Couplings and conformal invariance

The coupling $g$ has zero engineering dimension, but it can still depend on the energy at which it is measured. We will first test that dependence, then identify the additional reason that the specified $\mathcal N=4$ interactions remain conformal.

Continue with the action and multiplet conventions: four-dimensional $SU(N)$, $N\geq2$, ordinary fundamental trace $\operatorname{Tr}_N(t^at^b)=\delta^{ab}/2$, canonical gauge coupling $g>0$, and no masses. There are sixteen real Poincaré supercharges; an $\mathcal N=1$ presentation displays four. Initially use flat spacetime and the vacuum with zero scalar expectation values. No large-rank or strong-coupling approximation is assumed. The meaning of a beta function and an improved stress tensor is recalled in conformal fixed points.

First calculate the one-loop coefficient

For a representation $R$, define its index by $\operatorname{Tr}_R(t^at^b)=T(R)\delta^{ab}$. The adjoint index of $SU(N)$ is $T(\mathrm{adj})=N$, whereas $T(\mathrm{fund})=1/2$. These are indices of a representation, not its dimension. In particular, one adjoint fermion contributes $N$, not $N^2-1$, to the following coefficient.

Write $\beta_g=\mu\,dg/d\mu$, with $\mu$ a renormalization scale. Counting independent left-handed Weyl fields and real scalars,

$$\beta_g=-\frac{g^3}{16\pi^2}b_0+\text{higher loops}, \qquad b_0=\frac{11}{3}N -\frac23\sum_{\text{Weyl}}T(R_f) -\frac16\sum_{\text{real}}T(R_s). \tag{N04.1}$$

This is the standard vacuum-polarization result, used here as a QFT input. Its scalar coefficient doubles if one instead counts complex scalars. Tachikawa, (3.1.1)–(3.1.3), uses that complex-scalar convention.

The full $\mathcal N=4$ multiplet supplies

Contribution	Number of fields	Contribution to $b_0$
Gauge field, including its gauge-fixing ghosts	one adjoint	$11N/3$
Weyl fermions	four adjoints	$-8N/3$
Real scalars	six adjoints	$-N$

Thus

$$b_0=N\left(\frac{11}{3}-\frac83-1\right)=0. \tag{N04.2}$$

The conjugate of a Lorentzian Weyl field is not another independent Weyl field in this sum. Minahan’s (2.1) writes the same cancellation with coefficient $1/3$ and a paired count of eight; translating that convention is essential.

Regrouping provides a useful check: an $\mathcal N=1$ vector contributes $3N$, and each adjoint chiral contributes $-N$. Alternatively, an $\mathcal N=2$ vector contributes $2N$, and a full adjoint hyper contributes $-2N$. Both descriptions give zero.

This establishes a one-loop cancellation. The formula does not contain the cubic coupling, so it cannot yet test whether that coupling has the special value required by sixteen supercharges.

A nearby theory that passes the first test

Keep the same kinetic terms but replace the cubic coupling by $h$:

$$W_h=2\sqrt2\,h\,\operatorname{Tr}_N \bigl(\Phi_1[\Phi_2,\Phi_3]\bigr) =i\sqrt2\,h\,f^{abc}\Phi_1^a\Phi_2^b\Phi_3^c. \tag{N04.3}$$

Here $h$ is a dimensionless superpotential coupling, not a soft-breaking parameter. The original theory is $h=g$ in the chosen phase convention. Generic $h$ preserves only the four real supercharges of $\mathcal N=1$.

We can test the interaction using the canonical chiral wavefunction anomalous-dimension matrix. Use dimensional reduction with modified minimal subtraction, denoted $\overline{\mathrm{DR}}$. Define its sign by $\gamma=-\frac12\mu\,d\log Z/d\mu$ for a diagonal wavefunction factor $Z$ in holomorphic field normalization. A canonically normalized cubic then obeys $\beta_h=h(\gamma_1+\gamma_2+\gamma_3)$. These $\gamma_i$ describe elementary chiral renormalization in this calculation; they are not the dimensions of arbitrary gauge-invariant composite operators.

Combine a chiral label and color index into $I=(i,a)$, and write $W=Y_{IJK}\Phi_I\Phi_J\Phi_K/6$. The symmetric tensor for (N04.3) is $Y_{(ia)(jb)(kc)}=i\sqrt2h\,\epsilon_{ijk}f^{abc}$, with $\epsilon_{123}=1$. Exchanging two combined indices changes both antisymmetric factors, leaving $Y$ symmetric. The one-loop result is

$$\gamma_I{}^L= \frac1{16\pi^2} \left(\frac12\sum_{J,K}Y_{IJK}Y_{LJK}^* -2g^2N\delta_I{}^L\right) +\text{higher loops}. \tag{N04.4}$$

The tensor normalization, symmetry factor and beta-function relation are those of Martin–Vaughn, (1.1), (2.7), (2.10). No soft terms are turned on.

