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?


1 Answer 1


Knots are interpreted as worldlines of massive particles propagating in AdS. The mass of these particles are related to the representation in which we are calculating the trace of the Wilson line. In the gravitational path integral we should not include knots because the theory will not be of pure gravity anymore.

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    $\begingroup$ but the Wilson line is not made out of matter fields, it is only made out of the gravitational data, the spin connection and the vielbeine. Massive particles can only come into place if the Wilson line has endpoints (and that's also why their mass is related to the Wilson line representation). But a knot has, by definition, no endpoints. Hence there are no matter fields here. This also implies that including knots we remain in pure gravity. $\endgroup$ Commented Nov 2, 2021 at 9:17
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    $\begingroup$ You can look at 1306.4338 section IID where the authors show that the Wilson line for a closed curve in some highest weight representation can be reduced to the worldline path integral for a massive particle propagating in a closed loop in AdS. If the closed curve has the topology of a circle then it can be contracted to a point. But if there is a black hole, or if the curve has non trivial topology, then the path integral turns out to be more interesting. $\endgroup$ Commented Nov 2, 2021 at 9:41
  • $\begingroup$ You can also interpret Wilson lines as conical defects in AdS. If the opening angle of the defect is big enough it can backreact on the geometry and change the topology of the spacetime. But again, they are interpreted as matter fields and should not be considered while computing the path integral. $\endgroup$ Commented Nov 2, 2021 at 9:49
  • $\begingroup$ Right, so if I understand correctly the content of 1306.4338, one integrates out the quantum mechanical system on the Wilson line (as also explained in a paper by Witten which they cite). Then at low energies, i.e. below the scale set by the particle that we integrated out, the theory with the Wilson loop insertion is still an effective theory for pure gravity. (It is ofc not a problem that this is only an effective theory since so is the CS formulation in general) $\endgroup$ Commented Nov 2, 2021 at 10:07
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    $\begingroup$ 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). $\endgroup$ Commented Nov 2, 2021 at 13:09

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