT^b)=\frac12\delta^{ab},$$

where $\operatorname{Tr}$ is the fundamental trace. Use the site’s $(+---)$ metric and Hermitian adjoint-valued fields. The connection convention is

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

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

field	Lorentz representation	$SU(4)_R$ representation	real on-shell degrees of freedom per generator
$\mathcal A_\mu$	vector	$\mathbf1$	$2$
$\lambda^A_\alpha$	left Weyl spinor	$\mathbf4$	$8$ fermionic in total
$\bar\lambda_{\dot\alpha A}$	right Weyl conjugate	$\overline{\mathbf4}$	—
$X^I$	scalar	$\mathbf6$	$6$

Thus the multiplet has eight bosonic and eight fermionic on-shell degrees of freedom for each adjoint generator.

A compact way to display all sixteen supersymmetries is to begin in ten dimensions. Let $M,N=0,\ldots,9$, let $\Psi$ be a Majorana–Weyl adjoint spinor, and take

$$S_{10} =\frac1{g_{10}^2}\int d^{10}x\, \operatorname{Tr}\!\left( -\frac12F_{MN}F^{MN} +i\bar\Psi\Gamma^M D_M\Psi \right).$$

One consistent phase convention has

$$\delta_\varepsilon \mathcal A_M =i\bar\varepsilon\Gamma_M\Psi, \qquad \delta_\varepsilon\Psi =-\frac12F_{MN}\Gamma^{MN}\varepsilon.$$

After using the fermion equation of motion, two transformations close into a translation plus a gauge transformation. Reversing the phase of $\Psi$ changes both displayed transformation signs without changing the theory. The ten-dimensional construction and its reduction are given in Brink, Schwarz, and Scherk 1977, §§2–3.

Deriving the four-dimensional action

For a formal reduction on a flat six-torus of volume $V_6$, retain only fields independent of the internal coordinates. Integrating those coordinates defines the four-dimensional coupling by

$$\frac1{g_{\rm YM}^2}=\frac{V_6}{g_{10}^2}, \qquad\text{or equivalently}\qquad g_{\rm YM}^2=\frac{g_{10}^2}{V_6},$$

in the unrescaled zero-mode convention used here. Then set all internal derivatives to zero and identify

$$\mathcal A_{3+I}=X^I, \qquad I=1,\ldots,6.$$

The mixed and internal components of the ten-dimensional field strength become

$$F_{\mu,3+I}=D_\mu X^I, \qquad F_{3+I,3+J}=-i[X^I,X^J].$$

Substitution into $-F_{MN}F^{MN}/2$ fixes the relative coefficients. In the overall-coupling convention, the bosonic Minkowski action is

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

With $\operatorname{Tr}(T^aT^b)=\delta^{ab}/2$, the gauge term is $-F^a_{\mu\nu}F^{a\mu\nu}/(4g_{\rm YM}^2)$. This component normalization is the one compatible with the complex coupling below and with the D3-brane relation $g_{\rm YM}^2=4\pi g_s$ derived in the AdS$_5\times S^5$ parameter map.

Repeated $I,J$ are summed. Because $X^I$ are Hermitian, $[X^I,X^J]$ is anti-Hermitian and

$$V(X) =-\frac{1}{2g_{\rm YM}^{2}} \operatorname{Tr}[X^I,X^J][X^I,X^J] \geq0.$$

The vacuum condition is therefore $[X^I,X^J]=0$ for every pair. The fermion kinetic and Yukawa terms descend together from $i\bar\Psi\Gamma^M D_M\Psi$; the six internal gamma matrices furnish the $SU(4)_R$-equivariant maps between the $\mathbf4$, $\overline{\mathbf4}$, and $\mathbf6$. This derivation is a stronger normalization check than matching the bosonic potential alone, because it fixes the Yukawa couplings as well.

Translating to canonically normalized fields

Many perturbative calculations place $g_{\rm YM}$ inside the covariant derivative and give the quadratic kinetic terms unit normalization. Define

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

Then

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

and the quadratic gauge term becomes

$$-\frac{1}{2g_{\rm YM}^2}\operatorname{Tr}F(\mathcal A)^2 \longrightarrow -\frac12\operatorname{Tr}(dA^{\rm can})^2 =-\frac14(dA^{a,{\rm can}})^2.$$

So cubic and quartic vertices carry $g_{\rm YM}$ and $g_{\rm YM}^2$. Two physical quantities provide quick round-trip 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, 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 quoted scalar expectation value is not meaningful across conventions until the kinetic normalization is stated.

