nLab
Rozansky-Witten weight system

Contents

Contents

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 4n\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 4n\mathcal{M}^{4n} and depend only on the underlying holomorphic symplectic structure (Kapranov 99). Generally, they are defined for 4n\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 4n\mathcal{M}^{4n} and composed with an integration over 4n\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]X^{[n+1]} (for nn \in \mathbb{N}) for XX either

  1. a K3-surface

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

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


knotsbraids
chord diagram,
Jacobi diagram
horizontal chord diagram
1T&4T relation2T&4T relation/
infinitesimal braid relations
weight systemhorizontal weight system
Vassiliev knot invariantVassiliev braid invariant
weight systems are associated graded of Vassiliev invariantshorizontal weight systems are cohomology of loop space of configuration space

References

General

Original articles:

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

Review:

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)

Last revised on January 3, 2020 at 08:31:30. See the history of this page for a list of all contributions to it.