Einstein-Maxwell-Maxwell-Dilaton Theory

• EMMD theory is a gravitational framework that extends Einstein-Maxwell-Dilaton models by incorporating two independent U(1) gauge fields, multiple neutral scalars, and an axion.
• It arises through generalized Kaluza-Klein reduction from higher-dimensional AdS gravity, leading to rich black-brane solutions and precise holographic thermodynamics.
• The theory offers a versatile platform for studying non-conformal charged plasmas, transport coefficients, and hydrodynamic universality in string-inspired models.

Einstein-Maxwell-Maxwell-Dilaton Theory (EMMD) constitutes a class of gravitational theories extending the traditional Einstein-Maxwell-dilaton (EMD) framework by incorporating two independent Abelian gauge fields, a dilaton, multiple neutral scalars, and, in specific constructions, an axion. Such theories provide a versatile arena for analyzing non-conformal charged plasmas, BH thermodynamics, transport phenomena, and dimensional reduction/holography in higher-dimensional gravity. They also play a fundamental role in string-inspired models and the study of dilatonic compactifications.

1. Fundamental Action and Field Content

Einstein-Maxwell-Maxwell-Dilaton (EMMD) theories arise as the low-energy effective description for certain classes of toroidal Kaluza-Klein reductions. The canonical form of the EMMD action in $(d+1)$ dimensions, originating from generalized Kaluza-Klein reduction of %%%%1%%%%-dimensional AdS gravity, is

$$\begin{aligned} S_{(d+1)} = L \int d^{d+1}x\,\sqrt{-g}\, \Big[\, &R - \tfrac12 (\partial\phi)^2 - \tfrac12 (\partial\varphi)^2 - \tfrac12 (\partial\zeta)^2 \ & - \tfrac12 e^{\alpha\phi + \beta\varphi} F^{(2)}_{MN} F^{(2)\,MN} - \tfrac12 e^{\gamma\phi + \delta\zeta} F^{(1)}_{MN} F^{(1)\,MN} \ & - \tfrac12 e^{-2\phi-2\zeta} (\partial\chi)^2 - V(\phi,\varphi,\zeta)\, \Big], \end{aligned}$$

where:

• $g_{MN}$: Einstein-frame metric,
• $F^{(1)}_{MN}, F^{(2)}_{MN}$: Maxwell field strengths for two $U(1)$s,
• $\phi, \varphi, \zeta$: neutral scalar fields,
• $\chi$: axion (apparent from toroidal reduction),
• $V(\phi,\varphi,\zeta)$: single-exponential (in simplest constructions) dilaton potential induced by dimensional reduction,
• $Z_1, Z_2$: field-dependent gauge couplings, exponential in $\phi,$ $\varphi,$ and $\zeta$ with explicit exponents determined by the reduction scheme,
• $Q_{IJ}$: scalar kinetic metric (diagonal except for $\chi)$.

All kinetic terms are positive for $2\sigma > 1$, with $2\sigma = 20 - d$ parameterizing the toroidal compactification (Smolic, 2013).

2. Origin via Generalized Dimensional Reduction

EMMD theory arises through a non-diagonal toroidal Kaluza-Klein reduction of higher-dimensional pure AdS gravity. The process involves:

• Compactification on a torus $T^{20-d}$ to reach $(d+1)$ dimensions,
• Non-diagonal reduction introduces two Kaluza-Klein $U(1)$ gauge fields (associated with torus isometries),
• Weyl rescalings and field redefinitions yield a consistent lower-dimensional action featuring three neutral scalars, an axion, and two $U(1)$s,
• The effective scalar potential $V(\phi,\varphi,\zeta)$ encodes the (20+1)-dimensional cosmological constant.

This construction generates a large class of EMMD frameworks with universal couplings and facilitates the systematic study of holographic duals to non-conformal charged plasmas (Smolic, 2013).

3. Black Brane Solutions and Thermodynamics

The EMMD framework accommodates exact charged black-brane solutions parameterized by two boost parameters (wave numbers) $\omega_1, \omega_2$ describing the charges under both Maxwell fields. In Fefferman-Graham-like coordinates: $$ds_{d+1}^2 = \frac{d\rho^2}{4\rho^2 f(\rho) K_1(\rho) K_2(\rho)} + \frac1\rho \left[ -\frac{f(\rho)}{K_1 K_2} u_\mu u_\nu + \Delta_{\mu\nu} \right] dz^\mu dz^\nu,$$ with

$$f(\rho) = 1 - (m^2)^\sigma\rho^\sigma,\quad K_I(\rho)=1+Q_I \rho^\sigma,$$

where $Q_1, Q_2$ are related to the two charges, and all scalar and axion backgrounds are:

• $e^{2\phi} = K_1 K_2$,
• $e^{2\varphi} = K_1/K_2$,
• $e^{2\zeta} = K_2$,
• $\chi = (K_2^{-1} - 1)\sinh\omega_2$.

