nLab Rozansky-Witten weight system

Idea

Rozansky-Witten weight systems are weight systems on Jacobi diagrams (equivalently on round chord diagrams) given by Rozansky-Witten invariants on hyperkähler manifolds, with coefficients in certain Dolbeault cohomology groups.

For asymptotically flat and for compact hyperkähler manifolds the induced RW-weight systems take values in the ground field and hence are actual weight systems.

Properties

Rozansky-Witten weight systems depend only on the hyperkähler manifold $\mathcal{M}^{4n}$ which is the (classical) Coulomb branch of the RW-twisted D=3 N=4 super Yang-Mills theory, and in fact they are independent of the Riemannian geometry of $\mathcal{M}^{4n}$ and depend only on the underlying holomorphic symplectic structure (Kapranov 99). Generally, they are defined for $\mathcal{M}^{4n}$ any hyperkähler manifold which is either asymptotically flat (ALE spaces) or compact topological space (compact hyperkähler manifolds).

Via the equivalent reformulation by Kapranov 99 one finds (Roberts-Willerton 10) that the Rozansky-Witten invariants are structurally Lie algebra weight systems themselves, but internal to the derived category of coherent sheaves of $\mathcal{M}^{4n}$ and composed with an integration over $\mathcal{M}^{4n}$ which makes the resulting Dolbeault cohomology-valued weights become ground field-valued.

Examples

Ground-field valued weight systems

In order for Rozansky-Witten weight systems to take values in the ground field, hence to be actual weight systems, the hyperkähler manifold has to be asymptotically flat or compact (i.e. closed). The only known examples of compact hyperkähler manifolds are Hilbert schemes of points $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

1. a K3-surface

2. a 4-torus (in which case the compact hyperkähler manifolds is really the fiber of $(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4$)

(Beauville 83) and two exceptional examples (O’Grady 99, O’Grady 03 ), see Sawon 04, Sec. 5.3.

Related concepts

• D=3 N=4 super Yang-Mills theory

chord diagrams	weight systems
linear chord diagrams, round chord diagrams Jacobi diagrams, Sullivan chord diagrams	Lie algebra weight systems, stringy weight system, Rozansky-Witten weight systems

knots	braids
chord diagram, Jacobi diagram	horizontal chord diagram
1T&4T relation	2T&4T relation/ infinitesimal braid relations
weight system	horizontal weight system
Vassiliev knot invariant	Vassiliev braid invariant
weight systems are associated graded of Vassiliev invariants	horizontal weight systems are cohomology of loop space of configuration space

References

General

Original articles:

• Lev Rozansky, Edward Witten, Hyper-Kähler geometry and invariants of 3-manifolds, Selecta Math., New Ser. 3 (1997), 401–458 (arXiv:hep-th/9612216, doi:10.1007/s000290050016, MR98m:57041)

• Mikhail Kapranov, Rozansky–Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71–113, MR2000h:57056, doi, alg-geom/9704009

Unified description of Rozansky-Witten weight systems with Lie algebra weight systems, and unified Wheels theorem via Lie algebra objects:

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

Review:

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

Relation to Seiberg-Witten invariants

On relation between Rozansky-Witten invariants and Seiberg-Witten invariants of 3-manifolds:

• Matthias Blau, George Thompson, On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants, Adv. Theor. Math. Phys. 5 (2002) 483-498 (arXiv:hep-th/0006244)