Perturbative Chiral and Gauge Anomalies

In four dimensions, the perturbative anomaly of a chiral fermion factors into a universal spacetime calculation and a symmetric trace over its internal representation. The trace is the part that distinguishes one matter spectrum from another. After every fermion has been written with the same chirality, the full coefficient is obtained simply by adding those representation invariants with the correct spectator multiplicities.

For a compact gauge algebra built from simple and $U(1)$ factors, four kinds of local coefficient can survive: $G^3$, $G^2U(1)$, $U(1)^3$, and mixed $U(1)$–gravity. If every displayed factor is dynamical and no additional cancelling sector is present, every one of those coefficients must vanish. When all named connections are fixed backgrounds for genuine exact global symmetries, a nonzero class instead records an ‘t Hooft anomaly. These local tests are necessary for gauging, but they do not decide large-gauge, torsion, or global-form obstructions.

Required background. Regulated Jacobians and Measure Variation supplies the regulated chiral measure density, the representative-versus-class distinction, and the local/global stopping rule. Representations, Intertwiners, Invariants, and Tensor Decomposition supplies conjugate representations, tensor products, and invariant tensors.

Helpful background. Characteristic Classes and Chern–Weil Theory supplies the trace-sensitive Chern-character normalization. Fredholm and Dirac Index Theorems and Zero-Mode Counting supplies the twisted index behind the six-form used below.

A chiral triangle exposes the representation tensor

The calculation will use one convention throughout. In particular, the phenomenological all-left-handed convention is not the same as the preceding page’s positive-Euclidean-chirality example.

datum	convention	consequence
spacetime for the coefficient calculation	closed oriented Euclidean spin four-manifold $X$, with $\epsilon_E^{1234}=+1$	no boundary heat coefficient or Ward flux is included
chirality bookkeeping	list every physical Lorentzian field as left-handed; with the inherited Wick map, $P_L=(1-\gamma_5)/2=P_-$	a right-handed field in $(R,q)$ is replaced by a left-handed field in $(\bar R,-q)$
gauge algebra	$\mathfrak g=\bigoplus_\alpha\mathfrak g_\alpha\oplus\mathfrak u(1)^r$, with compact simple $\mathfrak g_\alpha$	distinct simple factors act on separate tensor factors
generators	Hermitian $T^a$, $[T^a,T^b]=if^{abc}T^c$	the conjugate representation uses $T_{\bar R}^a=-(T_R^a)^T$
Abelian normalization	compact $U(1)$ charges $q^I\in\mathbb Z$ in a chosen minimal-charge normalization, $c_I=f_I/(2\pi)$	a rescaling of the $U(1)$ generator rescales coefficients but not their vanishing
gravitational form	$p_1(TX)=-(8\pi^2)^{-1}\operatorname{tr}_{\mathrm{vec}}(\mathcal R\wedge\mathcal R)$	the mixed term is compared in a fixed diffeomorphism- and local-Lorentz-covariant representative
regime	perturbative local fermion anomaly, with no compensating inflow or Green–Schwarz sector	cancellation below is a statement about the declared four-dimensional spectrum

Let $R$ be a representation of one simple factor. Define its quadratic and cubic invariants by

$$\operatorname{tr}_R(T^aT^b)=T(R)\,\delta^{ab},$$

and

$$\mathcal D_R^{abc} :=\operatorname{tr}_R\!\left(T^{(a}T^bT^{c)}\right) =\frac12\operatorname{tr}_R\!\left(T^a\{T^b,T^c\}\right).$$

The two orientations of a chiral triangle have the internal group factor

$$\begin{aligned} &\operatorname{tr}_R(T^aT^bT^c) +\operatorname{tr}_R(T^aT^cT^b) \\ &\hspace{4em} =\operatorname{tr}_R\!\left(T^a\{T^b,T^c\}\right) =2\mathcal D_R^{abc}. \end{aligned}$$

The loop integral and Lorentz tensor are universal; the chiral spectrum enters through $\mathcal D_R^{abc}$. In a triangle with two gravitons, the gravitational vertices act as the identity on internal indices, so the group factor instead reduces to $\operatorname{tr}_R T^a$. It vanishes for a simple Lie algebra but equals $q\dim R$ for a $U(1)$ generator.

