Examples/classes:
Types
Related concepts:
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.
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.
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
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.
chord diagrams | weight systems |
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linear chord diagrams, round chord diagrams Jacobi diagrams, Sullivan chord diagrams | Lie algebra weight systems, stringy weight system, Rozansky-Witten weight systems |
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:
Review:
On relation between Rozansky-Witten invariants and Seiberg-Witten invariants of 3-manifolds:
Last revised on January 3, 2020 at 13:31:30. See the history of this page for a list of all contributions to it.