Mixed Axial-Gravitational Anomaly

• Mixed axial-gravitational anomaly is a four-dimensional phenomenon where the divergence of the chiral current includes a curvature term proportional to the Pontryagin density.
• Perturbative techniques, including triangle diagram and gradient-flow methods, reveal that the anomaly influences hydrodynamic transport through T²-dependent vortical and energy currents.
• Condensed-matter systems like Weyl semimetals provide experimental access to the anomaly via measurable thermal, electrical, and mechanical responses.

The mixed axial-gravitational anomaly is the four-dimensional anomalous nonconservation of an axial or chiral current in the presence of gravity, with the anomalous divergence proportional to the gravitational Pontryagin density. In the standard field-theoretic formulation, the curvature term appears in the divergence of the axial current rather than in the divergence of the stress tensor; accordingly, several sources emphasize that in $3+1$ dimensions the precise object is a mixed gauge-gravitational anomaly, not a pure gravitational anomaly of the energy-momentum tensor (Landsteiner et al., 2011). For a Dirac fermion, a standard normalization used in several analyses is

$$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$

while more general chiral-current formulas organize the anomaly in terms of the gauge-anomaly coefficients $d_{ABC}$ and mixed coefficients $b_A=\mathrm{Tr}(T_A)$ (Morikawa et al., 2018).

1. Definition, anomaly polynomial, and four-dimensional meaning

In the notation used for relativistic chiral fermions with global symmetry group $G$, the anomaly coefficients in $3+1$ dimensions are

$$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$

and the anomalous current divergence is

$$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$

The curvature-dependent term is the defining local expression for the mixed axial-gravitational anomaly in that framework (Landsteiner et al., 2011).

A closely related flat-space characterization is through the three-point function

$$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$

because the curved-space density $R\widetilde R$ is second order in metric perturbations. In that language, the anomaly is encoded in the anomalous longitudinal part of the axial-current leg in the $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$0 correlator (Corianò et al., 2024).

Several papers distinguish this anomaly sharply from a pure gravitational anomaly. The relevant violated current is the axial $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$1 current, while gravity supplies the external background. This is why the same effect is also described as the gravitational contribution to the axial anomaly or the $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$2-gravity-gravity anomaly (Morikawa et al., 2018). A useful consequence of this distinction is that it separates the mixed axial-gravitational anomaly from diffeomorphism anomalies of the stress tensor and from unrelated mixed gravitational anomalies of non-invertible or duality symmetries.

2. Perturbative derivations and Ward-identity structure

A standard perturbative derivation proceeds from the $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$3 triangle. In the gradient-flow analysis of the axial $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$4 anomaly in a gravitational field, the curved-space formula first obtained by Kimura is reproduced by expanding around flat space and studying the renormalized three-point function with two symmetric stress tensors. In that treatment, the universal flowed energy-momentum tensor reproduces the correct nonlocal structure of the triangle diagram and the correct anomaly coefficient, but does not automatically reproduce the translation or general-coordinate Ward-Takahashi relation at coincident insertions; local counterterms are required to restore that Ward identity (Morikawa et al., 2018).

This yields a precise lesson about scheme dependence. The anomaly coefficient is fixed, but contact terms matter when comparing different regularizations or composite-operator prescriptions. In the gradient-flow construction, the physically correct result emerges only after enforcing the appropriate gravitational Ward identities by local counterterms, while the parity-odd piece remains proportional to the curvature Pontryagin density (Morikawa et al., 2018).

The same $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$5 correlator has also been analyzed directly at finite temperature and density. In that perturbative treatment, the gravitational chiral anomaly coefficient is packaged in the longitudinal form factor $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$6, with

$$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$7

and the main result is that neither finite temperature nor finite fermion density affects the anomalous divergence. The medium-dependent terms in the real-time thermal computation vanish after contraction with the axial-current momentum, so the anomaly coefficient remains exactly the vacuum one (Corianò et al., 2024).

This non-renormalization by $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$8 and $$\nabla_\mu J_5^\mu = 2im\,\bar\psi\gamma_5\psi + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^\rho{}_{\sigma\mu\nu} R^\sigma{}_{\rho\alpha\beta},$$9 is consistent with the standard interpretation of anomalies as ultraviolet data. A plausible implication is that thermal and density effects reorganize finite parts of the amplitude while leaving the anomalous Ward identity itself unchanged. In the perturbative language of (Corianò et al., 2024), the anomaly still appears as an anomaly pole in the longitudinal sector of the axial-current leg.

3. Hydrodynamic transport and anomaly-induced currents

The mixed axial-gravitational anomaly has an especially important hydrodynamic manifestation in parity-odd transport. In the Kubo and constitutive-relation analysis of anomaly-induced transport, the chiral magnetic conductivity and the chiral vortical or chiral gravito-magnetic conductivity are

$d_{ABC}$0

$d_{ABC}$1

$d_{ABC}$2

The $d_{ABC}$3 terms are the direct transport signature of the mixed gauge-gravitational anomaly coefficient $d_{ABC}$4 (Landsteiner et al., 2011).

