Chiral and Gravitational Anomalies

For a chiral fermion, classical current conservation can be incompatible with a quantum measure that preserves the background gauge and gravitational symmetries. The obstruction is measured by an index density: in even dimension $d=2n$, the relevant anomaly polynomial is the $(2n+2)$-form built from the Dirac genus and the gauge Chern character. Descent produces the local consistent anomaly, while a Bardeen–Zumino polynomial converts it to a covariant anomaly at the cost of losing direct integrability as the variation of one effective action.

Required background. Gravitational Actions and Boundary Terms provides metric variation and boundary flux; Curvature supplies curvature two-forms and Pontryagin densities. Helpful background. Flat-Spacetime Quantum Field Theory provides chiral currents and fermion measures; Partial Differential Equations and Green Functions supplies elliptic Dirac operators and regulated traces; Causality and Global Structure establishes the Lorentzian background; Quantum Fields in Curved Spacetime provides spin structures and renormalized currents.

Statement and hypotheses

Let $M$ be a smooth, oriented, even-dimensional Riemannian spin manifold, initially compact and without boundary, with a unitary background connection of anti-Hermitian curvature two-form $\mathcal F$ in a representation $\mathcal R$. Define $\operatorname{ch}_{\mathcal R}(\mathcal F)=\operatorname{tr}_{\mathcal R}\exp(i\mathcal F/2\pi)$ and choose the positive chirality so that the index theorem reads

$$\operatorname{index}D_+ =\int_M \left[\widehat A(TM)\,\operatorname{ch}_{\mathcal R}(\mathcal F)\right]_{d}.$$

The same characteristic classes define the anomaly polynomial two form-degrees higher for one complex Weyl species of this chirality:

$$I_{d+2} =\left[\widehat A(TM)\,\operatorname{ch}_{\mathcal R}(\mathcal F)\right]_{d+2}.$$

Opposite chirality contributes the negative polynomial; reality conditions can change multiplicities. This polynomial concerns perturbative local anomalies. Its cancellation does not exclude global anomalies under large transformations, including those possible for pseudoreal representations. A boundary adds spectral and flux data. For a vectorlike Dirac theory one can preserve the vector gauge symmetry and locate the anomaly in the axial current; for an anomalous chiral gauge theory there is no regulator preserving every claimed gauge Ward identity. A change of counterterm can redistribute the anomaly but cannot erase its nontrivial class.

Alvarez-Gaumé and Witten derive the gauge, mixed, and gravitational anomaly polynomials and identify the dimensions and chiral field content in which they occur in Alvarez-Gaumé and Witten 1984, §§3–5.

Fujikawa’s regulated Jacobian

Consider a Euclidean Dirac field and an infinitesimal axial rotation

$$\psi\mapsto e^{i\alpha(x)\gamma_*}\psi, \qquad \bar\psi\mapsto\bar\psi e^{i\alpha(x)\gamma_*}.$$

The classical massless action is invariant for constant $\alpha$; a spacetime-dependent $\alpha$ exposes the current Ward identity. The expansion coefficients of $\psi$ and $\bar\psi$ in an eigenbasis of the Dirac operator change, and the formal Jacobian contains an ill-defined trace of $\gamma_*$. Let $\mathscr D_E=i\gamma_E^\mu D_\mu$ be the self-adjoint Euclidean Dirac operator, with Hermitian gamma matrices satisfying $\{\gamma_E^\mu,\gamma_E^\nu\}=2g_E^{\mu\nu}$. Its square is nonnegative. The regulated axial-divergence density is

$$\mathcal A(x) =2\lim_{M\to\infty} \operatorname{tr}\!\left[ \gamma_*e^{-\mathscr D_E^2/M^2} \right](x,x).$$

The heat-kernel expansion of $\mathscr D_E^2$ selects the coefficient of total dimension $d$. Tracing with $\gamma_*$ removes parity-even contractions and leaves epsilon-tensor contractions of gauge and Riemann curvatures. Nonzero eigenvalues pair with opposite chirality in the integrated trace; zero modes do not, so the integral equals twice the index for a Dirac axial rotation. This factor two comes from rotating both independent Grassmann measures, $\psi$ and $\bar\psi$. Fujikawa’s original measure argument is given in Fujikawa 1979, complete article, pp. 1195–1198; the heat-kernel and index connections are Vassilevich 2003, §§7.2–7.3, Eqs. (7.28), (7.32), and (7.46), PDF.

