What is the role of the dilaton in Jackiw-Teitelboim 2D gravity? I read that the Einstein Hilbert action is topological in 2 dimensions. (What does that mean?). To write down a non-trivial action one introduces the dilaton field in JT gravity. Does this field have a physical interpretation?
I read some references but did not really understand them.
 A: For the following, I refer to https://arxiv.org/abs/1711.08482
Dilatonic gravity models often originate from some higher dimensional parent theory. For example, the following action
\begin{equation}
I = \frac{1}{16 \pi G_N} \int d^2x \sqrt{-h} \left[\Phi^2 R_h + \lambda (\partial \Phi)^2 - U \left( \frac{\Phi^2}{L^2} \right) \right]
\end{equation}
is a result of dimensionally reducing the Einstein-Maxwell action for a magnetically charged BH in 4d. The induced metric in this case is $h$ and $U$ is just some potential.
When counting dimensions, the dilaton $\Phi^2$ has a dimension of length$^2$ and is often seen as a measure for the area of the transverse 2-sphere, i.e. the 2 spatial directions over which we integrated.
When specified for JT gravity
\begin{equation}
I_G[g, \phi] = \frac{1}{16 \pi G_N} \left[\int_\mathcal{M} \sqrt{-g} \phi (R + 2) + 2 \int_{\partial \mathcal{M}} \sqrt{- \gamma} \phi_b (K - 1) \right] \\
\end{equation}
The action is topological in the sense that when evaluating the path integral, the dilaton $\phi$ acts as a Lagrange multiplier, setting $R=-2$. So we need to sum over all possible geometries with that curvature. However, the second term contains the dilaton on the boundary $\phi_b$ and when we avert our attention to the boundary, the dilaton becomes dynamical.
In a sense, this is the result of introducing a spacetime cutoff which moves the boundary inwards a bit. Because only at the boundary at spatial infinity gravity can be neglected, this cutoff boundary feels the effect of gravity and the dilaton becomes dynamical. This is often called the boundary particle.