In $\mathcal N=1$ language the same multiplet is one vector superfield $V$ and three adjoint chiral fields $\Phi_i$. After canonical normalization and a phase choice for the chirals, the superpotential can be written

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

Anti-Hermitian generators or a different chiral-field phase can move a sign or a factor of $i$. The invariant statement is that the superpotential coefficient is not an independent parameter: its magnitude and its relation to the gauge coupling are fixed by the hidden twelve supercharges.

Minkowski and Euclidean continuations

With the Lorentzian path-integral weight $e^{iS_M}$ and the topological sign above, Wick rotation gives the bosonic Euclidean action

$$\begin{aligned} S_{E,{\rm bos}} ={}&\frac1{g_{\rm YM}^2}\int d^4x_E\, \operatorname{Tr}\!\left[ \frac12F_{\mu\nu}F_{\mu\nu} +D_\mu X^I D_\mu X^I -\frac12[X^I,X^J][X^I,X^J] \right]\\ &-i\theta\nu, \qquad \nu=\frac1{8\pi^2}\int\operatorname{Tr}(F\wedge F). \end{aligned}$$

Every term in the first line is nonnegative for Hermitian fields. The theta term is imaginary, so $S_E$ is generally complex even though its real part is bounded below. Euclidean left- and right-handed spinors are independent complex variables; the Lorentzian Majorana conjugation cannot be imposed unchanged. A localization or instanton formula must therefore state its Euclidean reality condition and contour rather than infer them from the Minkowski card.

Complex coupling and global completion

For the simply connected $SU(N)$ theory on a spin four-manifold, the chosen trace gives $\nu\in\mathbb Z$. The theta angle is then $2\pi$ periodic, and the local complex coupling is

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

For a unit (anti-)self-dual instanton, the real Euclidean action is $8\pi^2/g_{\rm YM}^2$, providing a direct check of the same normalization. These action and instanton conventions are displayed explicitly in Pestun 2012, §2, eq. (2.2), printed p. 9, and §5, printed pp. 36–37, PDF.

This local card is not yet a complete theory object. One must additionally specify a compact global form with Lie algebra $\mathfrak{su}(N)$, the allowed bundle sectors, a maximal mutually local spectrum of genuine Wilson–’t Hooft lines, and any discrete theta datum. For quotient groups, $\nu$ can be fractional on nontrivial bundles; a shift $\tau\mapsto\tau+1$ can then change the discrete datum rather than return to the same global theory. The exact finite charge lattices and their modular transport are developed on line operators, global forms, and discrete theta data; the global-theory orbit figure shows how one local $\mathfrak{su}(2)$ card splits into three globally distinct nodes.

It is useful to keep the dependency explicit:

$$\text{local algebra and action} \;\longrightarrow\; \text{global form and genuine lines} \;\longrightarrow\; \text{candidate duality arrow}.$$

The first item does not determine the second, and neither one proves the third.

Superconformal algebra and normalized local data

At the conformal point, the symmetry 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 supercharges $Q$ and sixteen conformal supercharges $S$. The scalar superconformal primary at the bottom of the stress-tensor multiplet has $\Delta=2$ and transforms in

$$\mathbf{20'}=[0,2,0]$$

of $SU(4)_R$. In the overall-coupling fields, a convenient unnormalized representative is

$$\mathcal O_2^{IJ} =\frac1{g_{\rm YM}^2} \operatorname{Tr}\!\left( X^I X^J-\frac{\delta^{IJ}}6X^KX^K \right).$$

The $1/g_{\rm YM}^2$ compensates $X^I=g_{\rm YM}X_{\rm can}^I$; an additional numerical factor is chosen when its two-point function is set to a particular CFT normalization. Descendants of this multiplet include the stress tensor, supercurrents, $SU(4)_R$ currents, and the exactly marginal operators tangent to the $\tau$ conformal manifold. The multiplet and its correlator constraints are analyzed in Dolan and Osborn 2002, §§2–6, while the representation-theoretic classification is fixed in Córdova, Dumitrescu, and Intriligator 2019, §§2.2.4 and 5.1.4.

