Hydrodynamics of perfect fluids with anomalies from the fermionic path integral

This paper derives hydrodynamic actions for perfect fluids with anomalies by analyzing the fermionic path integral near the infrared limit, demonstrating that integrating out fermions yields effective actions featuring transgression forms that provide a microscopic justification for anomaly incorporation and clarify the reduction to local hydrodynamic equations of motion.

The Gist

The Big Picture: From Tiny Particles to Flowing Fluids

Imagine you have a crowd of tiny, invisible dancers (fermions) moving around on a stage. In the world of high-energy physics, these dancers are usually described by complex quantum rules. However, sometimes, when you zoom out and look at the "big picture" (low energy), these dancers stop acting like individuals and start moving together like a single, flowing liquid.

This paper asks a fundamental question: How do we mathematically describe this "liquid" when the dancers have a special, quirky property called an "anomaly"?

In physics, an "anomaly" is like a glitch in the rules. Usually, if you have a symmetry (like balancing left and right), the laws of physics respect it. But in quantum mechanics, sometimes this balance breaks in a very specific way. The authors of this paper wanted to build a mathematical "instruction manual" (an action) that describes how this fluid flows while respecting these broken rules.

The Main Characters

1. The Dancers (Fermions): The fundamental particles.
2. The Stage (Background Fields): Invisible forces (like magnetic fields) that the dancers move through.
3. The Glitch (Anomalies): A specific way the dancers' movement breaks the usual symmetry between "left" and "right" spins.
4. The Fluid (Hydrodynamics): The collective flow of the dancers when they are crowded together.

The Journey: From Micro to Macro

The authors took a three-step journey to solve this puzzle:

Step 1: The "Microscopic" View (The Path Integral)

They started with the standard quantum description of the dancers. They used a mathematical tool called a "path integral," which is like summing up every possible way the dancers could move to find the most likely outcome.

• The Twist: They added a tiny, leftover interaction between the dancers (like a faint whisper between them). This interaction forces the dancers to organize into a fluid state.
• The Result: When they removed the individual dancers from the equation (mathematically "integrating them out"), they were left with a new set of rules describing the fluid itself.

Step 2: The "Glitch" Fix (Transgression Forms)

Here is where it gets tricky. The new fluid rules they found had a problem: they didn't look right when you changed the "stage" (the background fields). It was like a map that looked correct from one angle but fell apart from another.

To fix this, they discovered the fluid action needed a special mathematical ingredient called a Transgression Form.

• The Analogy: Imagine you are building a house. You have the walls (4D space) and the foundation (5D space). Usually, you just build the walls. But because of the "glitch" (anomaly), you need to connect the walls to a hidden basement (the 5th dimension) to keep the structure stable.
• The "Transgression": This is the mathematical bridge that connects the fluid's movement to the background forces. It's a generalization of something called a "Chern-Simons form." Think of it as a special type of glue that holds the fluid's behavior together, ensuring the "glitch" is handled correctly. This glue involves two sets of fields: the fluid's own momentum and the external forces acting on it.

Step 3: Three Types of Fluids

The paper didn't just find one set of rules; they found three distinct ways the fluid could behave, depending on how the dancers are grouped:

1. The Single-Fluid Theory: This describes a standard fluid where the dancers are mixed together. It's the "barotropic" fluid (where pressure depends only on density).
2. The Two-Fluid Theory: This describes a fluid where the "left-spinning" dancers and "right-spinning" dancers act like two separate streams flowing side-by-side, interacting but distinct.
3. The Weyl Fluid: This describes a fluid made of only one type of dancer (only left-spinning or only right-spinning).

The Secret Sauce: Restricted Variations

The final and perhaps most surprising part of the paper is about how to read the instruction manual.

In standard physics, to find how a system moves, you usually wiggle every part of it in every possible direction and see what happens. But the authors argue that for fluids, you can't just wiggle everything randomly. You must only wiggle the parts that respect the fluid's natural symmetries (like sliding the fluid along a surface or rotating it).

• The Analogy: Imagine a river. If you try to push the water up the riverbank (a direction the water can't naturally go), you aren't describing the river's flow; you're describing a wall. To understand the river, you must only look at how the water moves along the riverbed.
• The Result: By restricting their "wiggles" to only these natural fluid movements, they were able to turn their complex, 5-dimensional mathematical formulas into simple, local 4-dimensional equations that describe the actual flow of the fluid. This proves that the "glitch" (anomaly) doesn't break the fluid; it just adds a specific, predictable twist to its motion.

Summary of Claims

• Origin: They derived these fluid rules directly from the quantum behavior of fermions, rather than guessing them.
• Structure: The rules for these fluids naturally involve a mix of 4-dimensional space (our world) and 5-dimensional space (a mathematical tool to handle the anomalies).
• New Terms: They found that the "glue" (transgression form) connecting the fluid to the background forces isn't unique; it can include extra, flexible terms (like adjustable knobs) that don't break the rules but might change how the fluid transports energy.
• Method: They showed that to get the correct physical laws, you must treat the fluid action as a "constrained system," only allowing variations that look like natural fluid motions.

In short, the paper provides a rigorous "microscopic" proof for how quantum particles with broken symmetries turn into flowing fluids, and it gives us the precise mathematical blueprint for how those fluids move.