The calculation is performed in positive-definite signature, where the elliptic spectrum and chirality trace are controlled. Only after fixing $\gamma_*$ and the Euclidean orientation do we continue to the site’s Lorentzian $(-,+,+,+)$ convention with $\epsilon_{0123}=+\sqrt{-g}$. The continuation of $\gamma_*$ and the epsilon tensor fixes the overall sign; importing a Euclidean Pontryagin formula without this translation is not reliable.

Four-dimensional index density and axial charge

For a massless Dirac fermion of charge $q$, write $D_\mu=\nabla_\mu-iqA_\mu$ and $F=dA$, so $\mathcal F=-iqF$. Let $\mathcal R^a{}_b=\tfrac12R^a{}_{b\mu\nu}dx^\mu\wedge dx^\nu$ be the tangent-bundle curvature. With the chosen Euclidean orientation,

$$\begin{aligned} p_1(TM)&=-\frac{1}{8\pi^2}\operatorname{tr}(\mathcal R\wedge\mathcal R),\\ \widehat A&=1-\frac{p_1}{24}+\cdots, \qquad \operatorname{ch}=e^{qF/(2\pi)}. \end{aligned}$$

Taking the degree-four term and multiplying by the Dirac axial factor two gives

$$\mathcal A_E\,d\mathrm{vol}_E =\frac{q^2}{4\pi^2}F\wedge F +\frac{1}{96\pi^2}\operatorname{tr}(\mathcal R\wedge\mathcal R).$$

In a positively oriented Euclidean orthonormal frame, this chirality is $\gamma_*=-\gamma_E^1\gamma_E^2\gamma_E^3\gamma_E^4$. Thus the component density is

$$\mathcal A_E =\frac{q^2}{16\pi^2}\epsilon_E^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma} +\frac{1}{384\pi^2}\epsilon_E^{\mu\nu\rho\sigma} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\sigma}.$$

The factors of four follow from the two factors of $1/2$ in the curvature two-forms. This is a Euclidean local index formula with $\int\mathcal A_E\,d\mathrm{vol}_E=2\operatorname{index}D_+$, not a Lorentzian component equation with only the symbol $E$ removed. For example, with $\gamma_E^1=i\gamma_L^{\hat0}$ and the other three Euclidean matrices equal to the spatial Lorentzian matrices, this $\gamma_*$ equals $-i\gamma_L^{\hat0}\gamma_L^{\hat1}\gamma_L^{\hat2}\gamma_L^{\hat3}$. A Lorentzian current defined with the opposite sign of that chirality has the opposite axial charge. The continuation must also translate the volume/current form and the site’s $\epsilon^{0123}=-1/\sqrt{-g}$. The index-normalized expression above fixes the density without conflating these sign choices.

The second term is the gravitational contribution to the axial anomaly. It is not a four-dimensional failure of stress-tensor conservation.

One can check the gauge coefficient without memorizing an anomaly table. In a flat Euclidean frame, the Laplace-type endomorphism is $\mathcal E=-iq\gamma_E^\mu\gamma_E^\nu F_{\mu\nu}/2$. The chosen chirality obeys $\operatorname{tr}(\gamma_*\gamma_E^\mu\gamma_E^\nu\gamma_E^\rho\gamma_E^\sigma)=-4\epsilon_E^{\mu\nu\rho\sigma}$. The $\mathcal E^2/2$ term of $b_4$, multiplied by the heat-kernel factor $(4\pi)^{-2}$ and the axial factor two, then gives $q^2\epsilon_EFF/(16\pi^2)$. The minus sign from $(-i)^2$ cancels the chirality-trace minus sign. Reversing chirality reverses the result, as the index theorem requires.

That distinction prevents an important category error. Perturbative pure gravitational anomalies—nonconservation of the energy-momentum tensor under diffeomorphisms—occur for chiral matter in dimensions $4k+2$. In four dimensions one can instead have gauge anomalies and mixed gauge–gravitational anomalies in chiral currents. A consistent quantum gauge theory must sum the anomaly polynomial over all left- and right-handed species and make every gauged anomaly cancel.

