nLab Rozansky-Witten Wilson loop of unknot is A-hat genus

Statement

Proposition

(Rozansky-Witten Wilson loop observable of unknot is square root of A-hat genus)

For $\mathcal{M}^{4n}$ a hyperkähler manifold (or just a holomorphic symplectic manifold) the Rozansky-Witten invariant Wilson loop observable associated with the unknot in the 3-sphere is the square root $\sqrt{{\widehat A}(\mathcal{M}^{4n})}$ of the A-hat genus of $\mathcal{M}^{4n}$.

This is Roberts-Willerton 10, Lemma 8.6, using the Wheels theorem (Bar-Natan, Thang, Thurston 03) and the Hitchin-Sawon theorem (Hitchin-Sawon 99).

References

• Dror Bar-Natan, Le Tu Quoc Thang, Dylan Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. Volume 7, Number 1 (2003), 1-31 (euclid.gt/1513883092)

• Nigel Hitchin, Justin Sawon, Curvature and characteristic numbers of hyperkähler manifolds, Duke Math.J. 106 (2001) 599-615 (arXiv:math/9908114)

• Justin Roberts, Simon Willerton, On the Rozansky-Witten weight systems, Algebr. Geom. Topol. 10 (2010) 1455-1519 (arXiv:math/0602653)

Review in

• Justin Roberts, Section 9 of: Rozansky-Witten theory (arXiv:math/0112209)