Einstein–Maxwell–Dilaton (EMD) and Thermodynamic Ensembles

A frequently used generalization of the Einstein–scalar system is the Einstein–Maxwell–dilaton (EMD) model

$$S_{\rm EMD} =\int_M d^4x\,\sqrt{-g}\left( R-\frac14\,e^{-\alpha\phi}F_{\mu\nu}F^{\mu\nu} -\frac12(\partial\phi)^2 - V(\phi) \right), \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$

To have a well-posed and finite variational problem in asymptotically AdS$_4$, one works with

$$S_{\rm tot}=S_{\rm EMD}+S_{\rm GHY}+S_{\rm ct}+S_{\rm bdy}^{(A)}, \qquad S_{\rm GHY}=\frac{1}{8\pi G}\int_{\partial M} d^3x\,\sqrt{-\gamma}\,K,$$

and chooses the Maxwell boundary term $S_{\rm bdy}^{(A)}$ according to the thermodynamic ensemble.

Coordinates and cutoff

Near the AdS boundary it is convenient to use Poincaré coordinates

$$ds^2=\frac{dr^2+\eta_{ij}dx^i dx^j}{r^2},\qquad r\to 0.$$

We regulate the boundary at $r=\varepsilon$. On $r=\varepsilon$,

$$\sqrt{-\gamma}=r^{-3},\qquad n_\mu dx^\mu=\frac{dr}{r}.$$

A convenient gauge choice is $A_r=0$, with near-boundary expansion (for $d=3$)

$$A_i(r,x)=a_i(x)+r\,b_i(x)+\cdots.$$

1. Maxwell variation and the conserved radial flux

Introduce the dilaton-dependent gauge coupling

$$Z(\phi)\equiv e^{-\alpha\phi}.$$

The Maxwell part of the action is

$$S_A=-\frac14\int_M d^4x\,\sqrt{-g}\,Z(\phi)\,F_{\mu\nu}F^{\mu\nu}.$$

Varying $A_\nu$ and integrating by parts gives

$$\delta S_A =\int_M d^4x\,\sqrt{-g}\,\big[\nabla_\mu(ZF^{\mu\nu})\big]\delta A_\nu +\int_{\partial M} d^3x\,\sqrt{-\gamma}\,Z(\phi)\,n_\mu F^{\mu i}\,\delta A_i.$$

So the Maxwell equation is

$$\nabla_\mu\!\left(Z(\phi)F^{\mu\nu}\right)=0,$$

and the on-shell boundary variation is

$$\delta S_A\Big|_{\rm EOM}=\int_{\partial M} d^3x\,\pi_A^{\,i}\,\delta A_i, \qquad \pi_A^{\,i}\equiv \sqrt{-\gamma}\,Z(\phi)\,n_\mu F^{\mu i}.$$

For static, homogeneous charged backgrounds (typical in RG-flow thermodynamics), only the time component matters. In Lorentzian signature one often uses $A_t(r)$; in Euclidean signature the thermal circle is parameterized by $\tau\sim\tau+\beta$ and the relevant boundary datum is $A_\tau$.

A key simplification of EMD is that the radial electric flux is conserved:

$$\partial_r\big(\pi_A^{\,\tau}\big)=0 \qquad(\text{for homogeneous solutions}),$$

so $\pi_A^{\,\tau}$ can be evaluated at the horizon or at the boundary and gives the same result.

2. Grand canonical ensemble (Dirichlet on $A_\tau$): fixed chemical potential $\mu$

The grand canonical ensemble fixes the chemical potential:

$$Z_{\rm GC}=\mathrm{Tr}\,\exp\!\big[-\beta(H-\mu Q)\big],\qquad \Omega(T,\mu) = -\frac{1}{\beta}\log Z_{\rm GC}.$$

Holographically, this corresponds to Dirichlet boundary conditions on the Euclidean time component of the gauge field:

$$A_\tau\big|_{\partial M}=\mu \quad\Longleftrightarrow\quad \delta A_\tau\big|_{\partial M}=0.$$

Then the Maxwell boundary variation vanishes automatically because $\delta A_\tau=0$.