Conjugation gives the useful sign check

$$T(\bar R)=T(R), \qquad \mathcal D_{\bar R}^{abc}=-\mathcal D_R^{abc}.$$

Consequently, real and pseudoreal representations have zero local cubic tensor. This explains, for example, why an $SU(2)$ doublet has no perturbative $SU(2)^3$ anomaly even though it can still have a global anomaly. Where a nonzero reference tensor exists, define the cubic index by

$$\mathcal D_R^{abc}=A(R)\mathcal D_F^{abc}, \qquad A(F)=1.$$

Among compact simple Lie algebras, only $\mathfrak{su}(N)$ with $N\geq3$ admits the relevant cubic invariant. The symbol $A(R)$ is therefore not defined by dividing by a vanishing reference tensor for $SU(2)$ or another factor without that invariant. A complex representation alone is not sufficient—complex representations of $E_6$, for example, still have $\mathcal D_R^{abc}=0$. By contrast, the mixed coefficient $G^2U(1)$ uses $T(R)$ and can occur for any simple factor. The trace classification, the conversion of right-handed fields to left-handed conjugates, and the spectator factors used below are derived in Bilal 2008, §§ 7.1–7.2, arXiv v1, pp. 45–50, eqs. (7.1)–(7.16), Open PDF. Bilal absorbs couplings into the generators and uses a different metric convention, so only the invariant trace statements—not an unconverted component sign—are imported here.

The six-form organizes the same coefficients

The same group theory is packaged by the index polynomial. For species $i$, let

$$\mathcal A_i =-i\left( \sum_\alpha g_\alpha A_\alpha^aT_{i\alpha}^a +\sum_I q_i^I a_I\,\mathbf 1 \right)$$

be the anti-Hermitian bundle connection and define the normalized Hermitian curvature

$$\begin{aligned} X_i:=\frac{i\mathcal F_i}{2\pi} &=\sum_\alpha \frac{g_\alpha F_\alpha^a}{2\pi}T_{i\alpha}^a +\sum_I q_i^Ic_I\,\mathbf 1, \\ \mathcal F_i&=\mathrm d\mathcal A_i+\mathcal A_i^2. \end{aligned}$$

Here

$$F_\alpha^a =\mathrm dA_\alpha^a +\frac{g_\alpha}{2}f_\alpha^{abc} A_\alpha^b\wedge A_\alpha^c, \qquad f_I=\mathrm da_I.$$

If $B_I$ is instead a canonically normalized Abelian gauge field with coupling $g_I$, then $a_I=g_IB_I$ and $f_I=g_I\mathrm dB_I$. This keeps the charge lattice in $q_i^I$ and makes every coupling placement explicit.

All products of differential forms in this section are wedge products. A positive-Euclidean-chirality fermion has $[\widehat A(TX)\operatorname{ch}(E_i)]_6$. The physical left-handed field used here has $P_L=P_-$ and therefore the opposite sign:

$$\boxed{ I_{6,i}^{L} =-\frac16\operatorname{tr}_{\mathcal R_i}(X_i^3) +\frac{p_1(TX)}{24}\operatorname{tr}_{\mathcal R_i}(X_i) }.$$

Choosing all right-handed fields instead would reverse the whole polynomial, but none of the zero conditions below. The normalization follows from Álvarez-Gaumé and Vázquez-Mozo 2024, § 3, arXiv v2, pp. 5–8, eqs. (11), (17)–(19), and (22)–(23), Open PDF, after translating their positive-chirality formula through the inherited Wick convention. This page uses the six-form only to generate coefficients. Its index proof, descent to a consistent four-dimensional Ward identity, and inflow interpretation are separate questions.

Although the physical background is four-dimensional, $I_6$ is a universal formal degree-six characteristic form—equivalently, it is evaluated on an auxiliary extension or family with enough form degree. It is not an ordinary nonzero six-form integrated directly over the four-manifold $X$. This $D+2$-dimensional role and its descent interpretation are stated in Álvarez-Gaumé and Vázquez-Mozo 2024, § 2, arXiv v2, pp. 4–5, eqs. (4)–(10), Open PDF.

