Hydrodynamics of perfect fluids with anomalies from the fermionic path integral

Abstract: The path integral of the Dirac fermion with vector and axial gauge backgrounds is analyzed near the infrared limit in the presence of residual irrelevant current-current interaction. After integrating out fermions, a semiclassical low-energy effective action is obtained, written in terms of currents. Its expression is found to correspond to the hydrodynamic action previously proposed for perfect barotropic fluids with anomalies at zero temperature. This approach also leads to two further hydrodynamic actions to be associated, respectively, with the Weyl fermion, and the Dirac fermion having independent vector and axial currents. These actions feature four- and five-dimensional bulk-boundary terms, owing to anomaly inflow, which are identified as being the so-called transgression forms. These are generalizations of Chern--Simons forms that involve two gauge fields: the dynamical field and the background field. The path-integral argument provides a ``microscopic'' explanation for several ingredients of the action formulation of hydrodynamics that are necessary to incorporate anomalies. It also clarifies the infrared reduction required to pass from the effective field theory to a local hydrodynamic description. This reduction is implemented by considering restricted variations of the action, familiar from hydrodynamics, which at the same time lead to four-dimensional equations of motion from the five-dimensional transgression terms.

• The paper presents a rigorous derivation of anomalous hydrodynamics for perfect fluids via semiclassical integration of Dirac and Weyl fermions.
• It employs a Hubbard–Stratonovich transformation to recast current dynamics and restore gauge invariance with topological transgression forms akin to Chern–Simons theory.
• The study clarifies the variational principles that yield local hydrodynamic equations, bridging 4d fluid behavior with 5d bulk-boundary anomaly inflow.

Hydrodynamical Effective Action Construction via Fermionic Path Integrals

Overview

The paper "Hydrodynamics of perfect fluids with anomalies from the fermionic path integral" (2606.18779) presents a formal derivation of the anomalous hydrodynamics for perfect barotropic fluids, starting directly from the fermionic path integral for Dirac and Weyl fermions in the presence of vector and axial gauge backgrounds. The analysis proceeds via rigorous semiclassical integration over fermionic degrees of freedom, exploiting the structure of chiral anomalies and anomaly inflow. The derived low-energy effective actions reproduce previous hydrodynamic proposals and extend them by identifying explicit gauge-invariant forms, linking with transgression generalizations of Chern–Simons theory. The work also clarifies the variational principles required to obtain local hydrodynamic equations from these actions, especially concerning the interplay between bulk-boundary topological terms in the presence of anomalies.

Anomalies and Path Integral Structure

The starting point is the Dirac path integral with both vector ($A_\mu$) and axial ($\tilde{A}_\mu$) backgrounds. Upon choosing the appropriate gauge-preserving regularization, the path integral manifests the non-invariance of the axial symmetry due to the chiral anomaly. The partition function transforms as:

$$Z[\tilde{A} + d\tilde{\lambda}, A + d\lambda] = Z[\tilde{A}, A]\, \exp\left(i \int \tilde{\lambda}[\gamma\, dA\,dA + \alpha\, d\tilde{A}\,d\tilde{A}]\right)$$

with anomaly coefficients $\gamma=1$, $\alpha=1/3$ for a single Dirac fermion. The associated Ward identities yield conservation for the vector current and anomalous non-conservation for the axial current, precisely as expected for the consistent current structure.

Gauge invariance of the quantum theory is restored by introducing five-dimensional bulk Chern–Simons terms via anomaly inflow, yielding mixed $4d$–$5d$ actions. These bulk-boundary correspondences naturally motivate a field extension and the introduction of compensating fields such as a pseudo-scalar $\psi$, leading to the Wess–Zumino–Witten (WZW) construction.

Semiclassical Hydrodynamic Effective Action

The paper assumes an infrared regime induced by relevant interaction in the ultraviolet theory (e.g., residual current-current interactions), allowing for a hydrodynamic phase with a gap. The path integral is rewritten using a Hubbard–Stratonovich transformation, introducing dynamical current fields $j^\mu$. Upon integrating out the fermions in the semiclassical approximation, the effective action is recast in terms of fluid momentum variables, with the anomaly structure faithfully preserved due to 't Hooft anomaly matching.

Crucially, a Legendre transform relates the hydrodynamic pressure to the current interaction, yielding:

$$S_{\rm eff}[\psi, p] = \int \psi (d\pi\,d\pi + d\tilde{A}\,d\tilde{A}) + P(p) - \int_{\mathcal{M}_5} [\tilde{A}\,dA\,dA + \tilde{A}\,d\tilde{A}\,d\tilde{A}]$$

where $\tilde{A}_\mu$0 is the canonical fluid momentum. Gauge invariance of this low-energy action is restored by adding local polynomial terms, justified in detail via mathematical analysis of weighted traces and Wodzicki residues.

For $\tilde{A}_\mu$1, the theory reduces, via bosonization, precisely to free scalar hydrodynamics, matching known results. In $\tilde{A}_\mu$2, the action is characterized by extensive gauge structure and engineered anomaly terms.