Now evaluate the contraction rather than infer it from field counting:

$$\begin{aligned} \frac12\sum_{j,k,b,c} Y_{(ia)(jb)(kc)}Y_{(\ell d)(jb)(kc)}^* &=|h|^2 \left(\sum_{j,k}\epsilon_{ijk}\epsilon_{\ell jk}\right) \left(\sum_{b,c}f^{abc}f^{dbc}\right)\\ &=2N|h|^2\,\delta_{i\ell}\delta^{ad}. \end{aligned} \tag{N04.5}$$

The two is the number of ordered choices for the other two chiral labels. Hence

$$\gamma_1=\gamma_2=\gamma_3 =\frac{N}{8\pi^2}(|h|^2-g^2)+\cdots, \qquad \beta_h=\frac{3Nh}{8\pi^2}(|h|^2-g^2)+\cdots. \tag{N04.6}$$

For a concrete weakly coupled $SU(2)$ example, set $h=\sqrt2g$. Then $\beta_h/h=3g^2/(4\pi^2)$ at one loop, while (N04.2) still gives $\beta_g^{\text{one loop}}=0$. The theory runs. At $h=g$, the Yukawa and gauge contributions to (N04.4) cancel as well. This additional check is still a finite-order calculation, not an all-order proof.

What protects the full theory

The extra argument uses the interaction relations enforced by extended supersymmetry. A useful route is the $\mathcal N=2$ description: one vector multiplet and one massless adjoint hyper.

In a holomorphic Wilsonian scheme, the complex gauge coupling $\tau$ is the coefficient of a chiral gauge-kinetic integral. Wilsonian means that modes above a scale have been integrated out while lower-energy fields remain. Supersymmetry constrains this coefficient to depend holomorphically on coupling superfields. Ordinary perturbation theory is insensitive to the constant theta angle. An $L$-loop correction to $\tau$ would scale as $(\operatorname{Im}\tau)^{1-L}$; for $L>1$ this is not holomorphic. Only the one-loop logarithm is allowed.

There is an essential second step. In a generic $\mathcal N=1$ theory, converting from holomorphic to canonically normalized fields introduces running wavefunction factors. The preceding example demonstrates why those factors cannot be ignored. With the $\mathcal N=2$ interaction and $SU(2)_R$ Ward identities maintained, the vector and hyper kinetic and interaction coefficients are tied together. They do not supply that independent running. The resulting perturbative coupling beta function is one-loop exact, so $b_0=0$ implies its vanishing to all orders. The assumptions and this distinction are spelled out in Tachikawa, §3.1 after (3.1.11).

For the full $\mathcal N=4$ theory, the established stronger result is a family of conformal theories with one complex exactly marginal coupling: varying it preserves conformality and all sixteen Poincaré supercharges. The operator statement behind the protection is that the complete supersymmetric coupling operator is a four-supercharge descendant of the protected dimension-two scalar multiplet containing the stress tensor. Each supercharge raises dimension by $1/2$, so that descendant has dimension four. Its gauge-field term is only part of the operator; matter terms and possible equation-of-motion or total-derivative terms must be retained. Minahan, (4.14)–(4.17), gives this protection argument.

Here the protected multiplet and its supersymmetric Ward identities are additional structural input. A generic dimension-four operator can develop a beta function through its interactions, so its engineering dimension—or even its dimension at one fixed point—would not suffice. Likewise, a perturbative cancellation by itself does not rule out nonperturbative effects. Such effects can affect observables without making this protected coupling run. We are using the $\mathcal N=4$ nonrenormalization result, not deriving it from the arithmetic in (N04.2).

Exactly marginal does not mean that correlation functions are independent of the coupling. Nor does $\beta_g=0$ force all composite anomalous dimensions to vanish. Conformal symmetry fixes the form of a correlator while allowing its dimensions and coefficients to depend on which conformal theory is chosen.

Normalize the theta angle

To specify the complex coupling, temporarily continue the gauge sector to oriented Euclidean four-space. Use the Hermitian connection $a=gA$ and its two-form curvature $f=da-ia\wedge a$. Thus the components satisfy $f_{\mu\nu}=gF_{\mu\nu}$ in the canonical convention of the action lesson.

