CritPt / Challenge_3_main / In the AdS_3/BCFT_2 correspondence, consider a setup where the bulk black…

# Problem setup:
In the AdS$_3$/BCFT$_2$ correspondence, consider a setup where the bulk black hole geometry—characterized by an inverse temperature β=(2π)/(r₀)—is given by
\begin{equation}
ds^2=f(r)d\tau_E^2+\frac{dr^2}{f(r)}+r^2 d\phi^2, \qquad f(r)=r^2-r_0^2,
\end{equation}
with ϕ∼ϕ+2π and τ_(E)∼τ_(E)+β=τ_(E)+(2π)/(r₀), and this geometry is terminated by a spherically symmetric brane of tension η, where 0<η<1, behind the horizon. Let O(x) be a scalar primary operator in the BCFT, whose bulk dual is a massive scalar field of mass m. For simplicity, assume the operator is inserted at the Euclidean time-reflection symmetric slice τ_(E)=0.

# Main problem:

Using the geodesic approximation, determine the form of the one-point function ⟨O(x)⟩. How does the result depend on the mass m of the bulk field, the black hole radius r₀ and the brane tension η?

# Problem setup:
In the AdS$_3$/BCFT$_2$ correspondence, consider a setup where the bulk black hole geometry—characterized by an inverse temperature $\beta = \frac{2\pi}{r_0}$—is given by
\begin{equation}
ds^2=f(r)d\tau_E^2+\frac{dr^2}{f(r)}+r^2 d\phi^2, \qquad f(r)=r^2-r_0^2,
\end{equation}
with $\phi \sim \phi + 2\pi$ and $\tau_E \sim \tau_E + \beta = \tau_E + \frac{2\pi}{r_0}$, and this geometry is terminated by a spherically symmetric brane of tension $\eta$, where $0<\eta<1$, behind the horizon. Let $\mathcal{O}(x)$ be a scalar primary operator in the BCFT, whose bulk dual is a massive scalar field of mass $m$. For simplicity, assume the operator is inserted at the Euclidean time-reflection symmetric slice $\tau_E = 0$.

# Main problem:

Using the geodesic approximation, determine the form of the one-point function $\langle \mathcal{O}(x) \rangle$. How does the result depend on the mass $m$ of the bulk field, the black hole radius $r_0$ and the brane tension $\eta$?