Chiral Transport in Metric-Affine Geometries

Abstract: Anomalous transport in equilibrium fermionic fluids chirally coupled to background Weyl-type nonmetricity is studied. A formal descent analysis is carried out in which the dependence of the anomaly polynomial on the nonmetricity tensor is encoded in a Weyl invariant four-form. The constitutive relation of the axial-vector current is evaluated from the equilibrium partition function obtained using transgression techniques, showing the existence of nonmetricity-mediated chiral separation effects driven by the fluid's vorticity and the Weyl magnetic field. A second nonminimal coupling of fermionic matter to metric-affine geometries proposed in the literature is also discussed.

• The paper introduces a rigorous geometric framework showing how nonmetricity induces anomalous chiral transport in fermionic fluids.
• It employs Weyl-invariant anomaly polynomials and transgression techniques to derive constitutive relations for axial-vector currents.
• Results demonstrate that nonmetricity, through both vorticity and magnetic effects, leads to distinct equilibrium transport phenomena relevant for materials like Weyl semimetals.

Chiral Transport Induced by Nonmetricity in Metric-Affine Geometry

Introduction

The paper "Chiral Transport in Metric-Affine Geometries" (2606.04841) presents a comprehensive analysis of anomalous chiral transport phenomena in fermionic fluids at equilibrium, with a focus on the effects of background spacetime nonmetricity within the framework of metric-affine geometry. The work generalizes the traditional study of chiral anomalies and transport, moving beyond Riemann-Cartan geometry by incorporating the Weyl-type nonmetricity tensor and its consequences for the axial-vector current. It systematically constructs Weyl-invariant anomaly polynomials and derives constitutive relations for chiral currents, employing the formal machinery of transgression and descent.

Theoretical Framework and Coupling Schemes

A central aspect of the analysis is the identification of how fermions couple to nonmetric backgrounds, highlighting the distinction between minimal, Kosmann, and nonminimal couplings. It is established that, unlike torsion, nonmetricity does not minimally couple to Dirac fermions due to the symmetry structure of the Fock-Ivanenko connection. However, the paper scrutinizes two nonminimal couplings: the Weyl gauge field coupling in the SU(2,2) context, and the scheme of Rigouzzo & Zell, which introduces couplings to both the trace and traceless parts of nonmetricity, as well as edge torsion. This emphasizes the freedom in constructing matter couplings in such generalized geometric backgrounds, each prescription leading to specific transport effects.

Nonmetricity Invariants and Anomaly Structure

A major technical development in the paper is the construction of a robust, Weyl-invariant four-form built from the nonmetricity tensor, serving as a nonmetricity analog of the Nieh-Yan invariant. This form, $I_4^Q$, emerges as a central object in the anomaly polynomial, capturing the interplay between curvature and nonmetricity. The anomaly descent procedure is adapted to include nonmetricity, resulting in a generalized Chern-Simons form $\mathcal{W}_3$ and its higher-dimensional analogs.

The anomaly polynomial for the axial-vector current then takes the form

$$\mathcal{P}_6(\mathcal{F}_A, I_4^Q) = c_W\, \mathcal{F}_A \wedge I_4^Q,$$

where $c_W$ is determined by the fermionic coupling. This formalism interpolates seamlessly between anomaly inflow prescriptions and covariant anomaly currents, providing a basis for evaluating both consistent and covariant forms of the chiral anomaly in the presence of nonmetricity.

Equilibrium Partition Function and Constitutive Relations

Employing transgression techniques, the equilibrium partition function is computed by interpolating between background connections parameterized by chemical potentials for vorticity, spin, and the axial charge. In stationary spacetimes, generic expressions for bulk and boundary contributions to the partition function are obtained. Functional differentiation yields the constitutive relations of the covariant axial-vector current.

For Weyl-type nonmetricity backgrounds (i.e., vanishing shear component), the chiral current admits the constitutive relation

$$\langle \star J_5 \rangle_{\rm cov} = 2c_W\, u \wedge (\mu_Q\, \mathbf{B}_Q + \mu_Q^2\, \omega),$$

where $u$ is the fluid velocity one-form, $\mu_Q$ is the chemical potential associated with the Weyl gauge field, $\mathbf{B}_Q$ is its corresponding magnetic field, and $\omega$ is the fluid vorticity. This encapsulates two independent chiral separation effects: one mediated by nonmetricity-induced vorticity, and the other by the magnetic component of the Weyl gauge field.

Strong numerical results are provided for these transport coefficients, showing their explicit dependence on the nonmetricity chemical potential—proportional to the time-component of the Weyl gauge field—and the electron charge via $c_W = -e^2/(4\pi^2)$. The theoretical framework is extended to previous proposals for nonminimal couplings, allowing a unified treatment where nonmetricity, edge torsion, and their associated chemical potentials contribute additively and can be dialed by suitable coupling parameters.

Implications and Outlook

The results rigorously demonstrate that background nonmetricity generically induces new classes of anomalous chiral transport effects at equilibrium, distinct from those sourced by curvature or torsion alone. In particular, both vorticity and Weyl magnetic fields, when coupled to a nonzero temporal component of nonmetricity, drive axial current separation, modifying anomaly-induced hydrodynamics.

From a practical perspective, the findings have direct implications for anomalous transport in condensed matter systems, such as Weyl semimetals with point defects. The identification of nonmetricity with effective long-wavelength descriptions of lattice defects provides a geometrically natural explanation for certain transport phenomena observed in those settings. Theoretical predictions regarding static and stationary regimes (with vanishing or nonvanishing $\mathcal{W}_3$0) yield experimentally testable distinctions, especially in nonequilibrium or rotating samples.

On a more fundamental level, this synthesis of anomaly descent, metric-affine geometry, and transgressive partition functions underscores the need to account for generalized backgrounds in hydrodynamic and quantum field theoretical anomaly calculations. As couplings to nonmetricity are not uniquely fixed, their study opens avenues for further investigation into the role of spacetime geometry in quantum transport, and for possible extensions to higher dimensions, non-Abelian settings, or out-of-equilibrium dynamics.

Conclusion

This paper establishes, through rigorous geometric and field-theoretic analysis, that metric-affine geometries with nontrivial background nonmetricity generate non-dissipative, equilibrium chiral transport in fermionic fluids. The relevance for materials with emergent Weyl or Dirac fermions is immediate, and the formal development of Weyl-invariant anomaly polynomials and constitutive relations sets a new standard for describing anomalous hydrodynamics in non-Riemannian backgrounds. Future work will likely extend these results to dynamical and topological phases of matter, explore constraints from generalized symmetry, and refine the connection between geometric defect theory and quantum transport.