Nieh–Yan Invariant in Torsional Gravity

• The Nieh–Yan invariant is the exact torsion four-form d(e^a∧T_a) in Riemann–Cartan geometry, crucial for distinguishing torsionful gravitational effects.
• It plays a controversial role in chiral anomalies and thermal T² responses, with its contributions often dependent on regularization schemes.
• The invariant serves as a framework for extensions in metric-affine and teleparallel gravity and appears in emergent phenomena in Weyl semimetals and chiral superfluids.

Searching arXiv for recent and foundational papers on the Nieh–Yan invariant to ground the article in the cited literature. {"query":"Nieh-Yan topological invariant torsion anomaly thermal response arXiv", "max_results": 10} The Nieh–Yan topological invariant is the four-dimensional torsional density

$$\mathcal N \;=\; T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab} \;=\; d(e^a\wedge T_a),$$

and, in the Cartan-form language emphasized in recent holographic work, it can also be written as $NY=\star_4 d\Omega_3$ with $\Omega_3=\eta_{ab}e^a\wedge T^b$ (Nissinen et al., 2019, Hoyos et al., 2024). It is topological in the sense of being an exact form in Riemann–Cartan geometry, but its physical and mathematical status is highly sensitive to torsion, nonmetricity, boundaries, and renormalization conditions. As a result, the Nieh–Yan invariant appears across several distinct settings: as a boundary term in torsionful gravity, as a disputed contribution to the chiral torsional anomaly, as a source of thermal $T^2$ responses and nondissipative transport, and as a structural ingredient in metric-affine and teleparallel extensions of gravity (Erdmenger et al., 2024, Bombacigno et al., 2021).

1. Standard geometric definition and equivalent formulations

In the standard first-order geometric setup, the basic objects are the tetrad $e^a$, torsion 2-form $T^a$, and curvature 2-form $R_{ab}$. The Nieh–Yan density is

$$\mathcal N = T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab}, \qquad \mathcal N = d\mathcal Q, \qquad \mathcal Q = e^a\wedge T_a.$$

This exactness is the defining structural property: the invariant is locally the exterior derivative of a torsional 3-form, and is therefore distinct from purely curvature-based densities such as $R\wedge R$ (Nissinen et al., 2019).

Several component expressions are used in the literature. One form is

$$C_{\rm NY} = \frac14 \epsilon^{\mu\nu\rho\lambda} \left( \eta_{\alpha\beta} T^\alpha_{[\mu\nu}T^\beta_{\rho\lambda]} - 2R_{\mu\nu\lambda\rho} \right) = \nabla_\mu\!\left(\frac{1}{3!}\epsilon^{\mu\nu\rho\lambda}T_{[\nu\rho\lambda]}\right),$$

which makes explicit that the density is a total divergence and is controlled by the totally antisymmetric part of torsion (Erdmenger et al., 2024). In the holographic reinterpretation of the torsional anomaly, the same object is written as

$NY=\star_4 d\Omega_3$0

with the Nieh–Yan term identified equivalently as $NY=\star_4 d\Omega_3$1 (Hoyos et al., 2024).

A recurring point is that the invariant is nontrivial only when torsion is present. In the metric formulation of gravity with torsion, it is the combination

$NY=\star_4 d\Omega_3$2

and without torsion it collapses to zero (Banerjee, 2010). This distinguishes the Nieh–Yan sector from topological densities that remain meaningful in torsion-free geometry.

2. Topologicity, boundaries, and the Holst relation

In torsionful gravity, the Nieh–Yan term is the exact divergence that can be added to the Einstein–Hilbert action without changing the classical bulk equations of motion. By contrast, the Holst term,

$NY=\star_4 d\Omega_3$3

is not a harmless total derivative once torsion is treated as an independent field and matter with spin is present (Banerjee, 2010).

The key identity is

$NY=\star_4 d\Omega_3$4

This shows that the Holst density differs from the Nieh–Yan divergence by a torsion-quadratic term. Consequently, if metric and torsion are varied independently, the Holst term modifies the torsion-spin relation and therefore the classical equations of motion in the presence of spinning matter, whereas the Nieh–Yan term does not (Banerjee, 2010).

