Dissipative non-Abelian fluids from Scherk-Schwarz dimensional reduction

Abstract: We construct a $d$-dimensional dissipative colored fluid by Scherk--Schwarz reduction of a neutral viscous conformal fluid in $D=d+n$ dimensions on an $n$-dimensional unimodular group manifold. The off-diagonal components of the higher-dimensional stress tensor become non-Abelian color currents, while the higher-dimensional shear tensor induces shear, bulk-like and vector-dissipative structures in the reduced theory. We derive the map for the equation of state, sound speed, color current, entropy current and first-order transport coefficients. In particular, [ η=\ee^{α\varphi}\coshξ\,\heta,\qquad τ=η\,\frac{n}{(D-1)(d-1)},\qquad κ=η\sinh^2ξ. ] We also spell out the hydrodynamic-frame issue induced by dimensional reduction, discuss the status of the internal rapidity field $ξ$, and give a detailed account of how the second law descends from the parent theory, including the roles of temperature-dependent viscosity, non-unimodular groups and possible choices for $ξ$. The construction should be regarded as a toy model for non-Abelian dissipative hydrodynamics with the potential of paving the way to direct phenomenological model of, for example, quark--gluon plasma.

• The paper derives non-Abelian colored hydrodynamics from a higher-dimensional neutral viscous conformal fluid using Scherk-Schwarz reduction.
• It maps thermodynamic quantities and transport coefficients, linking higher-dimensional viscosity to emergent lower-dimensional gauge currents.
• The methodology preserves the second law of thermodynamics and provides a controlled framework applicable to quark–gluon plasma studies.

Dissipative Non-Abelian Fluids from Scherk-Schwarz Dimensional Reduction

Overview

This work constructs $d$-dimensional dissipative non-Abelian (colored) hydrodynamics by a Scherk-Schwarz reduction of a $D$-dimensional neutral viscous conformal fluid over a unimodular group manifold. The reduction yields a lower-dimensional fluid theory with emergent non-Abelian color currents and a sector of dissipative transport coefficients geometrically determined by the higher-dimensional theory. The paper elucidates the precise mapping of the equation of state, sound speed, conservation laws, hydrodynamic frame, and the descent of the second law of thermodynamics from the parent neutral theory. This construction serves as a constrained and explicit toy model for non-Abelian dissipative hydrodynamics relevant for applications to systems such as the quark–gluon plasma.

Geometric Background and Reduction Mechanism

The Scherk-Schwarz dimensional reduction is deployed on an $n$-dimensional unimodular group manifold $G$. The geometric framework embeds a non-Abelian gauge structure in the lower-dimensional theory via the group manifold isometries and associated left-invariant forms. The core geometric ansatz for the metric,

$$d\hat{s}^2 = e^{2\alpha\varphi}g_{\mu\nu}(x) d x^\mu d x^\nu + e^{2\beta\varphi(x)} h_{mn}(x) \nu^m \nu^n \,,$$

organizes the lower-dimensional gravitational, Yang–Mills, and scalar moduli fields. The unimodularity condition $f_{mn}{}^n=0$ is essential for the covariance and conservation (in the absence of anomalies) of color and entropy currents in the lower-dimensional theory.

Fluid Ansatz and Kinematics

The higher-dimensional fluid velocity $\hat{u}^A$ is decomposed into external ($u^a$) and internal ($n^\alpha$) sectors with an internal rapidity $\xi$: $D$0 with standard normalization conventions. The internal rapidity $D$1 controls the projection of parent dynamics onto the internal group manifold directions, and the color orientation is encoded in $D$2. These parameters dictate the emergence and structure of non-Abelian charge and current in the lower-dimensional theory.

Mapping of Thermodynamic and Hydrodynamic Quantities

Equation of State and Sound Speed

The reduced equation of state and sound speed are algebraic functions of the parent theory data and the reduction parameters:

• The energy and pressure map as

$D$3

• The sound speed is given by

$D$4

• For a parent conformal fluid, conformality is lost generically after reduction; the ratio $D$5 interpolates as a function of $D$6 between the conformal value and lower, more non-conformal values.

Color Current and Charge

The color current is realized as the mixed internal-external component of the uplifted stress tensor and reads

$D$7

with conservation following from unimodularity and the Bianchi identities.

First-Order Dissipative Transport

A key result is the mapping of higher-dimensional shear viscosity to lower-dimensional transport coefficients: $D$8

• $D$9: shear viscosity
• $n$0: induced geometric bulk-like coefficient (absent in the parent, emerges from reduction)
• $n$1: vector-dissipative coefficient (emerges due to internal motion)

These coefficients are not independent; their values are entangled via the reduction geometry and the single parent viscosity $n$2. Notably, no independent color-diffusion coefficient arises, and vector-dissipative structures arise exclusively from internal projections.

Hydrodynamic Frame and Conservation Laws

The hydrodynamic frame of the parent is not generically conserved after reduction. Even if the parent is in the Landau frame, the lower-dimensional stress tensor acquires contributions from the color sector: $n$3 Frame ambiguities must be handled appropriately, and the lower-dimensional theory is naturally written in a general frame—consistent with modern treatments of first-order relativistic hydrodynamics stability.

Second Law and Entropy Production

The formalism ensures that the non-negative entropy production law in the higher-dimensional theory is preserved in the reduced, colored theory, provided unimodularity holds. The positivity is inherited directly from the quadratic form of the higher-dimensional viscous sector and the unimodular reduction of vector divergences. The descent of the second law is formalized as: $n$4 Non-unimodular reductions produce anomalous source terms without definite sign, breaking the automatic conservation and positivity in the lower-dimensional theory.

Promoting the internal rapidity $n$5 to a dynamical field would necessitate additional structure (e.g., a Josephson-type equation), otherwise, the hydrodynamic system is underdetermined.

Example: $n$6 with $n$7

The paper provides an explicit $n$8, $n$9 reduction (i.e., $G$0), showing the emergence of $G$1 non-Abelian colored hydrodynamics with all transport data controlled by the higher-dimensional $G$2 and the internal rapidity $G$3. The mapping renders

$G$4

with explicit expressions for all induced transport coefficients, demonstrating the practical efficacy and constraints of the formalism.

Implications and Future Directions

The construction provides a systematic, geometrically constrained embedding of non-Abelian, first-order dissipative hydrodynamics into higher-dimensional neutral systems. The main theoretical implication is a reduced freedom in the lower-dimensional transport data—a strong constraint compared to the general symmetry-allowed constitutive relations. Practically, the model provides a controlled laboratory for colored fluid dynamics and is directly extensible to the study of quark–gluon plasma, holography, and beyond.

Directions for further work include:

• Extension to second-order causal (Israel–Stewart-type) theories to systematically address stability and causality
• Inclusion of anomalous transport effects
• Analysis of non-unimodular reductions for hydrodynamic anomalies
• Studies of explicit black-brane/cosmological backgrounds in string-effective actions

These generalizations will clarify how much of the non-Abelian gradient expansion is structurally and quantitatively controlled by a geometric higher-dimensional embedding.

Conclusion

This paper establishes a transparent map from higher-dimensional neutral dissipative hydrodynamics to lower-dimensional non-Abelian colored fluids via Scherk-Schwarz reduction. The emergent non-Abelian structure and dissipative transport sectors are tightly controlled by the geometric parameters and the higher-dimensional viscosity. While restrictive compared to the general hydrodynamic effective theory, this framework provides a sharpened tool for the theoretical and phenomenological exploration of colored dissipative phenomena in gauge and gravity contexts (2605.23842).