Nieh–Yan Term in First-Order Gravity

• Nieh–Yan Term is an exact torsional 4-form defined as the exterior derivative of the torsional Chern–Simons 3-form, capturing key topological properties in gravity.
• It plays a central role in Einstein–Cartan, teleparallel, and metric-affine gravity by relating torsion and curvature without altering bulk field equations for constant couplings.
• It has practical implications in parity violation, gravitational-wave phenomenology, and torsional transport in quantum materials, bridging gravitational and condensed-matter physics.

The Nieh–Yan term is the exact torsional 4-form

$$N_{\mathrm{NY}} \equiv d\!\left(e^a \wedge T_a\right)=T^a\wedge T_a-e^a\wedge e^b\wedge R_{ab},$$

defined in first-order gravity with coframe $e^a$, torsion $T^a$, and curvature $R_{ab}$. In components it can be written as

$$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$

so its scalar density is parity-odd through the Levi–Civita tensor. In Riemann–Cartan geometry it is a total derivative, while in teleparallel geometry $R_{ab}=0$ and it reduces to $T^a\wedge T_a$. This exactness makes the term topological for constant coupling, but non-constant scalar couplings, matter-induced torsion, boundaries, singularities, or nonmetricity can convert it into a dynamically relevant structure. The term therefore occupies a central position in Einstein–Cartan theory, teleparallel gravity, metric-affine gravity, chiral anomalies, and torsional transport (Banerjee, 2010, Wu et al., 2021, Hoyos et al., 2024).

1. Geometric definition and algebraic structure

The standard setting uses tetrads $e^a=e^a{}_\mu dx^\mu$, a Lorentz connection $\omega^{ab}=-\omega^{ba}$, torsion

$T^a=De^a=de^a+\omega^a{}_b\wedge e^b,$

and curvature

$e^a$0

With these conventions, the Nieh–Yan density is the exterior derivative of the torsional Chern–Simons 3-form $e^a$1, and its exactness follows directly from the Cartan structure equations and the Bianchi identity $e^a$2 (Huang et al., 2019).

A useful component decomposition of torsion separates its trace, axial, and tensor parts: $e^a$3 where $e^a$4 is the torsion trace vector, $e^a$5 is the axial pseudotrace, and $e^a$6 is the traceless remainder. This decomposition is especially important because different exact torsional 4-forms isolate different irreducible pieces at the boundary: the standard Nieh–Yan term is controlled by $e^a$7, whereas the related exact form $e^a$8 is controlled by $e^a$9 (Giacomo, 2023).

In teleparallel gravity, the fundamental variable is again the tetrad, but the affine connection is chosen curvature-free. In the Weitzenböck gauge,

$T^a$0

so the Nieh–Yan density reduces to a parity-odd torsion contraction,

$T^a$1

which is the form most often used in teleparallel cosmology and gravitational-wave phenomenology (Zhang et al., 2024).

2. Exactness, boundary terms, and relation to Holst-type structures

Because $T^a$2, its integral reduces by Stokes’ theorem to a boundary functional: $T^a$3 On compact manifolds without boundary, this integral vanishes. In a holonomic basis, the integral takes the form

$T^a$4

so only the axial torsion pseudotrace contributes. The integral therefore vanishes if $T^a$5; full torsion-freeness at the boundary is not required (Giacomo, 2023).

A related exact 4-form introduced later,

$T^a$6

satisfies

$T^a$7

It is “Nieh–Yan-like” rather than the standard Nieh–Yan invariant, but it clarifies that inserting a Hodge dual in the generating 3-form exchanges boundary control from the axial torsion $T^a$8 to the torsion trace $T^a$9. Unlike the standard identity $R_{ab}$0, this new invariant cannot be rearranged in first-order form into a simple torsion–torsion plus tetrad–curvature combination (Giacomo, 2023).

The relation to the Holst term is structurally important. In metric variables, the Holst density differs from a total derivative by a torsion-squared term, so the Holst contribution modifies the classical torsion equations when matter with spin is present. By contrast, the Nieh–Yan term itself is exact and therefore does not alter bulk field equations. In this sense, the Nieh–Yan term is the correct topological completion if one wishes to add a torsional term without changing the classical equations of motion. In the Weyssenhoff-fluid example studied for a static sphere with torsion, the associated torsion charge is nonzero at the stellar boundary but vanishes once a horizon forms (Banerjee, 2010).