For finite-action fields extending to an $SU(N)$ bundle on the compactification $S^4$, define

$$\begin{aligned} k&=\frac1{8\pi^2}\int\operatorname{Tr}_N(f\wedge f)\in\mathbb Z,\\ S_{E,\mathrm{gauge}} &=\frac1{g^2}\int\operatorname{Tr}_N(f\wedge *f)-i\theta k,\\ \tau&=\frac{\theta}{2\pi}+\frac{4\pi i}{g^2}. \end{aligned} \tag{N04.7}$$

The orientation fixes the sign of $k$, and $*$ is the Euclidean Hodge star. Since $f=f_{\mu\nu}dx^\mu\wedge dx^\nu/2$, the first integral is $\int d^4x\,\operatorname{Tr}_N(f_{\mu\nu}\widetilde f_{\mu\nu})/(16\pi^2)$, where $\widetilde f_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}f_{\rho\sigma}/2$. The real action is therefore $\int d^4x\,\operatorname{Tr}_N(f_{\mu\nu}f_{\mu\nu})/(2g^2)$, agreeing with the canonical kinetic term.

A unit self-dual field has action $8\pi^2/g^2-i\theta$ and gauge-sector weight $e^{2\pi i\tau}$. Shifting $\theta$ by $2\pi$ leaves every integer-$k$ sector unchanged. These normalizations follow Tachikawa, (3.2.1)–(3.2.6). Fermion zero modes and the chosen observable still determine whether a sector contributes; its exponential is not a complete instanton amplitude.

Both $g$ and $\theta$ are dimensionless and nonrunning on the $\mathcal N=4$ family. Locally, $\tau$ lies in the upper half-plane. This description does not identify every point related by a proposed duality: global gauge group and allowed line operators must also be specified.

Replacing $SU(N)$ by $U(N)$ adds an identity-generator multiplet. All local commutator interactions vanish on that center, so it is a free $\mathcal N=4$ Abelian multiplet alongside the interacting traceless sector. Its free kinetic normalization is not generated by substituting $N$ into a non-Abelian beta coefficient. Moreover $U(N)=(SU(N)\times U(1))/\mathbb Z_N$ globally; local decoupling alone does not erase restrictions on bundles and line operators.

Separate the theory, vacuum, and background

The conformal statement concerns the massless theory with its specified couplings. At the zero-scalar vacuum in flat space, conformal symmetry is unbroken. A vacuum with a nonzero dimensionful scalar expectation value can break that symmetry spontaneously even though the beta function remains zero. A temperature, boundary, mass deformation, or external geometric radius introduces still other scales. They should not be mistaken for quantum running of $g$.

Finally, flat-space conformality gives a traceless improved stress tensor in separated-point Ward identities, with the usual contact terms at operator insertions. It does not imply that the partition function is invariant under every position-dependent rescaling of a curved background metric. In four dimensions there can be a Weyl anomaly: a local curvature contribution to the trace and to the metric response of the partition function. Simmons-Duffin, §3 footnote k and §6.4 footnotes w,y, explicitly separates these statements. This background response is compatible with a vanishing coupling beta function. Preserving supersymmetry on a chosen curved geometry requires a separate background construction.

Exercises

1. Verify the adjoint factor and locate the free center

For the normalized $SU(N)$ generators, use $\sum_a t^aMt^a=\frac12\operatorname{Tr}_N(M)1_N-M/(2N)$. Derive $\sum_a[t^a,[t^a,M]]=NM$ for traceless $M$, and recover the adjoint factor in (N04.5). What changes for $M=1_N$ and the central generator of $U(N)$?

Hint 1

Set $M=1_N$ in the supplied completeness identity to obtain $\sum_a t^at^a$.

Hint 2

Expand the double commutator before imposing tracelessness. Then take $M=t^d$ and use the structure constants twice.

Solution

The sum of squares is $(N^2-1)1_N/(2N)$. Consequently

$$\begin{aligned} \sum_a[t^a,[t^a,M]] &=\frac{N^2-1}{N}M -2\left(\frac12\operatorname{Tr}_N(M)1_N-\frac{M}{2N}\right)\\ &=NM-\operatorname{Tr}_N(M)1_N. \end{aligned}$$

For traceless $M$ this is $NM$. Substituting $M=t^d$ and $[t^a,t^d]=if^{adc}t^c$ gives the adjoint quadratic operator $N\delta^{de}$, equivalently $\sum_{b,c}f^{abc}f^{dbc}=N\delta^{ad}$. Thus all three multiplet groupings in the main calculation use the same factor.

