# BF theory The action of a BF theory $D$ dimensions is given by$S_{B F}=\frac{1}{2 \pi} \int_M \operatorname{tr} B F_A,$where $B$ is a ($D-2$)-form. ## Duality The relation between 2D BF theory and 1D particle on a group is the dimensional reduced version of the [[0090 CS-WZW correspondence|relation]] between 3D [[0089 Chern-Simons theory|Chern-Simons]] and 2D [[0601 Weiss-Zumino-Witten models|WZW]]. If we constrain the dualities, we get the relation between [[0002 3D gravity|3D gravity]] and 2D [[0562 Liouville theory|Liouville]], which dimensionally reduces to the [[0655 JT-Schwarzian correspondence|relation]] between 2D [[0050 JT gravity|JT gravity]] and 1D Schwarzian theory. ## JT gravity as BF Similar to the CS formulation of 3D gravity, we can write JT gravity as a $\mathfrak{s l}(2, \mathbb{R})$ BF theory with action$S_{\mathrm{BF}}=\frac{k}{2 \pi} \int_M \operatorname{Tr}(B F)-\frac{k}{4 \pi} \oint_{\partial M} \operatorname{Tr}(B A).$To relate it to the gravity action$I_{\mathrm{JT}}=-\frac{1}{16 \pi G_N} \int d^2 x \sqrt{g} \varphi\left(R+\frac{2}{\ell^2}\right)-\int \lambda^a T_a-\frac{1}{32 \pi G_N} \oint d y \sqrt{h} \bar{\varphi}\left(K-\frac{1}{\ell}\right),$we have the relations$A=-\left(e^a P_a+\omega P_2\right), \quad B=-i\left(\lambda^a P_a+\varphi P_2\right).$Here we have the dilaton $\varphi$, the connection $\omega$, the zweibein $e^a$, the torsion 2-form $T$, and$k=\frac{1}{2 G_N}.$ - [[1985#Fukuyama, Kamimura]]