In that framework, the “chiral gravito-magnetic effect” is the same transport phenomenon usually called the chiral vortical effect: a current parallel to vorticity, equivalently to an external gravito-magnetic field generated by metric perturbations of the form

$d_{ABC}$5

The mixed axial-gravitational anomaly contributes

$d_{ABC}$6

to the current response and

$d_{ABC}$7

to the energy-current response, while the pure gauge anomaly contributes the $d_{ABC}$8 and $d_{ABC}$9 terms through $b_A=\mathrm{Tr}(T_A)$0 (Landsteiner et al., 2011).

That analysis also clarifies a common subtlety. The curvature term in the anomaly equation is naively fourth order in derivatives, yet it affects first-order hydrodynamics because the derivative counting is performed on connections $b_A=\mathrm{Tr}(T_A)$1 and $b_A=\mathrm{Tr}(T_A)$2; in that counting, the Riemann tensor is treated analogously to $b_A=\mathrm{Tr}(T_A)$3 (Landsteiner et al., 2011).

Later hydrodynamic work extends anomaly matching beyond first-order CVE. In the “Kinematical Vortical Effect” program, the gravitational chiral anomaly is argued to constrain a third-order axial current of the form

$b_A=\mathrm{Tr}(T_A)$4

with the anomaly fixing

$b_A=\mathrm{Tr}(T_A)$5

for a massless Dirac fermion (Prokhorov et al., 2022). In Einstein backgrounds with $b_A=\mathrm{Tr}(T_A)$6, a related analysis further derives

$b_A=\mathrm{Tr}(T_A)$7

so that the same anomaly coefficient controlling the Pontryagin density also fixes a scalar-curvature contribution to the third-order equilibrium current (Khakimov et al., 2024). This suggests that the mixed axial-gravitational anomaly constrains both first-order $b_A=\mathrm{Tr}(T_A)$8-vortical transport and later, higher-derivative equilibrium structures.

4. Condensed-matter realizations and experimental signatures

A central condensed-matter route to the mixed axial-gravitational anomaly uses Weyl semimetals. In a Weyl semimetal, the node-separation field $b_A=\mathrm{Tr}(T_A)$9 acts as an axial gauge field, and its spatial variation generates an axial magnetic field

$G$0

The axial magnetic effect is then

$G$1

so that at $G$2,

$G$3

for a single Dirac fermion. That $G$4 term is identified explicitly with the mixed axial-gravitational anomaly (1311.0878).

In the cylindrical geometry proposed there, the axial magnetic field is localized near the boundary and the edge states carry thermal angular momentum

$G$5

which at $G$6 becomes

$G$7

The corresponding mechanical response under heating or cooling is

$G$8

and the proposal is that observing this $G$9-dependent angular-momentum response would provide evidence for the gravitational contribution to the axial anomaly (1311.0878).

An experimental thermoelectric realization was reported in NbP. In that setting, the mixed axial-gravitational anomaly is tied to the anomalous energy current

$3+1$0

which enters the continuity equation as

$3+1$1

The measured observable is a positive longitudinal magnetothermoelectric conductance for $3+1$2, quadratic in $3+1$3 at low field and predicted to vanish in the ultra-quantum limit, where

$3+1$4

The reported positive longitudinal magnetothermoelectric conductance in NbP is interpreted as consistent with the mixed axial-gravitational anomaly (Gooth et al., 2017).

A different condensed-matter proposal dispenses with magnetic field altogether. In the second-order dc response

$3+1$5

the conductivity decomposes as

$3+1$6

where $3+1$7 is the deformation density of states. Because $3+1$8 is $3+1$9, fails the relevant permutation symmetry, and induces a continuity equation balanced only by the relaxation term, that work identifies it with an emergent mixed axial-gravitational anomaly in nonlinear charge transport at zero magnetic field (Holder et al., 2021).

5. Alternative formulations, inflow pictures, and detector-level realizations

One variational reformulation is Wiegmann’s fluid-mechanical construction of a perfect fluid deformed by a chiral phase $$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$0 and the gravitational Chern-Simons form. In that approach, the anomaly deformation of the Hamilton functional is

$$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$1

equivalently

$$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$2

Variation with respect to $$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$3 yields the anomalous continuity equation

$$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$4

while the gravitational contribution also enters a fluid spin current and an anomaly-induced spin-orbit term in the Euler equation (Wiegmann, 2024).

