$R_{\alpha }^{\beta }(\mathrm{\Gamma })$ is the curvature 2-form of the $GL(2)$ connection ${\mathrm{\Gamma }}_{\beta \mu }^{\alpha }$, and $g$ is the determinant of the internal metric $g_{\alpha \beta }$. From now on, $GL(2)\equiv GL(2,\mathbf{R})$, $SL(2)\equiv SL(2,\mathbf{R})$ since we deal only with the real linear groups. Furthermore, we take for definiteness the explicit realization of the group elements in terms of $2\times 2$ real matrices (and not any other representation).

With respect to the standard $B\text{−}F$ system, the theory has an extra (quadratic) constraint that would not arise by dimensional reduction from three-dimensional Chern-Simons theory. This constraint implies that the theory has no degree of freedom in the absence of boundaries, even if the spatial topology is nontrivial (circle). This is because the holonomies are then fixed up to conjugation [9]. The presence of this extra constraint makes the analysis of the Palatini Gauss-Bonnet theory different from that pursued in the papers [14, 15]. This constraint leads to the Klein-Gordon equation in the quantum theory. Note also that we do not implement any $sl(2)$ Hamiltonian reduction through boundary conditions at the edges of the strip. In fact, we impose no boundary condition at $r_1$ and $r_2$ on the canonical variables ${\mathrm{\Gamma }}_1$ and $B$.

Our notations are as follows. (i) The matrix $B\equiv B_{\beta }^{\alpha }{T}_{\alpha }^{\beta }$ is the $B$-field transforming as $B\to S^{-1}BS$ under $GL(2)$ transformations $S$, where $T_a^b$ is a basis of generators of the Lie algebra $gl(2)$. A definite choice is not needed at this stage. (ii) $\mathrm{\Gamma }\equiv {\mathrm{\Gamma }}_{\beta 1}^{\alpha }{T}_{\alpha }^{\beta }$ is the spatial component of the $GL(2)$-valued connection 1-form ${\mathrm{\Gamma }}_{\beta \mu }^{\alpha }{T}_{\alpha }^{\beta }d{x}^{\mu }$, transforming as $\mathrm{\Gamma }\to S^{-1}dS+S^{-1}\mathrm{\Gamma }S$. (iii) ${\mathrm{\Gamma }}_0\equiv {\mathrm{\Gamma }}_{\beta 0}^{\alpha }{T}_{\alpha }^{\beta }$ is its temporal component, which acts as a Lagrange multiplier for the Gauss constraint $\nabla B=0$. (iv) $\nabla B\equiv {\nabla }_{1}B={\partial }_{1}B+[{\mathrm{\Gamma }}_1,B]$ is the spatial covariant derivative of $B$. (v) ${\lambda }_1$ and ${\lambda }_2$ are Lagrange multipliers for the constraints $\mathrm{Tr}B=0$ and $\mathrm{Tr}(B^2-k^2\eta I_{2\times 2})=0$. (vi) $I_{2\times 2}$ is the identity in two dimensions. In the following we not systematically write it. (vii) $\eta =\pm 1$ is the sign of the determinant of the metric $g_{\alpha \beta }$.

Note that ${\mathrm{AdS}}_3$ means here “wrapped” AdS space (with closed timelike lines) and not its covering space $C{\mathrm{AdS}}_3$. The isometry group acts by left and right multiplication as $A\to LAR$, $A,L,R\in SL(2)$ and is isomorphic to the quotient $\{[SL(2)\times SL(2)]/{\mathbf{Z}}_2\}$, where ${\mathbf{Z}}_2$ is the center of $SL(2)$ (obtained by simultaneously replacing $L$ by $-L$ and $R$ by $-R$). Since the original action involved only Lie algebra elements, the previous considerations extend straightforwardly to the covering group $\overline{SL(2)}$ and the unwrapped covering space $C{\mathrm{AdS}}_3$. Going in the opposite direction (taking a quotient rather than unwrapping), one can consider $PSL(2)\simeq (SL(2)/{\mathbf{Z}}_2)$, i.e., ${\mathrm{AdS}}_3$ quotientized by the isometry $T_0\to -T_0$, $T_3\to -T_3$, $X_1\to -X_1$, $X_2\to -X_2$. All these groups share the same Lie algebra.

