# Divergence of gauge kinetic coupling at the AdS boundary

This is the Einstein-Maxwell-Dilaton Gravity action: $$\begin{eqnarray*} S_{EM} = -\frac{1}{16 \pi G_5} \int \mathrm{d^5}x \sqrt{-g} \ [R - \frac{f(\phi)}{4}F_{MN}F^{MN} -\frac{1}{2}D_{M}\phi D^{M}\phi -V(\phi)]\,, \end{eqnarray*}$$ where $$R$$ is the Ricci scalar, $$G_5$$ is the Newton's constant in five dimensions, $$F_{MN}$$ is the field strength tensor of $$U(1)$$ gauge field $$A_M$$, and $$L$$ is the AdS radius. Moreover, $$f(\phi)$$ is the gauge kinetic function that acts as coupling between $$U(1)$$ gauge field and dilaton field $$\phi$$, and $$V(\phi)$$ is the potential of the dilaton field.

To introduce the magnetic field, we're taking the following ansatz: $$\begin{eqnarray*} & & ds^2=\frac{L^2 e^{2A(z)}}{z^2}\biggl[-g(z)dt^2 + \frac{dz^2}{g(z)} + dx_{1}^2+ e^{B^2 z^2} \biggl( dx_{2}^2 + dx_{3}^2 \biggr) \biggr]\,, \nonumber \\ & & \phi=\phi(z), \ \ F_{MN}=B dx_{2}\wedge dx_{3}\,,A(z)=-a z^2 \end{eqnarray*}$$ By solving the equation of motions, we obtained the metric function $$g(z)$$ and the gauge kinetic function $$f(\phi)$$: $$\begin{eqnarray*} g(z) &=& 1 - \frac{ \int_0^z , d\xi \ \xi^3 e^{-B^2 \xi^2 -3A(\xi) } }{\int_0^{z_h} , d\xi \ \xi^3 e^{-B^2 \xi^2 -3A(\xi) }}\\ f(z) &=& g(z)e^{2 A(z)+2 B^2 z^2} \left(-\frac{6 A'(z)}{z}-4 B^2+\frac{4}{z^2}\right)-\frac{2 e^{2 A(z)+2 B^2 z^2} g'(z)}{z} \end{eqnarray*}$$ In our model, everything is working fine except that $$f(z)$$ diverges at the boundary $$(z \rightarrow 0)$$.

Will the divergence of the gauge kinetic function at the boundary affect the completeness of our model? If it does, how to rectify this problem?

• If $-6zA' + 4 = 0 \implies A = 2 \ln (z) / 3$ then $f$ doesn't diverge as $z \to 0$ unless there are issues with $g$ or its derivative. Do you know anything about $A$? Aug 1 at 6:29
• My mistake, actually we take A(z) to be $-a z^{2}$ where $a$ is a constant defined by deconfinement temperature and for our model it turns out to be 0.15 (in GeV units) Aug 1 at 6:36