Product gauge groups reduce to four trace sums

Write the internal representation of species $i$ as

$$\mathcal R_i=\bigotimes_\alpha R_{i\alpha}, \qquad n_i:=\prod_\alpha\dim R_{i\alpha}, \qquad m_{i\alpha}:=\frac{n_i}{\dim R_{i\alpha}}.$$

Thus $m_{i\alpha}$ counts the spectator states seen by the factor $G_\alpha$. Set $x_\alpha=g_\alpha/(2\pi)$. Expanding the cubic trace gives

$$\begin{aligned} \operatorname{tr}_{\mathcal R_i}(X_i^3) ={}&\sum_\alpha m_{i\alpha}x_\alpha^3 \mathcal D_{R_{i\alpha}}^{abc} F_\alpha^aF_\alpha^bF_\alpha^c \\ &+3\sum_{\alpha,I}m_{i\alpha}q_i^Ic_Ix_\alpha^2 T_\alpha(R_{i\alpha})F_\alpha^aF_\alpha^a \\ &+n_i\sum_{I,J,K}q_i^Iq_i^Jq_i^Kc_Ic_Jc_K, \end{aligned}$$

while the linear trace is

$$\operatorname{tr}_{\mathcal R_i}(X_i) =n_i\sum_Iq_i^Ic_I.$$

A term with one generator from a simple factor vanishes because $\operatorname{tr}T^a=0$. That removes $G_\alpha U(1)^2$, mixed gravity–$G_\alpha$, and terms involving one generator from each of two different simple factors.

Summing over all left-handed species defines four coefficient families:

$$C_\alpha^{abc} :=\sum_i m_{i\alpha}\mathcal D_{R_{i\alpha}}^{abc},$$ $$C_{\alpha I} :=\sum_i m_{i\alpha}q_i^I T_\alpha(R_{i\alpha}),$$ $$C_{IJK} :=\sum_i n_iq_i^Iq_i^Jq_i^K,$$

and

$$C_I^{\mathrm{grav}} :=\sum_i n_iq_i^I.$$

In these definitions $C_{IJK}$ is symmetric in all three Abelian labels. Substitution into the six-form gives the convention-complete result

$$\begin{aligned} I_6^L ={}&-\frac16\sum_\alpha x_\alpha^3C_\alpha^{abc} F_\alpha^aF_\alpha^bF_\alpha^c \\ &-\frac12\sum_{\alpha,I}x_\alpha^2C_{\alpha I} c_IF_\alpha^aF_\alpha^a \\ &-\frac16\sum_{I,J,K}C_{IJK}c_Ic_Jc_K +\frac1{24}p_1(TX)\sum_I C_I^{\mathrm{grav}}c_I. \end{aligned}$$

The corresponding local tests are therefore:

channel	coefficient condition when all named factors are gauged
$G_\alpha^3$	$C_\alpha^{abc}=0$ for every $a,b,c$
$G_\alpha^2U(1)_I$	$C_{\alpha I}=0$ for every $\alpha,I$
$U(1)_IU(1)_JU(1)_K$	$C_{IJK}=0$ for every symmetric triple $I,J,K$
gravity–gravity–$U(1)_I$	$C_I^{\mathrm{grav}}=0$ for every $I$

The positive couplings $g_\alpha$ do not affect whether these tensors vanish. There is no pure perturbative gravitational anomaly for spin-$1/2$ fields in four dimensions: the degree-six part of $\widehat A(TX)$ has no term without a gauge curvature. The complete trace classification is given in Bilal 2008, §§ 7.1–7.2, arXiv v1, pp. 45–50, eqs. (7.1)–(7.16), Open PDF. A modern regulator and counterterm cross-check of the pure-gauge trace conditions appears in Cohen, Lu, and Zhang 2023, § 4.2 and §§ 5.1–5.4, arXiv v1, pp. 24–36, eqs. (4.15)–(4.20), (5.10)–(5.14), and (5.25)–(5.33), Open PDF. That second source is restricted to flat four-dimensional gauge backgrounds and does not supply the gravitational or global claims.

