Classification of Irreducible Holonomies of Torsion-Free Affine Connections

Abstract

Abstract. We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of BatalinVilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and Ass as rather non-obvious extensions of Com ∞ and Ass ∞ , involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich's complex of ribbon graphs.

Projective holonomy II: cones and complete classifications

“…This connection is Ricci-flat and torsion-free; thus we may appeal to paper [Arm3] which, building on [MeSc1], gives all possible reductive holonomies for Ricci-flat torsion-free affine connections, and use various tricks and theorems to construct either Ricci-flat cones with the required holonomies, or projective manifolds with the required properties.…”

Holonomy Groups and Algebras

“…First special symplectic holonomies were given by Bryant ([17]) and by Q.-S.Chi, S.Merkulov and the author ( [34,35]). Finally, these holonomies were classified by Merkulov and the author ( [56], see also [65]), and the possible holonomies are listed in Table 3. …”

“…A complete classification was obtained by Merkulov and the author ( [56], see also [65]). We shall not deal much with the geometric content of these holonomies here, but rather conclude this survey with the classification table, referring the interested reader to the cited references.…”

AHS–structures and affine holonomies

“…The algebras g 0 in Table 1 and their semisimple parts exhaust all of Berger's original list of non-metric holonomies (Table 2 of [7]) except the full symplectic algebras, as well as some exotic holonomies (Table 3 of [7]). …”

“…It took quite a long time until the existence of all these holonomies was proved, often case by case and uncovering interesting relations to several areas of differential geometry and related fields. The program was finally completed by S. Merkulov and L. Schwachhöfer in [7].…”