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Rozansky-Witten invariants via Atiyah classes. (English) Zbl 0993.53026
This paper deals with the invariants introduced by L. Rozansky and E. Witten [Sel. Math., New Ser. 3, No. 3, 401-458 (1997; Zbl 0908.53027)]. Starting from a hyper-Kähler manifold \(X^{4n}\), they described a sigma-model, based on maps from \(3\)-manifolds into \(X\), which gives some kind of topological quantum field theory in \(2+1\) dimensions. Precisely what axioms it should satisfy remains unclear, but it should at least define an invariant of closed orientable \(3\)-manifolds \(M^3\).
Rozansky and Witten further explained that because their theory has an exact perturbation expansion, the invariant of \(M^3\) can be calculated by combining information derived separately from \(M\) and from \(X\). The information derived from \(X^{4n}\) amounts to a weight system of degree \(n\): it is a complex-valued function, defined on vertex-oriented trivalent graphs with \(2n\) vertices, and satisfying the IHX relations which arose originally in the theory of Vassiliev knot invariants. (In Kontsevich’s terminology, a weight system is an element of graph cohomology.) M. Kontsevich [Compos. Math. 115, No. 1, 115-127 (1999; Zbl 0924.57017)], reformulated the Rozansky-Witten weight systems using Gelfand-Fuchs cohomology and formal geometry to give a vast generalization of the construction. He showed that a Rozansky-Witten-type weight system can be defined whenever one has a smooth manifold with a foliation which has a transverse symplectic structure. The author was inspired by an early version of Kontsevich’s paper. The parts of the paper under review dealing with Gelfand-Fuks cohomology and formal geometry certainly overlap with Kontsevich’s work, though here the results are formulated much more algebraically, in the language of operads. However, the main aim of his paper is to study in detail the geometry and algebra underlying the original Rozansky-Witten weight systems of hyper-Kähler manifolds. There are several different and extremely useful points of view offered.
The standard weight systems used in Vassiliev theory come from metrized Lie algebras: one starts by using the (non-degenerate, symmetric) metric to turn the Lie bracket into a skew trilinear form; then, placing it at the vertices of a trivalent graph and contracting along edges with the inverse of the metric, one obtains a map from trivalent graphs to complex numbers for which the IHX relation is a direct consequence of the Jacobi identity.
The author explains that for a Kähler manifold with holomorphic tangent bundle \(T\) one should view the curvature not as a \((1,1)\)-form but as a \(\bar \partial\)-closed \((0,1)\)-form with values in \(\text{Hom}(S^2T, T)\). In this guise, it satisfies a Jacobi-type identity up to a \(\overline \partial\)-coboundary. In fact, using higher covariant derivatives of this curvature, the Dolbeault complex acquires the structure of an \(L_\infty\)-algebra. This demonstrates clearly that the “Lieness” arising here has nothing to do with hyper-Kähler geometry; it is actually far more basic.
If the manifold also has a holomorphic symplectic form (for example, it is hyper-Kähler), one may use this form, just as one uses the metric on a Lie algebra, to make weight systems acting on trivalent graphs, and valued in the Dolbeault cohomology of the manifold. Combining with the \(L_\infty\)-structure it produces invariants of graphs with higher-valence vertices, corresponding to higher degree graph cohomology classes. This can be expressed using the language of modular operads.
It is important to note that these weight systems are constructed in effect from a symmetric trilinear and skew bilinear tensor, whereas in the Lie algebra case there is a “switch of statistics” to a skew trilinear and symmetric bilinear form. The author explains this rather odd fact by showing that the two relevant systems for orienting trivalent graphs (respectively, skew edges with ordered vertices, and skew vertices) are actually equivalent. (Orderings of the vertices are necessary in the first case because the curvature, in its guise as a \(1\)-form, anticommutes with itself.)
Turning away from the Hermitian differential geometry point of view, the author shows that the above constructions work at a purely cohomological level. He explains how the Atiyah class of a holomorphic vector bundle, which is the obstruction to existence of a global holomorphic connection, is a cohomological version of the curvature the bundle would acquire if given a smooth Hermitian metric. Using this, the weight systems coming from a holomorphic symplectic manifold can be constructed using only sheaf cohomology, removing the need for any choices of metrics.

MSC:
53D35 Global theory of symplectic and contact manifolds
57R57 Applications of global analysis to structures on manifolds
55P48 Loop space machines and operads in algebraic topology
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
17B70 Graded Lie (super)algebras
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
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