Technical Summary

Problem Statement
The paper addresses the challenge of deriving the action formulation of hydrodynamics for perfect fluids with chiral anomalies directly from a microscopic fermionic path integral. While previous works had proposed hydrodynamic actions for "single-fluid" (Dirac fermion) and "two-fluid" (independent vector and axial currents) systems based on anomaly inflow arguments, these proposals lacked a rigorous derivation from the underlying quantum field theory. Furthermore, the presence of four- and five-dimensional topological terms (Chern-Simons and transgression forms) in these actions necessitated a specific variational principle to recover local four-dimensional equations of motion, a connection that required clarification. The authors aim to bridge the gap between the effective field theory of fermions in the infrared limit and the macroscopic hydrodynamic description, specifically for barotropic fluids at zero temperature.

Methodology
The authors employ a path-integral approach starting from the Dirac fermion theory in the presence of vector ($A_\mu$) and axial ($\tilde{A}_\mu$) background gauge fields. The methodology proceeds through the following steps:

1. Infrared Limit and Interaction: They assume the system undergoes a renormalization group (RG) flow driven by a relevant interaction into a low-energy fluid phase. This phase is modeled by adding a residual, irrelevant current-current interaction ($S_{int} \propto j_\mu j^\mu$) to the free Dirac action.
2. Hubbard-Stratonovich Transformation: To reformulate the theory in terms of currents suitable for hydrodynamics, they introduce Lagrange multipliers ($p_\mu$ and $\tilde{p}_\mu$) to enforce the definition of vector and axial currents. This allows the integration of fermionic degrees of freedom.
3. Semiclassical Integration: The fermions are integrated out in a semiclassical approximation. This involves computing the fermionic determinant using heat-kernel regularization. The authors utilize the concept of "weighted traces" to relate the determinant with the new dynamical background fields ($\pi_\mu = p_\mu + A_\mu$) to the standard regularization, identifying the difference as a local polynomial term.
4. Anomaly Matching and Gauge Invariance: The resulting effective action must satisfy 't Hooft anomaly matching conditions. The authors demonstrate that the naive effective action is not locally gauge invariant. To restore gauge invariance, they add specific local polynomial terms. These terms, combined with the bulk-boundary topological terms, are identified as transgression forms—generalizations of Chern-Simons forms involving two gauge fields (the dynamical fluid momentum and the background field).
5. Variational Principle: To extract local four-dimensional equations of motion from the mixed 4d-5d action, the authors apply a restricted variational principle. Instead of arbitrary variations, they vary the action only with respect to "admissible" variations: gauge transformations and diffeomorphisms of the fluid variables. This restriction is motivated by the physical requirement that hydrodynamics describes the slow modes remaining after equilibration.

Key Contributions and Results

• Derivation of the Single-Fluid Action: The authors derive the hydrodynamic action for a Dirac fermion (single-fluid theory) directly from the path integral. They identify the dynamical pseudo-scalar field $\psi$, previously introduced phenomenologically, as the Abelian Wess-Zumino field arising from the anomaly. The action is shown to be composed of a pressure term and transgression forms.
• Derivation of the Two-Fluid and Weyl Actions: Extending the construction to include independent axial currents, they derive the "two-fluid" hydrodynamic action and the "Weyl-fluid" action (for a single Weyl fermion). These actions naturally involve dynamical fields extending into a five-dimensional bulk via Chern-Simons terms.
• Identification of Transgression Forms: The paper explicitly identifies the anomalous terms in the hydrodynamic actions as transgression forms. These are shown to be the unique gauge-invariant objects connecting the dynamical fluid momentum and the background gauge fields.
• New Gauge-Invariant Terms: The derivation reveals the existence of additional gauge-invariant four-dimensional polynomial terms in the action, parameterized by two dimensionless couplings ($\beta, \gamma$). While these terms do not alter the anomaly structure (which is fixed by topology), they affect the constitutive relations of the currents and potentially transport coefficients.
• Resolution of the Variational Problem: The paper clarifies how to obtain local equations of motion from the 5D topological terms. By restricting variations to admissible gauge and diffeomorphism transformations, the authors recover the standard Carter-Lichnerowicz form of the Euler equations and the correct anomalous conservation laws.
  • For the two-fluid theory, they show that a consistent local variational principle requires independent diffeomorphisms acting on the chiral sectors ($\pi_+, \pi_-$), rather than common diffeomorphisms on the vector/axial fields.
  • For the single-fluid theory, the reduction $\tilde{\pi} \to d\psi$ leads to a consistent variational problem involving the pseudo-scalar field.

Significance and Claims
The paper claims to provide a "microscopic" explanation for the ingredients of the action formulation of hydrodynamics with anomalies. Specifically:

• It justifies the introduction of the Wess-Zumino field and the specific structure of the anomalous terms (transgression forms) as necessary consequences of gauge invariance and anomaly matching in the infrared limit of the fermionic path integral.
• It clarifies the "infrared reduction" required to pass from a general effective field theory to a local hydrodynamic description. The authors argue that the restricted variational principle is not merely a mathematical trick but corresponds to the physical projection onto the subspace of slow, symmetry-determined motions that survive hydrodynamic relaxation.
• It confirms and extends previous proposals based on anomaly inflow, adding the specific gauge-invariant terms with free couplings that were previously undetermined.
• The work suggests that these effective actions serve as candidates for anomalous bosonic theories in four dimensions, potentially relevant for the study of bosonization beyond two spacetime dimensions.

The authors remain modest regarding future applications, noting that the inclusion of temperature, entropy, and transport coefficients (and their dependence on the new gauge-invariant terms) are subjects for future investigation. They also acknowledge that the treatment of mixed 4d-5d actions and their quantization remains an open technical challenge.