More physically, after holographic renormalization the (Euclidean) on-shell action satisfies

$$I_E^{\rm (Dir)}\Big|_{\rm on\text{-}shell}=\beta\,\Omega(T,\mu),$$

and its variation reproduces the thermodynamic relation

$$\delta I_E^{\rm (Dir)}\Big|_{\rm on\text{-}shell} = -\beta\,Q\,\delta\mu \qquad\Longleftrightarrow\qquad d\Omega = -S\,dT - Q\,d\mu.$$

Sign conventions

The sign in $\delta I_E=-\beta Q\,\delta\mu$ matches standard thermodynamics. In terms of the canonical momentum $\pi_A^{\,\tau}$ above, one often defines the (renormalized) charge density as

$$\rho \equiv -\,\pi_{A,\rm ren}^{\,\tau},$$

so that $\delta I_E^{\rm (Dir)}=-\int d^3x\,\rho\,\delta\mu$. Depending on whether one chooses an inward/outward radial normal (and Lorentzian vs Euclidean continuation), these minus signs can shift; the ensemble mapping below is unchanged.

3. Canonical ensemble (Neumann on $A_\tau$): fixed charge $Q$ (or density $\rho$)

The canonical ensemble fixes the charge:

$$Z_{\rm C}(Q)=\mathrm{Tr}_{Q}\,\exp(-\beta H),\qquad F(T,Q)=-\frac{1}{\beta}\log Z_{\rm C}(Q).$$

Thermodynamics gives the Legendre transform

$$F(T,Q)=\Omega(T,\mu)+\mu Q, \qquad dF=-S\,dT+\mu\,dQ.$$

Holographically, fixing $Q$ means fixing the (renormalized) radial electric flux, i.e. a Neumann-type condition:

$$\delta \pi_{A,\rm ren}^{\,\tau}=0 \quad (\text{equivalently } \delta\rho=0).$$

To make the variational principle compatible with this, we add a Legendre boundary term for the gauge field,

$$I_{\rm Leg}^{(A)} \equiv -\int_{\partial M} d^3x\,A_\tau\,\pi_A^{\,\tau} \;=\; \int_{\partial M} d^3x\,\mu\,\rho \quad(\text{using }\rho=-\pi_A^{\,\tau} \text{ as above}).$$

Then the total boundary variation becomes

$$\delta\big(I_E^{\rm (Dir)}+I_{\rm Leg}^{(A)}\big)\Big|_{\rm EOM} = +\int d^3x\,\mu\,\delta\rho,$$

so it vanishes for fixed $\rho$ (canonical ensemble).

On-shell, the Legendre term implements the expected thermodynamic transformation:

$$I_E^{\rm (Neu)}\Big|_{\rm on\text{-}shell} = I_E^{\rm (Dir)}\Big|_{\rm on\text{-}shell} + \beta\,\mu Q \qquad\Longleftrightarrow\qquad F=\Omega+\mu Q.$$

Rule of thumb

• Grand canonical (fixed $\mu$): Dirichlet on $A_\tau$ and no extra gauge boundary term.
• Canonical (fixed $Q$): fix the electric flux (Neumann) and add the Legendre term $-\int A_\tau \pi_A^{\,\tau}$ (equivalently $+\int \mu\rho$).

4. Mixed boundary conditions (intermediate ensembles / current deformations)

More generally, one can impose a mixed (Robin) condition relating $\rho$ and $\mu$, which is the gauge-field analogue of multi-trace scalar boundary conditions. Add a finite boundary functional $W[A]$ and obtain

$$\delta(I_{\rm ren}+W)\Big|_{\rm EOM} =\int d^3x\,\Big(\pi_{A,\rm ren}^{\,i}-\frac{\delta W}{\delta A_i}\Big)\delta A_i \qquad\Rightarrow\qquad \pi_{A,\rm ren}^{\,i}=\frac{\delta W}{\delta A_i}.$$

For homogeneous electric backgrounds this becomes a relation between $\rho$ and $\mu$. Such mixed conditions correspond to deformations in the dual CFT involving the conserved current $J^i$ (and, in $d=3$, are closely tied to $SL(2,\mathbb{Z})$ operations when Chern–Simons terms are included).

5. What changes because of the dilaton?

Compared to Einstein–Maxwell, the only structural change is that the conserved flux is the dilaton-dressed one:

$$\pi_A^{\,i}\propto Z(\phi)\,n_\mu F^{\mu i}, \qquad Z(\phi)=e^{-\alpha\phi}.$$

Equivalently, the Maxwell equation is conservation of the “electric displacement” $Z(\phi)F^{r\tau}$. This means:

• the holographic charge/current is proportional to $Z(\phi)$ evaluated in the solution, and
• changing the boundary condition for $\phi$ (standard/alternative/mixed quantization) can indirectly affect the gauge-sector thermodynamics through the profile of $Z(\phi)$.

(Scalar boundary terms and counterterms are handled exactly as in the Einstein–scalar discussion.)