A complementary detector-based realization treats the mixed anomaly through flat-space scattering rather than explicit curved backgrounds. In that framework, the anomaly equation

$$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$5

is probed by two-graviton $$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$6 scattering and higher-spin helicity or zilch detectors. The resulting normalized flux one-point function vanishes at fixed non-collinear angle in the $$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$7 limit, but localizes distributionally at the beam directions as

$$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$8

which is interpreted as the detector-level analogue of the mixed anomaly pole and its finite energy-weighted sum rule (Barata et al., 7 Jul 2026).

The holographic inflow picture provides another formulation. In five bulk dimensions, the Chern-Simons term

$$d_{ABC}=\frac12 \mathrm{Tr}\!\left(T_A\{T_B,T_C\}\right), \qquad b_A=\mathrm{Tr}(T_A),$$9

induces a four-dimensional mixed gauge-diffeomorphism anomaly for the boundary $$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$0 current, which is the mixed axial-gravitational anomaly when that current is interpreted as axial. In that setting, the entanglement first law survives, but quadratic mixed $$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$1 contributions appear in relative entropy, and the graviton time-delay analysis suggests that the time delay can take either sign, potentially violating causality for any finite value of the Chern-Simons coupling (Bhattacharyya et al., 2016).

A related but distinct formal device is axial gravity in a metric-axial-tensor background,

$$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$2

That construction does not directly compute $$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$3, but it yields parity-odd trace anomalies proportional to the same Pontryagin density and, in a collapsing limit, reproduces the already found odd-parity trace anomaly of a Weyl fermion with coefficient

$$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$4

Its relevance is therefore structural rather than direct: it supplies a nonperturbative framework for split chiral-gravitational anomalies involving the same curvature pseudoscalar (Bonora et al., 2018).

6. Conceptual boundaries, analogues, and common confusions

Several nearby topics are not the standard mixed axial-gravitational anomaly. The distinctions are substantive rather than terminological.

Topic	Violated quantity or symmetry	Defining structure
Standard mixed axial-gravitational anomaly	Axial or chiral current	$$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$5 in $$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$6
Mixed gravitational anomaly of self-duality symmetry	Non-invertible self-duality defect	Curvature-dependent phase on curved 4-manifolds
Mixed axial-torsional anomaly	Axial current in torsional backgrounds	$$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$7 and axial torsion
Crystalline elastic analogue	Electromagnetic current in elastic background	$$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$8 with elasticity tetrads
Analog gravitational anomaly in chiral superconductors	Bogoliubov quasiparticle axial charge	Electromagnetic field acting as spin connection

The mixed gravitational anomaly of the Cardy–Rabinovici model is a mixed anomaly between a non-invertible self-duality symmetry and gravity, detected through curvature-dependent phases on manifolds such as $$\nabla_\mu J_A^\mu= \epsilon^{\mu\nu\rho\lambda} \left( \frac{d_{ABC}}{32\pi^2}F^B_{\mu\nu}F^C_{\rho\lambda} + \frac{b_A}{768\pi^2} R^\alpha{}_{\beta\mu\nu}R^\beta{}_{\alpha\rho\lambda} \right).$$9. It is explicitly not the anomaly of an axial current, and there is no equation of the form $$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$0 in that analysis (Hayashi et al., 2022).

The mixed axial-torsional anomaly in Weyl semimetals is also distinct. Its anomaly equation is

$$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$1

and the paper emphasizing it is explicit that this is not the standard curvature-based mixed axial-gravitational anomaly but a condensed-matter torsional analogue with regulator-dependent coefficients (Ferreiros et al., 2018).

Likewise, the crystalline response based on elasticity tetrads

$$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$2

produces a mixed Chern-Simons term

$$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$3

and in the presence of dislocations the resulting nonconservation law is proportional to $$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$4. That is a crystalline elasticity analogue of a mixed gauge-geometric anomaly, not the conventional relativistic mixed axial-gravitational anomaly (Nissinen et al., 2018).

Volovik’s “analog of gravitational anomaly” in topological chiral superconductors is also an analogue rather than a literal mixed gauge-gravitational anomaly. There the electromagnetic field serves as an emergent spin connection, so the curvature-squared anomaly reduces to an $$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$5-type term with an extra factor $$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$6 relative to the ordinary ABJ coefficient in the conventions used there (Volovik, 2021).

These distinctions matter because the mixed axial-gravitational anomaly, in the strict four-dimensional quantum-field-theoretic sense, remains the statement that the axial-current divergence acquires the curvature Pontryagin density. Its most robust signatures across the literature are the $$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$7 anomalous Ward identity, the $$\langle T^{\mu_1\nu_1}(p_1)\,T^{\mu_2\nu_2}(p_2)\,J_5^{\mu_3}(p_3)\rangle, \qquad p_1+p_2+p_3=0,$$8 contribution to vortical and energy transport, and, in selected condensed-matter settings, anomaly-controlled thermoelectric or nonlinear electrical responses (Landsteiner et al., 2011).