# Calculating the Bekenstein-Hawking entropy for 1+1 black hole with dilaton background

According to this paper the Bekenstein-Hawking entropy of a 1+1 black hole which described by the $$SL_k(2,\mathbb{R})/U(1)$$ WZW cigar geometry is given by the following formula appearing in eq. (5.7):

$$S=2\pi e^{-2\Phi_0}$$

But according to Bekenstein-Hawking formula the entropy is given by:

$$S_{BH}=\frac{k_Bc^3A}{4G\hbar}$$

We use natural units $$c=\hbar=k_B=1$$, So: $$S_{BH}=\frac{A}{4G}$$

The $$d$$ dimensional Newton constant $$G$$ is given by the string scale $$l_s$$ and string coupling $$g_s$$ in the following way: $$G_d=l_s^{d-2}g_s^2$$ for the 1+1 case we left with: $$G_2 = g_s^2.$$ Under dilaton field background, the coupling that measured from infinity is given by the asymptotic value of the dilaton $$\Phi$$ which we denote by $$\Phi_0$$ so: $$g_s=e^{\Phi_0}$$ Now, togather with a surface area of a 1d spatial line that is given by: $$A=2$$ The entropy should be:

$$S_{BH}=\frac{1}{2}e^{-2\Phi_0}$$

What is the source of the extra $$4\pi$$ factor such that $$S_{BH}$$ will fit exactly $$S$$?