---
title: Einstein-Maxwell-Maxwell-Dilaton Theory
url: https://www.emergentmind.com/topics/einstein-maxwell-maxwell-dilaton-theory-emmd
type: topic
---

# Einstein-Maxwell-Maxwell-Dilaton Theory

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 $(20+1)$-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 [1301.6020].

## 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 [1301.6020].

## 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 [1301.6020].

## 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 [1301.6020].

## 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 [1301.6020]:

| 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 [1301.6020].

## 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 [1502.00951, 1805.08621, 1706.09875].

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.

Source: https://www.emergentmind.com/topics/einstein-maxwell-maxwell-dilaton-theory-emmd