These identities also underlie the Barbero–Immirzi sector. The standard relation

$R_{ab}$1

is the algebraic bridge between Holst-type couplings and topological torsional terms, and several later constructions use it when promoting the Immirzi parameter to a field or when comparing parity-odd operators in first-order gravity (Giacomo, 2023).

3. Chiral anomaly, index theory, and the status of the Nieh–Yan coefficient

The axial anomaly in a torsionful background is often written schematically as

$R_{ab}$2

but the torsional coefficient $R_{ab}$3 is dimensionful and has long been debated. One line of work identifies a universal thermal contribution. In a torsional Landau-level analysis, only the lowest torsional Landau level contributes to the axial current, effectively reducing the problem from $R_{ab}$4 to $R_{ab}$5 dimensions. The resulting thermal Nieh–Yan coefficient is

$R_{ab}$6

where $R_{ab}$7 is the effective $R_{ab}$8-dimensional central charge. In the same framework, the anomalous thermal Hall response of a Weyl semimetal is proportional to that central charge, providing an experimental fingerprint of the thermal Nieh–Yan anomaly (Huang et al., 2019).

A closely related condensed-matter calculation for Kramers–Weyl semimetals finds the finite-temperature coefficient

$R_{ab}$9

with $$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$0 scheme-dependent and the $$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$1 term universal. In that formulation, acoustic phonons generate an emergent teleparallel frame field, and the Nieh–Yan effective action induces a parity-odd phonon term that mixes the transverse phonon modes into circular polarizations (Liu, 2021).

At the same time, several analyses argue that smooth torsion backgrounds do not yield a genuine topological chiral anomaly from the Nieh–Yan term. One approach rewrites the axial torsion as $$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$2 and shows that the smooth pseudoscalar part can be removed by a local chiral rotation, so the Nieh–Yan contribution disappears from the anomaly or index in nonsingular backgrounds. In that picture, nontrivial contributions arise only in the presence of singularities, boundaries, nontrivial topology, or an infrared scale such as positive cosmological constant or finite temperature, with coefficients

$$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$3

after imposing a quantization condition motivated by expressing the Nieh–Yan term as a difference of Pontryagin classes (Rasulian et al., 2023).

A boundary-sensitive heat-kernel analysis sharpens that point further. It shows that on spacetimes without boundaries, the Nieh–Yan invariant vanishes if the Dirac index is well defined; in known examples with nonvanishing bulk Nieh–Yan charge and no boundary, the heat-kernel expansion breaks down and the index is ill-defined. With finite boundaries, bulk and boundary Nieh–Yan terms can appear in cutoff regularization, but explicit local counterterms cancel them. What remains physically meaningful are scheme-independent Pontryagin-type torsional terms and finite boundary Chern–Simons terms, the latter giving a torsional anomalous Hall effect (Erdmenger et al., 2024).

A holographic reformulation reinterprets the anomaly even more radically. In that construction the $$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$4 axial symmetry remains unbroken and the axial current coupled to an external gauge field is conserved. The “anomaly” is transferred to a breakdown of Hodge duality relations between fermion bilinears, with the thermal response again scaling as $$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$5 through

$$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$6

This suggests that at least in that class of theories, the physically robust content lies not in axial nonconservation itself but in torsion-induced violations of duality constraints among currents (Hoyos et al., 2024).

4. Teleparallel gravity and gravitational-wave phenomenology

In teleparallel gravity the Nieh–Yan term becomes especially transparent because curvature vanishes identically. The parity-violating modification most often studied adds to the TEGR action a scalar coupling

$$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$7

together with a canonical kinetic term and potential for $$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$8. The effective parity-violating scale is

$$N_{\mathrm{NY}}=\frac14\,\epsilon^{\mu\nu\rho\sigma}\left(T^a{}_{\mu\nu}T_{a\,\rho\sigma}-e^a{}_\mu e^b{}_\nu R_{ab\,\rho\sigma}\right),$$9

In an FRW background the tensor quadratic action yields helicity-dependent propagation,

$R_{ab}=0$0

so the theory produces velocity birefringence but no amplitude birefringence (Wu et al., 2021).

