Rozansky-Witten theory, Rozansky-Witten invariant, Rozansky-Witten class
Rozansky and Witten constructed an isometry invariant of hyper-Kähler manifolds, which depends also on a trivalent graph. Kontsevich has shown an approach to Rozansky-Witten invariants via characteristic classes of foliations and Gelʹfand-Fuks cohomology. He devised a formal construction, again depending on a trivalent graph of a cohomology class of the Lie algebra of formal (in the sense of formal power series) Hamiltonian vector fields on any arbitrary finite-dimensional symplectic vector space. Characteristic classes of foliations may induce examples where this construction applies; one of the examples yields Rozansky-Witten classes.
Stimulated by Kontsevich’s 1997 letters to Victor Ginzburg and Witten, Kapranov has approached the RW invariants for all trivalent graphs at once via a single Atiyah class. This construction gives essentially a clever repackaging of the Kontsevich’s construction.
L. Rozansky, E. Witten, Hyper-Kähler geometry and invariants of 3-manifolds, Selecta Math., New Ser. 3 (1997), 401–458, MR98m:57041
Jian Qiu, Maxim Zabzine, Odd Chern-Simons theory, Lie algebra cohomology and characteristic classes, arxiv/0912.1243