Overview of Solutions

Fields constant or absent: four classes.

1. One vector field Dilaton Black Holes [23]
2. Two vector fields dyonic Dilaton Black Holes
3. Dilaton-Modulus Black Holes
4. Axion Black Holes

Class 1: extra coupling parameter $a$ in action.

$$e^{\phi} F^2 \Rightarrow e^{a\phi} F^2$$

Solvable E.O.M. only in the first and second case.
Solution generating symmetries in case three (Lorentz boost) and four ($SL(2,R)$).

Solution for vector field(s) in first three cases:

$$F_{rt}=Q/r^2 \qquad F_{\theta\phi}=G\sin{\theta}$$

Class	E.O.M. constraints	Global structure
1	no $\Psi \rightarrow QG=0$	Ss
2	no $\Psi \rightarrow D=0$	RN
	no $\sigma \rightarrow Q_- ^2=G_- ^2$
3	no $\Psi \rightarrow D=0$	Ss
4	class $1 \rightarrow a=1$	Ss
	class $2 \rightarrow Q_- ^2=G_- ^2$	RN
	class $3 \rightarrow Q_A G_A=Q_B G_B$	Ss

Where $D=Q_A G_B +Q_B G_A$ and $Q_{\pm}^2=Q_A ^2 \pm Q_B ^2$.

Table of Stringy Solutions

Class	Scalars	Metric
1	$e^\phi =\left( \frac{r-r_-}{r} \right) ^{\frac{-2a}{1+a^2}}$	$\lambda =\left( \frac{r-r_+}{r} \right) \left( \frac{r-r_-}{r} \right) ^{\frac{1-a^2}{1+a^2}}$
		$R^2 =r^2 \left( \frac{r-r_-}{r} \right) ^{\frac{2a^2}{1+a^2}}$
2	$e^\phi =\frac{r+\Sigma}{r-\Sigma}$	$\lambda =\frac{(r-r_+)(r-r_-)}{(r-\Sigma)(r+\Sigma)}$
		$R^2=(r-\Sigma)(r+\Sigma)$
3	$e^\phi =\frac{\sqrt{(r-\Sigma_1)(r+\Sigma_1)}}{r-\Sigma_2}$	$\lambda =\frac{r-r_+}{\sqrt{(r-\Sigma_1)(r+\Sigma_1)}}$
	$\sigma =\frac{r-\Sigma_1}{r+\Sigma_1}$	$R^2=\sqrt{(r-\Sigma_1)(r+\Sigma_1)}(r-\Sigma_2)$

Class 4: Axion Black Holes.
Solution generating symmetry which leaves the (Einstein) metric invariant.

$${e^\phi}' = \frac{e^\phi}{a^2 +b^2 e^{2\phi}} \quad \Psi '=\frac{ac+bd e^{2\phi}}{a^2+b^2 e^{2\phi}}$$

$$F_A ' = a F_A -be^\phi \sigma^{-1} \tilde F_B$$

Area singularities are responsible for different properties w.r.t. General Relativity Black Holes.

Table of Parameter Relations

Class	Parameter relations
1	$Q^2 =\frac{4 r_+ r_-}{1+a^2}$
	$2M=r_+ + \left( \frac{1-a^2}{1+a^2} \right) r_-$
2	$\Sigma (r_+ + r_-)={\textstyle{1\over 4}}\left( Q_+^2 -G_+^2 \right)$
	$\Sigma ^2 +r_+ r_- ={\textstyle{1\over 4}}\left( Q_+^2 +G_+^2 \right)$
	$2M=r_+ +r_-$
3	$Q_A ^2=(\Sigma_1 -\Sigma_2 )(\Sigma_1 -r_+)$
	$Q_B ^2=(\Sigma_1 +\Sigma_2 )(\Sigma_1 +r_+)$
	$r_+ =2M$

Note: Dilaton-Modulus Black Hole solutions have been obtained with electrical charge only.

Axion solutions lead to new parameter relations for every class.