Moduli, Scalar Charges, and the First Law of Black Hole Thermodynamics

We show that under variation of moduli fields $\phi$ the first law of black hole thermodynamics becomes $dM=\frac{\kappa \mathrm{dA}}{8\pi }+\Omega dJ+\psi dq+\chi dp-\Sigma d\phi$, where $\Sigma$ are the scalar charges. Also the Arnowitt-Desner-Misner mass is extremized at fixed $A$, $J$, $(p,q)$ when the moduli fields take the fixed value ${\phi }_{\mathrm{fix}}(p,q)$ which depend only on electric and magnetic charges. Thus the double-extreme black hole minimizes the mass for fixed conserved charges. We can now explain the fact that extreme black holes fix the moduli fields at the horizon $\phi ={\phi }_{\mathrm{fix}}(p,q)$: ${\phi }_{\mathrm{fix}}$ is such that the scalar charges vanish: $\Sigma ({\phi }_{\mathrm{fix}},(p,q\left)\right)=0\mathrm{}$.