A full Bayesian analysis of 46 binary-black-hole events from GWTC-1 and GWTC-2, using \texttt{Bilby} and \texttt{dynesty}, found no significant evidence for such parity violation and placed the first observational constraint on this teleparallel Nieh–Yan modification,

$R_{ab}=0$1

The correction enters the circular-polarization waveform as a helicity-dependent phase shift $R_{ab}=0$2, rather than as the amplitude birefringence characteristic of dynamical Chern–Simons gravity (Wu et al., 2021). In the weak-field regime, the dynamical model has exactly the same PPN parameters as GR,

$R_{ab}=0$3

so Solar-System tests do not constrain the parity-violating coupling through standard PPN observables (Rao, 2021).

Teleparallel cosmology reveals additional structures once the affine connection is allowed to be irregular even while the metric remains FLRW. In that case the background evolution stays GR-like, but scalar and tensor perturbations couple already at linear order. The quadratic action acquires a mixed term

$R_{ab}=0$4

and the linearized perturbation equations show direct scalar–tensor sourcing. This produces nonzero scalar–tensor cross-correlations in addition to the usual helicity-dependent tensor propagation (Li et al., 2023).

Metric teleparallel gravity with a more general torsion scalar and a pseudoscalar coupling $R_{ab}=0$5 has also been used to compute scalar-induced gravitational waves during radiation domination. In that framework the parity-violating tensor propagation is present, but observational bounds imply $R_{ab}=0$6, so the dominant departures from GR arise from modified scalar transfer functions and from an additional tetrad scalar $R_{ab}=0$7, not from sizeable chirality. A notable result is that the familiar GR resonance at $R_{ab}=0$8 for a monochromatic primordial spectrum is absent; the induced GW spectrum is smooth and significantly different from the GR prediction (Zhang et al., 2024).

The same teleparallel coupling has also been used as an “audible axion” mechanism. During radiation domination, an axion-like field coupled to the Nieh–Yan term generates direct tachyonic instability in one GW helicity, producing a chiral stochastic background with potentially observable peaks in PTA, ASTROD-GW, LISA, Taiji, and related bands. The peak frequency scales as

$R_{ab}=0$9

and the present-day spectral density is obtained numerically from the helicity-resolved mode functions. In a different teleparallel setting, scattering of gravitational waves off axion domain walls yields transmitted waves with nonzero circular polarization, and the degree of circular polarization is independent of the domain-wall tension (Xu et al., 2024, Soda et al., 16 Dec 2025).

5. Einstein–Cartan and metric-affine generalizations

Outside teleparallel geometry, the Nieh–Yan term becomes intertwined with torsion, contorsion, nonmetricity, and projective symmetry. In generic metric-affine geometry the standard Riemann–Cartan expression loses both exactness and projective invariance once nonmetricity is allowed. This motivates a generalized two-parameter family,

$T^a\wedge T_a$0

Projective invariance holds iff $T^a\wedge T_a$1, while topologicity is restored iff $T^a\wedge T_a$2. These conditions are independent, so one can construct projectively invariant but nontopological theories, or fully topological ones (Bombacigno et al., 2021, Bombacigno et al., 2021).

In the corresponding $T^a\wedge T_a$3 models, the Legendre transform introduces a Palatini-like scalaron $T^a\wedge T_a$4 and a dynamical Immirzi field $T^a\wedge T_a$5. In the projectively invariant sector, a field redefinition $T^a\wedge T_a$6 yields an effective scalar–tensor action with a manifestly positive kinetic denominator,

$T^a\wedge T_a$7

so the Immirzi sector is ghost-free. Applied to Bianchi I cosmology, these models admit big-bounce solutions. For $T^a\wedge T_a$8, the initial singularity is replaced by a bounce, and in the viable class the resulting finite-time singularities do not spoil geodesic completeness or scalar-wave regularity (Bombacigno et al., 2021, Bombacigno et al., 2021).

Scalar-dependent couplings to the standard Nieh–Yan density also play an important role in inflation. In metric-affine natural inflation, a periodic coupling $T^a\wedge T_a$9 modifies the canonical normalization through

$e^a=e^a{}_\mu dx^\mu$0

with $e^a=e^a{}_\mu dx^\mu$1 receiving an explicit contribution from $e^a=e^a{}_\mu dx^\mu$2. A pure Nieh–Yan coupling drives the strong-coupling limit to linear inflation and cannot rescue natural inflation, but adding an analogous periodic non-minimal coupling to the Ricci scalar yields viable predictions even for sub-Planckian periodicity and order-one couplings (Kraiko et al., 22 May 2026).

