# Lattice QFT of the Jones Polynomial

Start with a gauge theory with Chern-Simons action

$$S[A] = \frac{k}{4 \pi} \text{Tr} \intop_{M} \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right)$$

and a Wilson loop observable in the fundamental representation for a knot $K$

$$W_K[A] = \text{Tr} \, \vec{\exp} \intop_K A^a \cdot \frac{i {\sigma^a}}{2}.$$

Attempt to calculate

$$\left< W_K \right>_k = \int DA \, e^{i S[A]} W_K[A]$$

by running a Metropolis-Hastings simulation on the lattice and plugging the lattice regularization of $S[A]$ and $W_K[A]$.

Naively, we expect the result to be

$$\left< W_K \right>_q = J_K(q)$$

where $J_K(q)$ is the Jones polynomial of the knot $K$ and

$$q = \exp \left(\frac{2\pi i}{k + 2}\right).$$

Question 1: has it been confirmed to work?

Question 2: how does the framing anomaly manifest itself? Lattice regularization of $W_K$ is straightforward, and doesn't depend on framing.

Question 3: ok, that might be too naive to expect the approach above to give finite results. Instead, do what Witten suggests: don't calculate $W_K$, instead restrict the monodromies around the lattice plaquettes which intersect with the knot to the $SU(2)$ orbit corresponding to the fundamental irrep and calculate the vacuum amplitude. Has it been confirmed to work? How does the framing anomaly manifest itself?

• As far as I know, it's an open question what the "right" lattice regularization is for even Abelian Chern-Simons theory. – user1504 Sep 7 '18 at 0:32
• Yes, hep-lat/0305006 is the best proposal I know for lattice abelian CS, but there is probably a reason it was never published. In the non-abelian case I think there may be problems with gauge invariance as well (at least for naive discretizations), perhaps similar to the issues faced by lattice regularizations of chiral gauge theories. This is a recollection from the last time I (briefly!) looked into this in 2015, so I'm not too sure about it. – David Schaich Sep 8 '18 at 12:27