Einstein-Dilaton-Three-Maxwell Action

• Einstein-Dilaton-Three-Maxwell Action is a five-dimensional gravitational model incorporating a dilaton and three U(1) gauge fields to represent finite chemical potential and anisotropic magnetic effects.
• The model employs independent, dilaton-dependent couplings for each Maxwell field and integrates anisotropic metric ansätze to simulate complex external field environments.
• It extends traditional holographic frameworks by facilitating systematic solution strategies and redundancy management via the Bianchi identity, relevant for modeling QCD-like plasmas.

The Einstein-dilaton-three-Maxwell action is a five-dimensional bulk gravitational model incorporating a scalar dilaton field and three U(1) gauge fields. This framework provides a holographic dual description of strongly coupled gauge theories exhibiting both finite chemical potential and spatial anisotropy, including scenarios relevant to QCD-like models in external fields. The model generalizes earlier Einstein-dilaton-Maxwell constructions by treating multiple Maxwell fields independently, with a particular focus on anisotropy and the roles of electric and magnetic sectors (Aref'eva et al., 2024).

1. Action and Field Content

The action for the Einstein-dilaton-three-Maxwell system is

$$S = \int d^5x\,\sqrt{-g_5}\left[ R - \tfrac12\,\partial_M\phi\,\partial^M\phi + V(\phi) - \sum_{i=0}^2\frac{1}{4}\,f_i(\phi)\,F_{(i)\,MN}\,F_{(i)}^{MN} \right]$$

where:

• $g_5 = \det g_{MN}$ is the determinant of the five-dimensional metric,
• $\phi$ is the dilaton field, with potential $V(\phi)$,
• $F_{(0)} = dA$ is the electric field strength corresponding to the chemical potential,
• $F_{(1)}$ and $F_{(2)}$ are magnetic field strengths, introduced to capture spatial and magnetic anisotropies,
• $f_i(\phi)$ are scalar-dependent gauge couplings modulating each field’s kinetic term.

This action enables independent and dilaton-dependent coupling of each gauge field, accommodating multiple axes of anisotropy in the dual gauge theory description.

2. Metric and Matter Ansätze

Metric Formulation

The metric ansatz distinguishes between zero and nonzero temperature regimes:

• Zero Temperature $(T=0)$: No blackening function,

$$ds^2 = B^2(z)\left[ -dt^2 + g_1(z)dx_1^2 + g_2(z)dx_2^2 + g_3(z)dx_3^2 + dz^2 \right]$$

• Finite Temperature $(T>0)$: Includes a blackening function $g(z)$ for a black-brane geometry,

$$ds^2 = B^2(z)\left[ -g(z)dt^2 + g_1(z)dx_1^2 + g_2(z)dx_2^2 + g_3(z)dx_3^2 + \frac{dz^2}{g(z)} \right]$$

where $B(z)$, $g(z)$, and $g_i(z)$ are functions of the holographic coordinate $z$ only.

Matter Field Configuration

• Electric Field ($i=0$): Standard ansatz,

$A_t = A_t(z), \qquad A_i=0 \quad (i=1,2,3,4)$

representing the chemical potential sector.

• Magnetic Fields ($i=1,2$): Constant fluxes,

$$F_{(1)} = q_1\,dx^2\wedge dx^3, \qquad F_{(2)} = q_2\,dx^1\wedge dx^3$$

where $q_{1,2}$ are fixed charges controlling the strength of the magnetic fields and their orientation in the spatial subspace. In full models, a third magnetic field may be included.

3. Equations of Motion and Redundancies

The equations of motion are derived via standard variational principles, yielding:

Einstein Equations

$$G_{MN} \equiv R_{MN} - \tfrac12 g_{MN}R = T^{\phi}_{MN} + \sum_{i=0}^2 T^{F_i}_{MN}$$

with explicit energy-momentum contributions from both the dilaton and gauge fields: $$T^\phi_{MN} = \partial_M\phi\,\partial_N\phi - \tfrac12 g_{MN}(\partial\phi)^2 - g_{MN} V(\phi)$$

$$T^{F_i}_{MN} = \tfrac12 f_i(\phi)\left[F_{(i)\,M}{}^{P}F_{(i)\,NP}\right] - \tfrac14 g_{MN} f_i(\phi) F_{(i)}^2$$

This yields five independent equations corresponding to metric components 00, 11, 22, 33, 44.