The boundary character of the invariant is physically relevant. In a collapsing Weyssenhoff fluid sphere, the Nieh–Yan contribution reduces to a boundary integral and yields

$NY=\star_4 d\Omega_3$5

which is nonzero while the surface lies outside the horizon and vanishes when the surface reaches $NY=\star_4 d\Omega_3$6 (Banerjee, 2010). This establishes that the term can carry a boundary torsion charge even though it leaves the bulk equations unchanged.

The same paper emphasizes a second structural distinction: metric holography survives in the presence of torsion, but torsion itself does not become holographic in the same sense. The Nieh–Yan term is a genuine torsion boundary invariant, yet it does not restore a bulk-boundary relation for torsion analogous to the usual Einstein–Hilbert one (Banerjee, 2010).

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

A standard anomaly formula in the literature inserts the Nieh–Yan density into the divergence of the axial current with a dimensionful coefficient: $NY=\star_4 d\Omega_3$7 or schematically $NY=\star_4 d\Omega_3$8 (Valle et al., 2021, Nissinen et al., 2019). In this form the coefficient is UV-sensitive and nonuniversal, and this nonuniversality is central to the long-standing controversy over whether the Nieh–Yan term is a genuine anomaly coefficient or a regularization artifact.

A heat-kernel analysis sharpens that point. In four dimensions the Nieh–Yan contribution appears in the $NY=\star_4 d\Omega_3$9 coefficient of the heat-kernel expansion and therefore carries a factor $\Omega_3=\eta_{ab}e^a\wedge T^b$0. Since the Dirac index is $\Omega_3=\eta_{ab}e^a\wedge T^b$1-independent, the conclusion drawn in recent work is that if the Dirac index is well-defined, then the Nieh–Yan invariant must vanish on manifolds without boundaries (Erdmenger et al., 2024). In zeta-function regularization the chiral variation depends on $\Omega_3=\eta_{ab}e^a\wedge T^b$2, so the Nieh–Yan term is absent; in cutoff regularization a $\Omega_3=\eta_{ab}e^a\wedge T^b$3 term appears, but explicit local counterterms remove the bulk and boundary Nieh–Yan pieces (Erdmenger et al., 2024).

This leads to a strong claim: the non-thermal Nieh–Yan coefficient is renormalization-scheme dependent and can be shifted away by local counterterms (Erdmenger et al., 2024). The same work argues that on manifolds with compact boundaries the Nieh–Yan density may arise both in bulk and on the boundary, but those terms remain removable, whereas finite bulk Pontryagin-type terms and a boundary Chern–Simons term survive. The boundary Chern–Simons term is then interpreted as a torsional anomalous Hall effect (Erdmenger et al., 2024).

A related proposal emphasizes a different criterion for nonzero contributions. In smooth torsion backgrounds the Nieh–Yan term can be removed by a chiral field redefinition and does not contribute to the chiral anomaly, but singular vierbeins, boundaries, or IR scales such as a cosmological constant or temperature can generate finite contributions (Rasulian et al., 2023). This suggests that the anomaly question is inseparable from global geometry, singular structure, and the choice of regulator.

A further reinterpretation departs from the conventional anomaly narrative altogether. Using holography, the 2024 analysis argues that the $\Omega_3=\eta_{ab}e^a\wedge T^b$4 axial symmetry remains unbroken and that the anomaly arises instead from a breakdown of Hodge-duality relations between fermion bilinears once torsion is present (Hoyos et al., 2024). In that framework, the three-form bilinear $\Omega_3=\eta_{ab}e^a\wedge T^b$5 acquires a transverse component that is no longer simply the Hodge dual of the conserved axial current, so the usual relation $\Omega_3=\eta_{ab}e^a\wedge T^b$6 fails. The conserved axial current and the torsion-coupled spin current then become distinct operators (Hoyos et al., 2024). This suggests that the phrase “Nieh–Yan anomaly” encompasses several non-equivalent mechanisms.

