# Knots in 3d pure gravity

Three-dimensional pure gravity on $$\mathrm{AdS}_3$$ can be described as a Chern-Simons theory with gauge group $$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$$,with action $$S[A,\bar{A}] = \text{CS}[A]-\text{CS}[\bar{A}],$$ where $$\text{CS}[A] := \frac{k}{4\pi}\int_{\cal M} \text{tr}\Big( A\wedge\mathrm{d}A + \frac{2}{3} A\wedge A\wedge A\Big),$$
and $$A$$ and $$\bar{A}$$ are gauge connections for the first and second factor of $$\mathrm{SL}(2,\mathbb{R})$$, respectively and $$k\in\mathbb{Z}$$. The connections $$A$$ and $$\bar{A}$$ are related to the vielbein, $$e$$, and the spin connection, $$\omega$$, by $$e = \frac{\ell_{\text{AdS}}}{2}(A-\bar{A}), \qquad \omega = \frac{1}{2}(A+\bar{A}).$$

Famously, a class of natural observables in a Chern-Simons theory consists of various knots of Wilson loops, which for $$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$$ take the form $${\cal W}_\mathbf{r}[\gamma] :=\mathrm{tr}_\mathbf{r}\left[\mathrm{P}\exp\left(\oint_\gamma A\right)\ \ \mathrm{P}\exp\left(-\oint_\gamma \bar{A}\right)\right],$$ where $$\gamma$$ is a closed curve, which generally can be a knot, and $$\mathbf{r}$$ is a representation of $$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$$.

Question 1: What is the gravitational interpretation/origin of $${\cal W}_\mathbf{r}[\gamma]$$, when $$\gamma$$ is a knot?

Question 2: Insertions of various links of knotted Wilson lines $$\displaystyle\prod\limits_{i=1}^n {\cal W}_\mathbf{r_i}[\gamma_i]$$ will change the topology of spacetime. If in the gravitational path integral we are supposed to sum over topologies, is the expectation that we should sum over all possible links of knots of Wilson line insertions? How would one attempt to do that systematically?

• This thing is to be integrated over $U$ and $A$. This is not an effective action. The effective action is what remains of that after you've integrated over $U$. This is pure gravity (with an insertion). Commented Nov 2, 2021 at 13:09