Transgression Forms and Bulk-Boundary Structure

A central technical achievement is the explicit identification of the anomaly terms as Abelian transgression forms, generalizing Chern–Simons constructions to pairs of gauge fields: one dynamical (fluid momentum) and one background. The transgression form for a pair $\tilde{A}_\mu$3 is:

$\tilde{A}_\mu$4

Extending this yields mixed transgression terms for both vector and axial couplings, underpinning the topological structure of the effective hydrodynamic actions.

Gauge-invariant freedom in these actions is exhaustively analyzed, showing that up to gauge-invariant polynomials, transport coefficients and physical observables may depend on additional non-anomalous couplings.

Generalizations: Weyl and Two-Fluid Hydrodynamics

Beyond the single-fluid Dirac theory, analogous path integral manipulations produce:

• Weyl fluid actions: Derived via chiral decomposition. Each sector corresponds to a unique transgression form with fixed anomaly coefficient. The five-dimensional terms are dynamical and not reducible to local $\tilde{A}_\mu$5 action structure.
• Two-fluid actions: Involving independent vector and axial degrees of freedom, with pressure function $\tilde{A}_\mu$6 and explicit bulk transgression terms. The structure admits chiral decomposition, linking directly to left/right Weyl fluids.

Both actions are derived rigorously from the fermionic determinant after semiclassical reduction and gauge-invariant polynomial additions.

Variational Principles and Local Hydrodynamic Equations

The derivation and consistency of hydrodynamic equations is contingent on the restricted variational principle: only gauge/diffeomorphism-induced variations are admissible, reflecting the constrained nature of Eulerian hydrodynamics. The interplay of these variational restrictions with bulk-boundary mixed $\tilde{A}_\mu$7–$\tilde{A}_\mu$8 actions is nontrivial; only under these admissible variations do the resulting equations of motion become local and reproduce the correct anomalous conservation laws and force-balance equations.

In two-fluid theory, independent variations of chiral sectors are necessary; common diffeomorphisms are insufficient. Reduction of degrees of freedom (e.g., setting the axial one-form to a pseudo-scalar gradient) produces the single-fluid theory and modifies the variational structure accordingly.

The admissible variation framework is also rigorously justified by an argument regarding infrared reduction: after hydrodynamic relaxation, only symmetry-preserving slow fluid modes survive.

Practical and Theoretical Implications

• Transport Properties: The explicit form of gauge-invariant polynomial terms allows for the investigation of their contribution to transport coefficients, beyond the purely anomalous effects.
• Hydrodynamic-Bosonization Correspondence: In $\tilde{A}_\mu$9, the hydrodynamic action coincides exactly with bosonization results. In $$Z[\tilde{A} + d\tilde{\lambda}, A + d\lambda] = Z[\tilde{A}, A]\, \exp\left(i \int \tilde{\lambda}[\gamma\, dA\,dA + \alpha\, d\tilde{A}\,d\tilde{A}]\right)$$0, the effective actions are candidate anomalous bosonic theories for higher-dimensional bosonization—a major subject in quantum field theory.
• Mixed Bulk-Boundary Systems: The treatment clarifies the construction of actions involving both local $$Z[\tilde{A} + d\tilde{\lambda}, A + d\lambda] = Z[\tilde{A}, A]\, \exp\left(i \int \tilde{\lambda}[\gamma\, dA\,dA + \alpha\, d\tilde{A}\,d\tilde{A}]\right)$$1 fluid dynamics and topological $$Z[\tilde{A} + d\tilde{\lambda}, A + d\lambda] = Z[\tilde{A}, A]\, \exp\left(i \int \tilde{\lambda}[\gamma\, dA\,dA + \alpha\, d\tilde{A}\,d\tilde{A}]\right)$$2 bulk inflow, essential for rigorous anomaly matching.

Future Directions

Open avenues include:

• Incorporating temperature, entropy, and non-barotropic effects, extending the formalism to real fluids or more complicated backgrounds.
• Including mixed axial–gravitational anomalies, relevant for gravitational responses in anomalous hydrodynamics.
• Investigating the role of gauge-invariant terms in measurable transport properties and their dependence on underlying microscopic interactions.
• Quantization and further exploration of $$Z[\tilde{A} + d\tilde{\lambda}, A + d\lambda] = Z[\tilde{A}, A]\, \exp\left(i \int \tilde{\lambda}[\gamma\, dA\,dA + \alpha\, d\tilde{A}\,d\tilde{A}]\right)$$3–$$Z[\tilde{A} + d\tilde{\lambda}, A + d\lambda] = Z[\tilde{A}, A]\, \exp\left(i \int \tilde{\lambda}[\gamma\, dA\,dA + \alpha\, d\tilde{A}\,d\tilde{A}]\right)$$4 mixed actions, especially their relation to bosonization and field-theoretic dualities in more than two dimensions.

Conclusion

This paper rigorously demonstrates the derivation of hydrodynamic actions with chiral anomalies from the fermionic path integral, elucidates their topological structure as transgression forms, and establishes the variational principles necessary for local hydrodynamic equations. The construction unifies geometric, effective field theory, and anomaly inflow perspectives, providing a robust platform for further developments in anomalous hydrodynamics, topological phases, and higher-dimensional bosonization.