### Gukov on Knots

One of the most celebrated connections between math and physics is the relation between knot theory and 3D Chern-Simons theory. If you consider a Chern-Simons gauge theory for a compact gauge group, $G$, the natural observables are the expectation-values of Wilson loops in unitary representations, $R_i$ of $G$. These compute the Jones polynomial (for $G=SU(2)$) and its generalizations.

3D gravity can also be cast in the form of a Chern-Simons theory,

Euclidean | Minkowski | |
---|---|---|

$\Lambda\gt 0$ | $SU(2)\times SU(2)$ | $SL(2,ℂ)$ |

$\Lambda = 0$ | $ISO(3)$ | $ISO(2,1)$ |

$\Lambda\lt 0$ | $SL(2,ℂ)$ | $SL(2,ℝ)\times SL(2,ℝ)$ |

At least, with Euclidean signature, you might ask whether there’s some relation between 3D quantum gravity and knot theory. Recently, Gukov has proposed that, with negative cosmological constant, 3D quantum gravity is related to the A-polynomial of knot-theory

Consider the complement of a knot, $\gamma$, in $S^3$. A theorem of Thurston states that “most” knots, $\gamma$, are “hyperbolic”: $S^3\setminus\gamma$ can be endowed with a hyperbolic metric. The exceptions are torus knots (knots which can be inscribed on the surface of a torus) and satellite knots (a torus knot, where the torus, as embedded in $S^3$, is itself knotted – with some restrictions to exclude trivial cases).

Classical solutions to $SL(2,ℂ)$ Chern-Simons theory

are flat $SL(2,ℂ)$ connections which translates into a hyperbolic metric and its associated Levi-Cevita connection, when we identify $\mathcal{A}=\omega +i e$. Under a reversal of orientation of $M$, $e\to -e$, and hence $\mathcal{A}\to \overline{\mathcal{A}}$. Thus the Euclidean action gets complex-conjugated under orientation reversal, as it should.

The boundary of $S^3\setminus\gamma$ is $\Sigma=T^2$, and a flat $SL(2,ℂ)$ connection is specified by a homomorphism from $\pi_1(\Sigma)=ℤ\oplusℤ$ to $SL(2,ℂ)$, up to conjugation. That is, it’s specified by a pair of elements,

(You can neglect the entries marked “*”: demanding that $g_a$ and $g_b$ commute generically constrains one linear combination and by a conjugation, we can rescale the other.) Not every pair, $(x,y)\in ℂ^*\times ℂ^*$, however, can be the boundary value of a flat $SL(2,ℂ)$ connection. The constraint that the connection does extend to $S^3\setminus\gamma$ can be expressed as a polynomial equation [D. Cooper, M. Culler, H. Gillet, D.D. Long, P.B. Shalen, “Plane curves associated to character varieties of 3-manifolds,” Invent. Math. 118 (1994) 47.]

which is the A-polynomial mentioned above.

The algebraic curve $L=\{A(x,y)=0, (x,y)\in ℂ^*\times ℂ^*\}$ has several components. In particular, there’s alway a factor of $x-1$, corresponding to the trivial representation. Quantizing the system can be done by breaking up the path integral into an integration over the fields on $S^3\setminus N(\gamma)$, where $N(\gamma)$ is a tubular neighbourhood of the knot, producing a state on the boundary $T^2$. This then gets paired with the state associated to the solid torus containing the Wilson line.

Gukov does the semiclassical (WKB) computation of the boundary state, and ties in the result with a number of conjectures in knot theory. For completely independent reasons, I’ve been interested in $SL(2,ℂ)$ Chern-Simons theory, so this was a very nice paper to see.