The Weyl-anomaly coefficients follow from the exact $\mathcal N=1$ anomaly relations. The gaugino has $R=1$, while the fermion in each of the three chiral multiplets 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

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

gives

$$a=c=\frac14\dim\mathfrak g =\frac{N^2-1}{4} \qquad\text{for }\mathfrak g=\mathfrak{su}(N).$$

The anomaly relations are derived in Anselmi et al. 1998, §§2–3. Because $a$ and $c$ are local anomalies, they depend on the Lie algebra but not on the global form. Partition functions with background flux and genuine-line sectors can still distinguish global theories with the same $a$ and $c$.

What this theory card does not establish

The displayed action is an electric-frame Lagrangian. It does not make magnetic lines local, choose a nonperturbative regulator preserving every desired structure, or prove S-duality. It also does not protect every operator. The Konishi multiplet is long, and its scaling dimension varies with $\tau$.

Nor does the flat-space card determine every curved-background counterterm or non-spin refinement. On a general four-manifold, fermion structure, global form, one-form backgrounds, quadratic refinements, and discrete theta phases must be supplied before a partition function is defined.

Common pitfalls

Mixing the two field normalizations. Writing $D=d-igA$ while retaining an overall $1/g^2$ action double-counts the coupling. Translate the fields, Wilson line, and scalar expectation value together.

Reading a positive potential from its displayed sign alone. For Hermitian scalars the commutator is anti-Hermitian. The Lorentzian Lagrangian contains $+[X,X]^2/2$, while the potential energy contains its negative.

Treating $\tau$ as the whole theory. The same local $\tau$ and Lie algebra can accompany different global forms, genuine lines, and discrete theta data.

Exercises

1. Recover the scalar terms from ten dimensions

Use $F_{\mu I}=D_\mu X^I$ and $F_{IJ}=-i[X^I,X^J]$ to explain the relative factors $1$ and $1/2$ in the four-dimensional bosonic action.

Solution

In $F_{MN}F^{MN}$, the mixed components occur twice, as $(\mu,I)$ and $(I,\mu)$, so the ten-dimensional coefficient $-1/2$ becomes a scalar kinetic coefficient of magnitude $1$. The internal pair is already summed over ordered $I,J$ and retains magnitude $1/2$. Because $F_{IJ}=-i[X^I,X^J]$, the extra factor $(-i)^2=-1$, together with the internal metric signs, yields the displayed Lorentzian commutator term and a nonnegative potential energy.

2. Check the normalization round trip

Starting with $\mathcal A=g_{\rm YM}A^{\rm can}$ and $X=g_{\rm YM}X_{\rm can}$, verify both the Wilson-loop exponent and the W-boson mass formula.

Solution

Substitution gives $i\oint\mathcal A=ig_{\rm YM}\oint A^{\rm can}$, so the two Wilson-loop expressions agree. Likewise $\alpha(X)=g_{\rm YM}\alpha(X_{\rm can})$, and squaring and summing over $I$ gives $M_{W,\alpha}^2=g_{\rm YM}^2\sum_I|\alpha(X_{\rm can}^I)|^2$. Both checks fail if only the gauge field or only the scalar is rescaled.

3. Reproduce the Weyl-anomaly coefficients

Insert the displayed values of $\operatorname{Tr}R$ and $\operatorname{Tr}R^3$ into the anomaly relations and specialize to $SU(N)$.

Solution

Since $\operatorname{Tr}R=0$,

$$a=\frac{9}{32}\frac89\dim\mathfrak g =\frac14\dim\mathfrak g, \qquad c=\frac{9}{32}\frac89\dim\mathfrak g =\frac14\dim\mathfrak g.$$

For $\mathfrak{su}(N)$, $\dim\mathfrak g=N^2-1$, so $a=c=(N^2-1)/4$.

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.
• Córdova, Clay, Thomas T. Dumitrescu, and Kenneth Intriligator. “Multiplets of Superconformal Symmetry in Diverse Dimensions.” Journal of High Energy Physics 03 (2019): 163. doi:10.1007/JHEP03(2019)163. Open PDF.
• Dolan, Francis A., and Hugh Osborn. “Superconformal Symmetry, Correlation Functions and the Operator Product Expansion.” Nuclear Physics B 629 (2002): 3–73. doi:10.1016/S0550-3213(02)00096-2.
• Pestun, Vasily. “Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops.” Communications in Mathematical Physics 313 (2012): 71–129. doi:10.1007/s00220-012-1485-0. Open PDF.