Dilaton and Maxwell Equations

• Dilaton:

$$\phi'' + \phi'\left(3\frac{B'}{B}+\frac{g'}{g}+\sum_{i=1}^3\frac{g_i'}{2g_i}\right) -\frac{B^2}{g}V'(\phi) + \sum_{i=0}^2\frac{f_i'(\phi)}{4}F_{(i)}^2 = 0$$

• Electric Maxwell:

$$\partial_z\left(\sqrt{-g_5} f_0(\phi)g^{zz}g^{tt}A_t' \right) = 0 \Rightarrow A_t'' + A_t'\left(\frac{B'}{B}+\frac{f_0'}{f_0}+\sum_{i=1}^3\frac{g_i'}{2g_i}\right) = 0$$

• Magnetic Maxwell: The constant flux ansatz ensures these equations are satisfied identically for $F_{(1)}$ and $F_{(2)}$.

Redundancy via Bianchi Identity

Including all equations yields a total of seven relations (five Einstein, one dilaton, and one Maxwell equation). However, due to the contracted Bianchi identity,

$\nabla^M T_{MN}=0,$

there exists a redundancy such that only six of these equations are independent. Specifically, once the Einstein equations and either the dilaton or Maxwell equation are satisfied, the remaining matter equation follows automatically. This is a structural feature arising from the coupling of gravity and matter fields in this system (Aref'eva et al., 2024).

4. Solution Strategy

Constructing solutions to this coupled system follows a systematic procedure:

1. Model Input: Select two gauge–dilaton couplings $f_i(\phi)$ and fix warp factors $B(z), g_1(z), g_2(z), g_3(z)$ to specify spatial anisotropy and scaling.
2. Resolve Einstein Equations: Use linear combinations of the five Einstein equations to algebraically solve for remaining unknowns, such as magnetic couplings and the blackening function $g(z)$.
3. Dilaton Extraction: Derive the dilaton profile $\phi(z)$ using a combination (trace equation) of Einstein equations to isolate $\phi'(z)$.
4. Potential Determination: Extract the scalar potential $V(\phi)$ from the "44" component of the Einstein equations.
5. Gauge Field Solution: Integrate the Maxwell equation for $A_t(z)$ subject to boundary conditions $A_t(0)=\mu$ (chemical potential) and $A_t(z_h)=0$ (horizon regularity).

This method yields a six-parameter family of anisotropic black-brane solutions, suitable for holographically modeling gauge theories at finite density and in external fields.

5. Physical Interpretation and Applications

The anisotropic Einstein-dilaton-three-Maxwell models serve as holographic toy models for QCD-like theories, where each gauge field encodes distinct physical features:

• $F_{(0)}$: Dual to chemical potential, controlling finite baryon density.
• $F_{(1)}, F_{(2)}$: Encode various spatial and magnetic anisotropies, simulating external fields and direction-dependent properties in the boundary theory.

The flexibility in the choice of anisotropy and coupling functions enables exploration of rich phase structures, responses to external fields, and anisotropic transport phenomena. The family of solutions thus provides a platform for investigating properties of hot, dense, and anisotropic quark-gluon plasma or similar strongly coupled media.

6. Relation to Generalized Einstein-Maxwell Models

The three-Maxwell model is a restricted example of the broader Einstein-dilaton-four-Maxwell family, as studied in (Aref'eva et al., 2024). The generalization involves additional gauge fields to further account for complex anisotropic configurations, including those induced by arbitrary external magnetic fields. In all such models, a key structural property is the redundancy in the set of equations of motion, traceable to the Bianchi identity and the specific energy-momentum coupling structure. The solution method for the three-Maxwell case extends directly to these more general configurations.

7. Mathematical Consistency and Degrees of Freedom

The analysis of independent equations reveals that the contracted Bianchi identity reduces the set of coupled differential equations to six independent ODEs. A plausible implication is that this reduction reflects an underlying gauge invariance in the combined gravity-matter sector, characteristic of diffeomorphism and gauge symmetry in higher-dimensional holographic models. This structural redundancy must be respected in constructing explicit solutions and in the stability analysis of resulting backgrounds (Aref'eva et al., 2024).