Descent and Wess–Zumino consistency

Locally, write the closed invariant polynomial as

$$I_{d+2}=dI_{d+1}^{(0)}.$$

Under a gauge or local Lorentz transformation,

$$\delta I_{d+1}^{(0)}=dI_d^{(1)}.$$

The consistent variation of the effective action is proportional to $\int_M I_d^{(1)}$. Because it comes from successive variations of one functional, it obeys the Wess–Zumino consistency condition

$$\delta_{\alpha_1}\mathcal A_{\alpha_2} -\delta_{\alpha_2}\mathcal A_{\alpha_1} =\mathcal A_{[\alpha_1,\alpha_2]}.$$

This algebraic condition is a discriminating check: an arbitrary covariant-looking polynomial need not be a consistent anomaly. The descent construction also shows that adding a local counterterm can move an anomaly between currents or between gauge and Lorentz descriptions, but cannot remove a nontrivial cohomology class.

Consistent versus covariant anomaly

The consistent current is defined by functional differentiation,

$$J_{\mathrm{cons}}^\mu =\frac{1}{\sqrt{-g}}\frac{\delta\Gamma}{\delta A_\mu}.$$

Its divergence satisfies Wess–Zumino consistency, yet the current itself need not transform covariantly. A local Bardeen–Zumino current $J_{\mathrm{BZ}}^\mu$, constructed from the connection and curvature, defines

$$J_{\mathrm{cov}}^\mu =J_{\mathrm{cons}}^\mu+J_{\mathrm{BZ}}^\mu.$$

The new divergence is manifestly covariant, but $J_{\mathrm{cov}}$ is generally not the functional derivative of a single effective action and its coefficient differs from the consistent one. Consequently, quoting an “anomaly coefficient” without naming the current is ambiguous. Bardeen and Zumino construct this polynomial and compare the two forms for gauge and gravitational anomalies in Bardeen and Zumino 1984, §§2–4.

Boundaries and the integrated theorem

The local heat-kernel density describes the bulk divergence. On a manifold with boundary, integrating it does not by itself give an integer index: one must impose an elliptic boundary condition and include the corresponding boundary or eta-invariant contribution. Physical current can also flow through the boundary. Thus a nonzero bulk integral may be balanced by boundary spectral asymmetry, and a vanishing integrated anomaly does not imply that the local anomaly density vanishes pointwise.

Likewise, anomaly cancellation is stronger than cancellation on one background. The full characteristic polynomial must cancel for arbitrary admissible background curvatures. Choosing a special geometry with $F\wedge F=0$ or $R\wedge R=0$ is a check of that configuration, not a proof that the representation is anomaly-free.

Diagnostic: which current can come from the action?

Suppose a calculation gives a consistent current $J_{\mathrm{cons}}$ and a local Bardeen–Zumino current $J_{\mathrm{BZ}}$. Explain why $J_{\mathrm{cov}}=J_{\mathrm{cons}}+J_{\mathrm{BZ}}$ can transform covariantly while failing the integrability test for being $\delta\Gamma/\delta A$, and state which anomaly satisfies Wess–Zumino consistency directly.

Solution

Functional derivatives commute, so a current obtained from one effective action has a symmetric functional curl: the variation of $J^\mu(x)$ with respect to $A_\nu(y)$ must match the exchanged variation, up to the standard graded qualifications. The consistent current has this property by construction, and its anomalous divergence therefore obeys the Wess–Zumino condition. Adding the local Bardeen–Zumino current changes the transformation law so the sum is covariant, but the added polynomial generally has a nonzero functional curl; therefore the covariant current need not be integrable to one action. This checks why consistent and covariant anomalies are both meaningful but answer different questions. A boundary would add further surface variations and must be included before applying the integrability test globally.

References

• Alvarez-Gaumé, Luis, and Edward Witten. “Gravitational Anomalies.” Nuclear Physics B 234 (1984): 269–330. DOI.
• Bardeen, William A., and Bruno Zumino. “Consistent and Covariant Anomalies in Gauge and Gravitational Theories.” Nuclear Physics B 244 (1984): 421–453. DOI.
• Fujikawa, Kazuo. “Path-Integral Measure for Gauge-Invariant Fermion Theories.” Physical Review Letters 42 (1979): 1195–1198. DOI.
• Vassilevich, Dmitri V. “Heat Kernel Expansion: User’s Manual.” Physics Reports 388 (2003): 279–360. DOI. Open PDF, arXiv:hep-th/0306138.