4. Thermal response and nondissipative transport

At finite temperature, a distinct line of work replaces the nonuniversal $\Omega_3=\eta_{ab}e^a\wedge T^b$7 coefficient by a $\Omega_3=\eta_{ab}e^a\wedge T^b$8 response. In topological Weyl materials the proposed thermal anomaly is

$\Omega_3=\eta_{ab}e^a\wedge T^b$9

with $T^2$0 dimensionless, and for a single chiral fermion the coefficient is reported as

$T^2$1

This is presented as a more robust low-energy response because $T^2$2 is an infrared physical scale rather than a regulator artifact (Nissinen et al., 2019).

The torsional Landau-level analysis makes the thermal coefficient more concrete. In a torsional magnetic field, all higher torsional Landau levels cancel in the axial current, and only the lowest torsional Landau levels contribute, reducing the effective problem from $T^2$3 dimensions to $T^2$4 dimensions (Huang et al., 2019). In that reduced theory the anomaly coefficient is the free energy density of the $T^2$5-dimensional conformal field theory. At finite temperature,

$T^2$6

so the thermal Nieh–Yan coefficient is proportional to the central charge $T^2$7 (Huang et al., 2019). The same framework yields an anomalous thermal Hall current

$T^2$8

which is proposed as an experimental fingerprint of the thermal Nieh–Yan effect in Weyl semimetals (Huang et al., 2019).

Recent holography reaches a parallel thermal conclusion. In an $T^2$9 black brane background with $e^a$0, the torsion-induced response obeys

$e^a$1

so the Nieh–Yan coefficient scales as $e^a$2 (Hoyos et al., 2024). In that interpretation the $e^a$3 law is a torsional/topological response in the fermion spin sector rather than evidence for literal axial-symmetry breaking.

The transport formulation systematizes these results in hydrodynamics. From the equilibrium partition function obtained by transgression, the Nieh–Yan sector induces nondissipative covariant axial-vector, heat, stress, and spin currents, with anomaly equation

$e^a$4

Representative constitutive relations are

$e^a$5

and

$e^a$6

showing that torsion, vorticity, spin chemical potential, and chiral imbalance can all enter the same anomaly-controlled response sector (Valle et al., 2021). Unlike the thermal $e^a$7 coefficient in Weyl materials, however, $e^a$8 is explicitly UV-sensitive and model dependent (Valle et al., 2021).

5. Emergent geometry and condensed-matter realizations

In Weyl materials, the Nieh–Yan invariant appears through emergent geometry. Near a Weyl point,

$e^a$9

and slowly varying deformations of the band structure are encoded in emergent tetrads $T^a$0, while nontrivial spatial dependence of the tetrads produces torsion (Nissinen et al., 2019). This makes the low-energy quasiparticles behave as chiral fermions in an effective curved and torsional background, providing a direct condensed-matter realization of the torsional density.

Chiral $T^a$1-wave superfluids and superconductors furnish a more detailed realization. In $T^a$2He-A and related systems, the low-energy Weyl quasiparticles experience an emergent Riemann–Cartan spacetime with tetrads built from the order-parameter triad and superflow, nonzero torsion, and in general nonzero curvature (Nissinen, 2019). In that setting the anomalous Galilean momentum conservation is identified with the gravitational Nieh–Yan chiral anomaly,

$T^a$3

with the nonuniversal cutoff fixed by the microscopic weak-coupling scale as $T^a$4 (Nissinen, 2019). A thermal version of the same response is then argued to take the form

$T^a$5

with $T^a$6 for a single complex chiral fermion in the explicit calculation adopted there (Nissinen et al., 2019).

Another condensed-matter route uses acoustic phonons in Kramers–Weyl semimetals of non-magnetic chiral crystals. There, strain does not act as a pseudo-gauge field; instead it modifies the local frame field $T^a$7 and produces torsion (Liu, 2021). Integrating out the Kramers–Weyl fermions yields a Nieh–Yan-induced phonon action

$T^a$8

which mixes the two transverse phonon polarizations into circularly polarized modes with

$T^a$9

The resulting phonon angular momentum forms a hedgehog texture in momentum space, and a temperature gradient drives a total phonon angular momentum response (Liu, 2021).