Gauge consistency depends on which currents are gauged

The equations above classify a local anomaly class. Their physical verdict depends on which connections are integrated over.

role of the connection	meaning of a nonzero class
fixed background for a genuine global symmetry	an ‘t Hooft anomaly; the standalone QFT may exist, but gauging that symmetry is obstructed and the class must be reproduced along the RG flow
dynamical gauge connection	a failure of the proposed redundancy unless the complete system contains another sector or mechanism that cancels the same class
some factors dynamical and others background	the representative must preserve every gauged Ward identity; any remaining mixed variation appears in the background/global current

A local Bardeen counterterm can move a mixed representative between Ward identities, but it cannot erase a nontrivial class while preserving all of the symmetries that are simultaneously gauged. The orientation-level redistribution of the mixed gauge–gravitational term is described in Bilal 2008, § 11.3.2, arXiv v1, pp. 92–93, eqs. (11.29)–(11.33), Open PDF. The background-versus-dynamical verdict is stated in Bhardwaj et al. 2024, § 4.2.1, arXiv v2, pp. 67–69, Open PDF.

There is one more mixed-case check. After the dynamical $G$ Ward identity has been preserved, a background $U(1)$ current with divergence proportional to $\operatorname{tr}(F_G\wedge F_G)$ may describe an ABJ-broken continuous symmetry, perhaps leaving only a subgroup. It should not automatically be renamed an ‘t Hooft anomaly of an exact continuous $U(1)$. The exact global symmetry must be identified before applying the background-field verdict.

The preceding page’s heat-kernel insertion was covariant. A gauge Ward identity is instead the variation of one effective action and therefore uses a consistent representative. The two representatives have different local normalizations, but the representation tensors whose vanishing removes the class are the same. Their explicit Bardeen–Zumino relation is developed in Consistent and Covariant Anomalies.

Controlled spectra fix signs and multiplicities

The fastest consistency check is a vectorlike pair. In all-left-handed notation it is

$$(R,q)\oplus(\bar R,-q).$$

Its four contributions cancel separately:

$$\begin{aligned} \mathcal D_R+\mathcal D_{\bar R}&=0, \\ qT(R)-qT(\bar R)&=0, \\ (\dim R)\bigl(q^3+(-q)^3\bigr)&=0, \\ (\dim R)\bigl(q+(-q)\bigr)&=0. \end{aligned}$$

Vectorlike matter is sufficient but not necessary. For a purely Abelian spectrum with no spectator multiplicity, the two conditions are

$$\kappa_1:=\sum_iq_i=0, \qquad \kappa_3:=\sum_iq_i^3=0.$$

Two short spectra separate them:

left-handed charges	$\kappa_1$	$\kappa_3$	local verdict
$\{1,1,-2\}$	$0$	$-6$	mixed gravitational term cancels, cubic gauge anomaly remains
$\{1,5,-7,-8,9\}$	$0$	$0$	both local $U(1)$ tests pass; there is no opposite-charge pairing

The second row is genuinely chiral as a charge spectrum. Its arithmetic is

$$1+5-7-8+9=0,$$

and

$$1+125-343-512+729=0.$$

Now consider a product group. Take four left-handed $SU(2)$ doublets with $U(1)$ charges

$$\{1,1,1,-3\}$$

and three $SU(2)$ singlets with charges

$$\{-2,-2,4\}.$$

For a doublet, $T(\mathbf2)=1/2$, the $U(1)$ multiplicity is $n_i=\dim\mathbf2=2$, and the spectator factor in $SU(2)^2U(1)$ is $m_{i,SU(2)}=1$. Since every $SU(2)$ representation has zero cubic tensor, the pure $SU(2)^3$ coefficient vanishes. The other three checks are

$$C_{SU(2),U(1)} =\frac12(1+1+1-3)=0,$$ $$C_{U(1)}^{\mathrm{grav}} =2(1+1+1-3)+(-2-2+4)=0,$$

and

$$\begin{aligned} C_{U(1)U(1)U(1)} &=2(1+1+1-27)+(-8-8+64) \\ &=-48+48=0. \end{aligned}$$

This is a chiral spectrum that passes every perturbative local test for $SU(2)\times U(1)$. It also has an even number of fundamental doublets, so it passes the familiar mod-two test on an ordinary spin background. That last sentence is deliberately not a complete classification of all $SU(2)$ global anomalies or global gauge-group forms.

