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arxiv: 2606.18779 · v1 · pith:ZVNDPWELnew · submitted 2026-06-17 · ✦ hep-th · cond-mat.str-el

Hydrodynamics of perfect fluids with anomalies from the fermionic path integral

Alexander G. Abanov , Andrea Cappelli This is my paper

Pith reviewed 2026-06-26 20:12 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-el
keywords fermionic path integralanomalous hydrodynamicstransgression formsperfect fluidsanomaly inflowDirac fermionWeyl fermionzero-temperature hydrodynamics
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The pith

Integrating out fermions from the path integral produces the effective action for anomalous hydrodynamics of perfect fluids at zero temperature.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes the path integral for Dirac fermions coupled to vector and axial gauge backgrounds near the infrared limit, including a residual current-current interaction. After integrating out the fermions, it obtains a semiclassical effective action expressed in currents that matches a previously proposed hydrodynamic action for perfect barotropic fluids with anomalies. The same procedure yields analogous actions for Weyl fermions and for Dirac fermions with independent vector and axial currents. These actions include four- and five-dimensional bulk-boundary terms identified as transgression forms arising from anomaly inflow. The derivation supplies a microscopic account of the ingredients needed for anomalous hydrodynamics and shows how restricted variations implement the infrared reduction to a local hydrodynamic description.

Core claim

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 dyn

What carries the argument

The semiclassical low-energy effective action in terms of currents obtained after integrating out fermions from the Dirac path integral, which matches the hydrodynamic action and incorporates transgression forms for anomaly inflow.

If this is right

  • The hydrodynamic action for perfect barotropic fluids with anomalies follows directly from integrating out fermions.
  • Transgression forms supply the necessary bulk-boundary terms that encode anomaly inflow in the effective actions.
  • Restricted variations of the five-dimensional transgression terms yield the four-dimensional hydrodynamic equations of motion.
  • The same path-integral procedure associates distinct hydrodynamic actions with the Weyl fermion and with Dirac fermions carrying independent vector and axial currents.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The derivation indicates that anomaly-induced transport in hydrodynamics originates in the underlying fermionic degrees of freedom rather than being added by hand.
  • Analogous microscopic derivations could be attempted for other anomalous systems, such as those appearing in condensed-matter realizations of chiral fermions.
  • The clarification of the infrared reduction may allow systematic inclusion of higher-order corrections while preserving the hydrodynamic structure.

Load-bearing premise

The infrared limit taken in the presence of the residual irrelevant current-current interaction produces the specific semiclassical effective action that matches the hydrodynamic formulation.

What would settle it

An explicit evaluation of the fermionic path integral in the infrared that produces an effective action differing from the proposed hydrodynamic form in its current dependence or anomaly terms would falsify the claimed correspondence.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper analyzes the path integral of Dirac fermions coupled to vector and axial gauge backgrounds near the infrared limit in the presence of a residual irrelevant current-current interaction. After integrating out the fermions, a semiclassical low-energy effective action is obtained in terms of currents; this is shown to match the hydrodynamic action previously proposed for perfect barotropic fluids with anomalies at zero temperature. Analogous constructions are given for the Weyl fermion and for the Dirac fermion with independent vector and axial currents. The resulting actions contain four- and five-dimensional bulk-boundary transgression forms arising from anomaly inflow. The path-integral derivation is used to explain several structural ingredients of the hydrodynamic formulation and to implement the infrared reduction via restricted variations that produce four-dimensional equations of motion from the five-dimensional terms.

Significance. If the explicit integration and matching hold, the work supplies a microscopic derivation of anomalous perfect-fluid hydrodynamics directly from the fermionic path integral. It thereby accounts for the appearance of transgression forms, the necessity of restricted variations, and the role of anomaly inflow within the hydrodynamic action, strengthening the link between quantum field theory anomalies and the effective classical description.

minor comments (2)
  1. The precise form of the residual irrelevant current-current interaction and the technical steps that implement the infrared reduction (restricted variations) should be stated explicitly in the main text, with the resulting effective action written out before the matching claim is asserted.
  2. The transgression forms are identified with anomaly inflow; a short appendix or paragraph comparing their explicit expressions to the standard Chern-Simons forms would improve readability for readers unfamiliar with the generalization to two gauge fields.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and detailed summary of our work, as well as the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation from path integral is independent

full rationale

The paper derives the semiclassical effective action by integrating out Dirac fermions from the standard path integral in gauge backgrounds, then identifies the result with a prior hydrodynamic formulation via explicit computation and anomaly inflow. The IR reduction via restricted variations is part of the stated procedure rather than a fitted input. No load-bearing step reduces by construction to a self-citation, ansatz, or renamed known result; the matching is presented as an outcome of the integration, not an input. This is the most common honest non-finding for a first-principles calculation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the assumptions explicitly invoked there.

axioms (2)
  • standard math The path integral of the Dirac fermion with vector and axial gauge backgrounds is well-defined near the infrared limit.
    Standard starting point of the analysis stated in the abstract.
  • domain assumption The residual irrelevant current-current interaction permits a semiclassical low-energy effective action after fermion integration.
    This assumption is required for the infrared reduction described in the abstract.

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discussion (0)

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Reference graph

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