These condensed-matter constructions all treat the Nieh–Yan sector as an emergent torsional response of low-energy chiral quasiparticles. A plausible implication is that the invariant is often more sharply observable as transport, momentum transfer, or collective-mode dynamics than as an abstract topological number.

6. Generalizations beyond Riemann–Cartan geometry

In generic metric-affine geometry, the standard topological interpretation is modified by nonmetricity. The usual Nieh–Yan density becomes

$R_{ab}$0

so it ceases to be a total divergence when $R_{ab}$1 (Bombacigno et al., 2021). This is the basis for the statement that the standard Nieh–Yan term breaks projective symmetry and loses its topological character in the presence of nonvanishing nonmetricity (Bombacigno et al., 2021).

To separate topologicity from projective invariance, a generalized metric-affine density is introduced: $R_{ab}$2 Under projective transformations, projective invariance is restored if $R_{ab}$3, whereas topologicity is recovered only for $R_{ab}$4 (Bombacigno et al., 2021, Bombacigno et al., 2021). In the associated scalar-tensor reformulation, the Immirzi parameter becomes a dynamical scalar field, and in Bianchi I cosmology negative $R_{ab}$5 yields a classical big bounce; some branches also exhibit finite-time singularities that do not necessarily spoil geodesic completeness or wave regularity (Bombacigno et al., 2021).

The generalization program is broader than the metric-affine case. A systematic transgression construction with enlarged $R_{ab}$6 or $R_{ab}$7 connections produces a family of exact four-forms $R_{ab}$8, with the standard Nieh–Yan form recovered as a special case, for example

$R_{ab}$9

This places the standard invariant inside a larger class of torsional topological densities (Montesinos et al., 2021).

A different extension introduces a “Nieh–Yan-like” invariant

$$\mathcal N = T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab}, \qquad \mathcal N = d\mathcal Q, \qquad \mathcal Q = e^a\wedge T_a.$$0

which reduces to

$$\mathcal N = T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab}, \qquad \mathcal N = d\mathcal Q, \qquad \mathcal Q = e^a\wedge T_a.$$1

Unlike the standard Nieh–Yan term, which is tied to the axial torsion $$\mathcal N = T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab}, \qquad \mathcal N = d\mathcal Q, \qquad \mathcal Q = e^a\wedge T_a.$$2, this new invariant depends on the trace torsion vector $$\mathcal N = T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab}, \qquad \mathcal N = d\mathcal Q, \qquad \mathcal Q = e^a\wedge T_a.$$3 (Giacomo, 2023). It is exact and does not affect the pure-gravity field equations under the appropriate boundary conditions, but it can matter once matter fields source torsion (Giacomo, 2023).

Teleparallel models provide yet another extension. In metric teleparallel gravity with a scalar-coupled Nieh–Yan term, the curvature vanishes identically and the Nieh–Yan density reduces to the parity-odd torsional invariant $$\mathcal N = T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab}, \qquad \mathcal N = d\mathcal Q, \qquad \mathcal Q = e^a\wedge T_a.$$4 (Zhang et al., 2024). In the radiation era this parity-violating sector modifies the scalar perturbation system and the source of scalar-induced gravitational waves, producing a present-day SIGW spectrum that is qualitatively different from the GR result and lacks the usual monochromatic resonance divergence (Zhang et al., 2024).

Taken together, these developments indicate that the Nieh–Yan invariant is not a single fixed object with a unique role. In Riemann–Cartan geometry it is the exact torsional four-form $$\mathcal N = T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab}, \qquad \mathcal N = d\mathcal Q, \qquad \mathcal Q = e^a\wedge T_a.$$5; in anomaly theory it is a disputed and often scheme-dependent contribution; in thermal and condensed-matter settings it supports robust $$\mathcal N = T^a\wedge T_a - e^a\wedge e^b\wedge R_{ab}, \qquad \mathcal N = d\mathcal Q, \qquad \mathcal Q = e^a\wedge T_a.$$6 responses; and in generalized gravity it becomes a template for broader torsional, projectively invariant, or parity-violating constructions (Hoyos et al., 2024, Erdmenger et al., 2024).