Thermodynamic quantities (in units $L = 1$) are given by:

• $$T = \frac{\sigma}{2\pi} m \cosh\omega_1 \cosh\omega_2$$,
• $s = 4\pi m^{2\sigma} \cosh\omega_1 \cosh\omega_2$ (entropy density),
• $\mu_1 = \tanh\omega_1\,\cosh\omega_2$, $\mu_2 = \tanh\omega_2$ (chemical potentials),
• $q_1, q_2$ (charge densities) explicitly as functions of $m$, $\omega_1$, $\omega_2$.

The full equation of state closes in analytic form, and the first law $d\varepsilon = Tds + \mu_1 dq_1 + \mu_2 dq_2$ holds. The speed of sound at fixed charges is

$$c_s^2 = \left( \frac{\partial P}{\partial \varepsilon} \right)_{q_1,q_2},$$

where $P$ and $\varepsilon$ are the pressure and energy density, respectively (Smolic, 2013).

4. Holographic Dictionary and Boundary Dynamics

Fefferman-Graham asymptotic expansions enable the holographic identification of sources and expectation values for all fields. The renormalized boundary action, complete with explicit counterterms, yields:

• $\langle T_{ij} \rangle$, $\langle J^i_{(1)} \rangle$, $\langle J^i_{(2)} \rangle$,
• Scalar and axion operator expectation values,
• Ward identities reflecting lower-dimensional symmetry and arising from the reduction of the higher-dimensional conformal Ward identity:

$$\nabla^i \langle T_{ij} \rangle = F^{(1)}_{ji} \langle J^i_{(1)} \rangle + F^{(2)}_{ji} \langle J^i_{(2)} \rangle + \cdots,$$

$$\langle T^i{}_i \rangle = (2\sigma - d) \langle \mathcal{O}_\phi \rangle + \cdots.$$

The thermodynamic ensemble (fixed $\mu_1, \mu_2$ or fixed $q_1, q_2$) is fully characterized holographically, and all first law and Euler relations are verified directly by the boundary variational calculus (Smolic, 2013).

5. First-Order Hydrodynamics and Transport Coefficients

The EMMD plasma supports first-order hydrodynamics with constitutive relations: $$T^{ij} = (\varepsilon+P)u^i u^j + P\eta^{ij} - 2\eta\,\sigma^{ij} - \zeta\,\Delta^{ij}\,\nabla\!\cdot\!u,$$

$$J_{(I)}^i = q_I u^i + \sigma_{IJ}(E^i_{(J)} - T\Delta^{ij}\partial_j (\mu_J/T))$$

for $I,J=1,2$. Key transport coefficients (as explicit functions of the parameters) include:

• Shear viscosity $\eta = \frac{s}{4\pi}$ (saturating the KSS bound),
• Bulk viscosity $\zeta$ given in closed form, with ratio $\zeta/\eta$ violating the Buchel bound for generic parameters but satisfying a sharper alternative bound,
• Conductivity matrix $\sigma_{IJ}$, thermal conductivity $\kappa_T$ with a generalized Wiedemann-Franz relation,
• All coefficients are universal functions of $(m, \omega_1, \omega_2) \sim (T, \mu_1, \mu_2)$.

The table below summarizes key hydrodynamic coefficients for the EMMD black-brane plasma (Smolic, 2013):

Coefficient	Expression	Physical Role
Shear viscosity	$\eta = \frac{s}{4\pi}$	Dissipation (shear)
Bulk viscosity	Explicit in $(m, \omega_i)$	Expansion damping
Conductivity $\sigma_{IJ}$	Explicit, symmetric in $IJ$	Charge transport

A notable result is that the modified viscosity bound

$$\frac{\zeta}{\eta} \ge 2\left(\frac{1}{d-1} - c_s^2 \right)^2$$

is always satisfied, while the standard Buchel bound can be violated.

6. Physical and Phenomenological Implications

EMMD theory provides a controlled laboratory to analyze charged, non-conformal holographic plasmas. Explicit solutions and transport coefficients allow for precise testing of hydrodynamic universality and bounds, including KSS and modified Buchel bounds. The theory incorporates:

• Two independent charge sectors (arising from the double Maxwell structure),
• Three neutral scalars and an axion with nontrivial couplings, encoding anisotropy and additional sectorial relaxation/response,
• Thermodynamic richness, including dual chemical potentials and intricate phase behavior,
• Theoretical relevance to string theory, generalized dimensional reductions, and top-down AdS/CFT correspondence.

These results demonstrate the power of generalized dimensional reduction in constructing lower-dimensional models capturing the essential physics of higher-dimensional gravity plus rich matter content (Smolic, 2013).

7. Connections to Einstein-Maxwell-Dilaton and Related Theories

EMMD generalizes EMD (single Maxwell, dilaton coupling) by introducing a second Maxwell field and additional scalars. In the single-field limit—with one Maxwell decoupled and restricted scalar sector—EMMD reduces to the standard EMD theory as studied in BH thermodynamics, black string entropy, and cosmological contexts (Ghezelbash, 2015, Setare et al., 2018, Hirschmann et al., 2017).

A plausible implication is that further extensions—such as including more $U(1)$s, non-Abelian fields, or scalar potentials with multiple exponential terms—remain amenable to similar reduction and holographic analysis, potentially expanding the landscape of integrable non-conformal plasma models. These frameworks also connect to generalized geometry, consistent truncations, and higher-dimensional black object solutions.