In Einstein–Cartan–Palatini inflation with a pseudoscalar inflaton, the Nieh–Yan interaction generates a pure axial torsion background,

$e^a=e^a{}_\mu dx^\mu$3

and integrating out torsion renormalizes the kinetic term,

$e^a=e^a{}_\mu dx^\mu$4

This enhances the effective decay constant without introducing new propagating torsion modes. In that setup natural inflation remains inconsistent with current data, but a squared quartic hilltop potential can be made consistent with Planck 2018 and BICEP–Keck for sub-Planckian $e^a=e^a{}_\mu dx^\mu$5, while simultaneously producing chiral gravitational waves (Adshead et al., 14 Jul 2025).

Related conclusions appear in Higgs inflation with non-minimal Ricci, Holst, and Nieh–Yan couplings. There, a non-minimally coupled Nieh–Yan term alone does not give successful inflation, whereas a Holst-only regime can reproduce the metric plateau predictions. With all three couplings active, the predictions for $e^a=e^a{}_\mu dx^\mu$6 and $e^a=e^a{}_\mu dx^\mu$7 span almost the whole range probed by forthcoming experiments, though many viable points are tuned and the allowed runnings are comparatively restricted (Långvik et al., 2020).

6. Condensed-matter realizations and torsional transport

Condensed matter supplies explicit realizations of effective torsion and therefore of Nieh–Yan physics. In Kramers–Weyl semimetals of chiral crystals, strain does not shift Weyl nodes as a pseudo-gauge field; instead it deforms the emergent frame field,

$e^a=e^a{}_\mu dx^\mu$8

so acoustic phonons induce torsion in a teleparallel background. Integrating out the fermions generates a Nieh–Yan effective action for the lattice deformation, which in turn yields the phonon term

$e^a=e^a{}_\mu dx^\mu$9

This mixes the two transverse phonon modes into circular polarizations with dispersions

$\omega^{ab}=-\omega^{ba}$0

and phonon angular momentum

$\omega^{ab}=-\omega^{ba}$1

A temperature gradient then induces a net phonon angular momentum whose low-temperature coefficient contains a universal $\omega^{ab}=-\omega^{ba}$2 piece governed by $\omega^{ab}=-\omega^{ba}$3 (Liu, 2021).

The same thermal coefficient reappears in the torsional Landau-level approach. There a uniform torsional magnetic field produces gapless torsional Landau levels, but all higher levels cancel in the axial current and only the lowest torsional Landau level survives. The anomaly coefficient is then identified with the $\omega^{ab}=-\omega^{ba}$4-dimensional CFT free-energy density, and the induced thermal Hall conductivity satisfies

$\omega^{ab}=-\omega^{ba}$5

so the response is proportional to the central charge $\omega^{ab}=-\omega^{ba}$6 and to the momentum-space Weyl-node separation $\omega^{ab}=-\omega^{ba}$7 (Huang et al., 2019).

A hydrodynamic and equilibrium-field-theory treatment based on transgression derives the nondissipative constitutive relations generated by the torsional Chern–Simons form $\omega^{ab}=-\omega^{ba}$8. In four dimensions, the covariant axial current, heat current, stress current, and spin current are

$\omega^{ab}=-\omega^{ba}$9

$T^a=De^a=de^a+\omega^a{}_b\wedge e^b,$0

$T^a=De^a=de^a+\omega^a{}_b\wedge e^b,$1

$T^a=De^a=de^a+\omega^a{}_b\wedge e^b,$2

These relations encode torsion-driven chiral separation, vortical separation, heat transport, stress response, and spin transport, all controlled by the nonuniversal coefficient $T^a=De^a=de^a+\omega^a{}_b\wedge e^b,$3 (Valle et al., 2021).

Taken together, these developments show that the Nieh–Yan term is no longer confined to formal discussions of first-order gravity. It functions as a boundary invariant, an organizing identity behind Holst and Immirzi sectors, a disputed but increasingly well-structured component of the axial anomaly, a parity-violating operator in teleparallel cosmology and gravitational-wave propagation, and an experimentally motivated descriptor of torsional transport in quantum materials. Its modern significance lies precisely in this breadth: the same exact 4-form governs questions of topology, renormalization, parity violation, and effective geometry across both gravitational and condensed-matter systems.