International Congress of Mathematicians

116 Dimitri Tamarkin3.2.1.To describe the bracket it suffices to define it on the submodule of generatorsG = (K v [—1] ® K) r . Define [] G P*(G,G;HM) to be zero when restricted ontoK r M K r and K Vr [^l] H Ä" Vr [-l]. Restriction onto K M K v [^l] takes values inu)x C H M and is given by the canonic *-pairing from [1](K v mK) r ^i 2 *üj x .Recall the definition. We have K Vr = hom(K r ,Vx ® w x ). For open U,V C X wehave the composition mapK(U) ® K V (V) -^V x ®oj x (Un V) = i 2 *oj x (U x V)which defines the pairing. This uniquely defines the *-Lie bracket.4. Formality ConjectureFollowing the logic of Kontsevich's formality theorem, one can formulate aformality conjecture in this situation.4.1. Quasi-isomorphismsA map / : g —¥ I) of differential graded pro-*-Lie algebras is called quasiisomorphismif it induces an isomorphism on cohomology. Call a pro-*-Lie algebraperfect if such is its underlying complex of pro-vector spaces. The morphism / iscalled perfect quasi-isomorphism if it is a quasi-isomorphism and both g and f) areperfect.Two perfect pro-*-lie algebras are called perfectly quasi-isomorphic if thereexists a chain of perfect quasi-isomorphisms connecting g and h.Conjecture 4.1 deign and H$K are perfectly quasi-isomorphic.The importance of this conjecture can be seen from the following theorem:Theorem 4.2 Any chain of perfect quasi-isomorphisms between deign* and H$K*establishes a bijection between the set of isomorphism classes of A-linear coissonbrackets on SK r ® A which vanish modulo the maximal ideal MA and the set ofisomorphism classes of all deformations of the chiral algebra SK r over A.References[1] A. Beilinson, V. Drinfeld, Chiral Algebras.[2] W. Goldman, J. Millson, Deformations of Flat Bundles over Kahler Manifolds,Geometry and Topology, 129^145, Lect. Notes in Pure and Applied Math. 105Dekker NY (1987).[3] M. Kontsevich, Quantization of Poisson Manifolds, preprint.