For $M=1_N$, the two terms cancel. The normalized central generator $t^0=1_N/\sqrt{2N}$ commutes with every generator, so its adjoint action and all cubic interactions vanish. It contributes a free Abelian sector, not an extra interacting color direction with Casimir $N$. This also explains why $SU(1)$ is not a nontrivial interacting example.

2. Follow the detuned coupling over a controlled interval

Take real $h>0$ and work only to the one-loop accuracy of (N04.6). Treat $g$ as constant at this accuracy, set $t=\log(\mu/\mu_0)$ and $u=h^2/g^2$, and solve the running for $u(0)=u_0>0$. Is $u=1$ attractive toward lower energies? State why this calculation is not an exact global flow.

Hint 1

Differentiate $h^2/g^2$ and separate variables in $du/[u(u-1)]$.

Hint 2

The integral is $\log|(u-1)/u|$. Lower energies have $t<0$. Check both $u_0<1$ and $u_0>1$.

Solution

Writing $c=3Ng^2/(4\pi^2)>0$ gives $du/dt=c\,u(u-1)$. Integration yields

$$\frac{u-1}{u}=\frac{u_0-1}{u_0}e^{ct}, \qquad u(t)=\frac1{1-(1-u_0^{-1})e^{ct}}.$$

For $t<0$, either positive initial value moves toward one within this truncated equation. Linearizing $u=1+\delta$ gives $\delta(t)=\delta(0)e^{ct}+\cdots$, confirming local infrared attraction. The exactly tuned solution stays at one.

For $u_0>1$ the formal ultraviolet solution has a pole at $t=c^{-1}\log[u_0/(u_0-1)]$. Perturbation theory fails before that pole, so it is not an established physical singularity. The approximation requires $Ng^2/(16\pi^2)$ and $Nh^2/(16\pi^2)$ small, and an interval over which neglected higher-loop effects remain controlled. In the detuned theory the gauge coupling itself can run beyond one loop. The mathematical limit $t\to-\infty$ of this truncated formula therefore does not by itself prove the exact infrared endpoint.

3. Check the instanton weight without introducing a running scale

Use the positive Euclidean norm $\|f\|^2=\int\operatorname{Tr}_N(f\wedge *f)$ and $*^2=1$ on two-forms to prove $\|f\|^2\geq8\pi^2|k|$. Find the gauge-sector weights of a self-dual field with $k>0$ and an anti-self-dual field with $k<0$. Finally rescale a localized field by $a_\rho(x)=\rho^{-1}a_1(x/\rho)$, with $\rho$ a positive length. Does its classical action depend on $\rho$?

Hint 1

Expand $\|f\mp*f\|^2$. The cross term is $\int\operatorname{Tr}_N(f\wedge f)$.

Hint 2

Keep $e^{i\theta k}$ for either sign of $k$. Under the rescaling, the curvature has two inverse powers of $\rho$ and the volume has four positive powers.

Solution

Since $*$ is an isometry, $\|f\mp*f\|^2=2\|f\|^2\mp16\pi^2k\geq0$. Choosing the appropriate sign proves the bound. Self-duality saturates it for $k>0$, giving

$$e^{-S_E}= \begin{cases} e^{-8\pi^2k/g^2+i\theta k}=e^{2\pi i\tau k},& k>0,\ f=*f,\\ e^{-8\pi^2|k|/g^2-i\theta|k|} =e^{-2\pi i\bar\tau|k|},& k<0,\ f=-*f. \end{cases}$$

Both weights are suppressed at weak coupling; the minus sign in the antiholomorphic exponent is needed for that check. Both are periodic under $\theta\mapsto\theta+2\pi$.

For the rescaled field, $f_\rho(x)=\rho^{-2}f_1(x/\rho)$ in components. The Jacobian $d^4x=\rho^4d^4(x/\rho)$ cancels the curvature factors in both action integrals. Thus the classical gauge action and $k$ are independent of $\rho$. The size is a configuration parameter, not a dynamically generated scale in the coupling. Quantum measures, zero modes, and operator insertions still require analysis; classical scale independence alone would not have proved $\beta_g=0$.

You can now distinguish three claims: the field content cancels one loop, the specified extended supersymmetry protects the coupling, and a particular vacuum or background can still introduce a scale. Commuting matrices and the Coulomb branch asks how scalar expectation values do that while leaving the underlying theory unchanged.

For a different continuation, electric–magnetic duality transforms the complex coupling together with quantized flux labels. Its first calculation uses compact Maxwell theory to distinguish a change of description from renormalization-group evolution.