A non-Abelian chiral cancellation can also occur without an Abelian factor. For traceless $X$ in the fundamental of $SU(N)$,

$$\operatorname{tr}_{\Lambda^2\mathbf N}e^{tX} =\frac12\left[ (\operatorname{tr}_{\mathbf N}e^{tX})^2 -\operatorname{tr}_{\mathbf N}e^{2tX} \right].$$

Comparing the $t^3$ coefficients gives

$$\operatorname{tr}_{\Lambda^2\mathbf N}X^3 =(N-4)\operatorname{tr}_{\mathbf N}X^3.$$

Thus the two-index antisymmetric $\mathbf{10}$ of $SU(5)$ has the same cubic index as the $\mathbf5$, whereas $\overline{\mathbf5}$ has the opposite one. The chiral combination

$$\mathbf{10}\oplus\overline{\mathbf5}$$

therefore cancels its local $SU(5)^3$ coefficient. No Standard Model hypercharge assignment or phenomenology is being inferred from this isolated group-theory check.

Computational companion. The Anomaly Descent and Inflow calculation has not been implemented or validated. The trace identities and exact arithmetic needed here are therefore worked directly on this page; no external computed output is being used as evidence.

Local cancellation is not the end of consistency

Vanishing of all four coefficient families has a precise but limited meaning: the local perturbative anomaly class for infinitesimal transformations of the declared Lie algebra vanishes. Several further questions remain logically independent.

further question	why the trace sums do not answer it
does every multiplet represent the actual global gauge group?	Lie-algebra generators do not encode center quotients, charge lattices, or bundle sectors
is there a large-gauge or torsion anomaly?	local curvature polynomials see infinitesimal data, not determinant phases around noncontractible loops
are boundary conditions and surface charges compatible?	the closed-manifold calculation contains neither boundary domains nor inflow or edge degrees of freedom
is the quantum gauge theory otherwise well defined?	anomaly cancellation does not prove renormalizability, positivity, unitarity, or nonperturbative existence

The decisive counterexample is one left-handed $SU(2)$ fundamental doublet. Its perturbative cubic tensor vanishes because the representation is pseudoreal, yet the theory has the familiar $\mathbb Z_2$ global gauge anomaly. This is Witten’s original result Witten 1982, pp. 324–328. For higher isospin and more general tangential structures, the parity statement requires additional qualifications; see Wang, Wen, and Witten 2019, Introduction, printed pp. 2–3, and § 2.2, printed pp. 6–9, eqs. (2.3)–(2.9), Open PDF. The page on Global and Torsion Anomalies develops those tests.

Likewise, if a manifold has a boundary, a bulk coefficient cannot be discarded silently. One must specify the allowed transformations, boundary conditions, and any compensating boundary or inflow system. The verdict then applies to the complete bulk–boundary problem rather than to the isolated bulk spectrum.

Common pitfalls

Using the quadratic index for a cubic anomaly. $T(R)$ controls $G^2U(1)$, whereas $\mathcal D_R^{abc}$ controls $G^3$. A representation can have nonzero $T(R)$ and zero cubic tensor, as the $SU(2)$ doublet does.

Forgetting spectator multiplicities. A field in $(R_\alpha,R_\beta)_q$ contributes $\dim R_\beta$ copies to a $G_\alpha$ trace and $\dim R_\alpha\dim R_\beta$ copies to a purely Abelian or mixed gravitational sum. Omitting those dimensions changes the theory being tested.

Mixing left- and right-handed lists. Either sum left minus right in the same representations, or conjugate every right-handed field and use one left-handed list. Doing both conjugation and an additional minus sign counts the chirality twice.

Treating every nonzero background anomaly as an inconsistency. A fixed background probes a global symmetry, so a nonzero class is allowed ‘t Hooft data. It becomes an obstruction when that symmetry is promoted to a dynamical gauge redundancy without a cancelling sector.

Calling a counterterm a cancellation of the class. A local counterterm can move a mixed variation among currents or remove a trivial representative. It cannot remove a nontrivial class while preserving every Ward identity one intends to gauge.

Equating local cancellation with full anomaly freedom. The four trace sums know only the perturbative Lie-algebra problem. Global form, large transformations, torsion, boundaries, and inflow require separate tests.

Check your understanding

1. Starting from $T_{\bar R}^a=-(T_R^a)^T$, show that the quadratic index is unchanged but the cubic tensor changes sign.