Note that the group coordinates are dimensionless, and $\mathrm{AdS}$ space can be taken to have unit radius of curvature.

Left translations, in terms of AdS coordinates, read as $$L_0=\frac{1}{2}(\frac{\partial }{\partial T}-\frac{\partial }{\partial \mathrm{\Psi }})$$(27)$$\begin{gathered} L_1=\frac{1}{2}\sin (T-\mathrm{\Psi })(-\frac{\sinh \rho }{\cosh \rho }\frac{\partial }{\partial T}+\frac{\cosh \rho }{\sinh \rho }\frac{\partial }{\partial \mathrm{\Psi }}) \\ +\frac{1}{2}\cos (T-\mathrm{\Psi })\frac{\partial }{\partial \rho } \end{gathered}$$(28)$$\begin{gathered} L_2=\frac{1}{2}\cos (T-\mathrm{\Psi })(-\frac{\sinh \rho }{\cosh \rho }\frac{\partial }{\partial T}+\frac{\cosh \rho }{\sinh \rho }\frac{\partial }{\partial \mathrm{\Psi }}) \\ -\frac{1}{2}\sin (T-\mathrm{\Psi })\frac{\partial }{\partial \rho }, \end{gathered}$$(29)while right translations $R_a^A$ become $$R_0=\frac{1}{2}(\frac{\partial }{\partial T}+\frac{\partial }{\partial \mathrm{\Psi }})$$(30)$$\begin{gathered} R_1=\frac{1}{2}\sin (T+\mathrm{\Psi })(-\frac{\sinh \rho }{\cosh \rho }\frac{\partial }{\partial T}-\frac{\cosh \rho }{\sinh \rho }\frac{\partial }{\partial \mathrm{\Psi }}) \\ +\frac{1}{2}\cos (T+\mathrm{\Psi })\frac{\partial }{\partial \rho } \end{gathered}$$(31)$$\begin{gathered} R_2=\frac{1}{2}\cos (T+\mathrm{\Psi })(-\frac{\sinh \rho }{\cosh \rho }\frac{\partial }{\partial T}-\frac{\cosh \rho }{\sinh \rho }\frac{\partial }{\partial \mathrm{\Psi }}) \\ -\frac{1}{2}\sin (T+\mathrm{\Psi })\frac{\partial }{\partial \rho }. \end{gathered}$$(32)

In Ref. [9], we pointed out that the diffeomorphism invariance of the action (1) was not an independent symmetry but could be expressed in terms of the gauge symmetries generated by the first class constraints. Of these, the Gauss constraint generates internal gauge transformations, while the trace constraint generates projective transformations. The transformation generated by the quadratic constraint was called “new symmetry” as it was not in the original list of original symmetries. Time diffeomorphisms could be expressed in terms of it, but we were reluctant to identify the new symmetry with time diffeorphisms because this involved dividing by a variable (here, $\lambda$) that was not always different from zero. It is interesting to note, however, that with $H=0$ and the understanding of the dynamics as that of a free particle in $\mathrm{AdS}$ space, the quadratic constraint, which is the mass shell condition, does generate reparametrizations in the time $t$. In that case, identifying the two symmetries is therefore meaningful (the condition $\lambda >0$ on the “lapse” $\lambda$ then makes sense). But if the boundary Hamiltonian $H$ is not zero, reparametrizations of the boundary time is not a gauge symmetry, yet the transformations generated by $\mathcal{K}$ are proper gauge symmetries. In fact, translations in $t$ at the boundary are, as we have seen, improper gauge symmetries generated by the Gauss constraint with, in general, a $B$-dependent parameter, but arbitrary reparametrizations are not symmetries at the boundary. It follows that there is no hope to express, at the boundary, the proper gauge symmetries generated by $\mathcal{K}$ in terms of the time diffeomorphisms.