   Solution

   Two minus signs occur in $\operatorname{tr}_{\bar R}(T^aT^b)$, and transposition reverses the order inside a trace without changing its value. Hence $T(\bar R)=T(R)$. Three generators give three minus signs; after using trace cyclicity and the symmetric definition of $\mathcal D$, one obtains $\mathcal D_{\bar R}^{abc}=-\mathcal D_R^{abc}$.

2. Verify the two local $U(1)$ tests for the left-handed charge spectrum $\{1,1,-2\}$. Which one fails?

   Solution

   The linear sum is $1+1-2=0$, so the mixed gravitational coefficient vanishes. The cubic sum is $1+1-8=-6$, so the $U(1)^3$ gauge coefficient does not vanish. The spectrum cannot define a standalone dynamical $U(1)$ gauge theory without additional cancellation.

3. A left-handed field transforms as $(R_A,R_B)_q$. What multiplicities accompany its $G_A^3$, $G_A^2U(1)$, and $U(1)^3$ coefficients?

   Solution

   The $G_A$ generator acts as the identity on $R_B$, so both $G_A^3$ and $G_A^2U(1)$ acquire the spectator factor $\dim R_B$. The Abelian generator acts on the full tensor product, so $U(1)^3$ acquires $\dim R_A\dim R_B$.

4. Use the character identity for $\Lambda^2\mathbf N$ to find its cubic index relative to the fundamental. What happens for $N=3,4,5$?

   Solution

   Comparing the $t^3$ coefficients gives $A(\Lambda^2\mathbf N)=N-4$ when the fundamental cubic tensor is nonzero and normalized to $A(\mathbf N)=1$. For $N=3$, the antisymmetric representation is $\bar{\mathbf3}$ and $A=-1$; for $N=4$, it is the real $\mathbf6$ and $A=0$; for $N=5$, it is the $\mathbf{10}$ and $A=1$. The symbol $A(R)$ should not be defined by division when the reference cubic tensor itself vanishes.

5. Why does $\mathcal D_{\mathbf2}^{abc}=0$ not make a single left-handed $SU(2)$ doublet anomaly-free?

   Solution

   The cubic tensor tests only the local perturbative anomaly under transformations connected to the identity. A single doublet has a mod-two phase under the nontrivial large-gauge class detected by Witten’s global test. The local polynomial therefore vanishes while the global determinant phase does not.

What to carry forward

• Consistent and Covariant Anomalies derives the current improvement and integrability distinction only oriented here.
• Wess–Zumino Consistency and Descent turns the six-form into a consistent local Ward variation.
• Anomaly Polynomials and Inflow develops extension dependence and cancellation by a complete bulk–boundary system.
• Global and Torsion Anomalies tests phases invisible to all four perturbative trace sums.
• Gauge-Anomaly Cancellation and Quantum Consistency performs the one-generation Standard Model hypercharge and multiplicity sums.
• Local Anomaly Descent and Wess–Zumino Consistency supplies the theorem-first local-cohomology classification.

References

• Álvarez-Gaumé, Luis, and Miguel Á. Vázquez-Mozo. “Anomalies and the Green–Schwarz Mechanism.” In Handbook of Quantum Gravity, edited by Cosimo Bambi, Leonardo Modesto, and Ilya L. Shapiro, 2241–2284. Singapore: Springer, 2024. DOI. Open PDF, arXiv v2.
• Bhardwaj, Lakshya, Lea E. Bottini, Ludovic Fraser-Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. “Lectures on Generalized Symmetries.” Physics Reports 1051 (2024): 1–87. DOI. Open PDF, arXiv v2.
• Bilal, Adel. “Lectures on Anomalies.” LPTENS-08/05, arXiv:0802.0634v1, 2008. Stable record. Open PDF.
• Cohen, Timothy, Xiaochuan Lu, and Zhengkang Zhang. “Anomalies from the Covariant Derivative Expansion.” Physical Review D 107 (2023): 116015. DOI. Open PDF, arXiv v1.
• Wang, Juven, Xiao-Gang Wen, and Edward Witten. “A New $SU(2)$ Anomaly.” Journal of Mathematical Physics 60, no. 5 (2019): 052301. DOI. Open PDF, arXiv v4.
• Witten, Edward. “An $SU(2)$ Anomaly.” Physics Letters B 117, no. 5 (1982): 324–328. DOI.