Manifolds with Multiplication on the Tangent Sheaf

This talk reviews the current state of the theory of F–(super)manifolds (M, ◦), first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here ◦ is an OM–bilinear multiplication on the tangent sheaf TM , satisfying an integrability condition. F–manifolds and compatible flat structures on them furnish a useful weakening of Dubrovin’s Frobenius structure which naturally arises in the quantum K–theory, theory of extended moduli spaces, and unfolding spaces of singularities. §1. Generalities: F–structure vs Poisson structure 1.1. Manifolds. Manifolds considered in this talk can be C, analytic, or formal, eventually with even and odd coordintes (supermanifolds). The ground field K of characteristic zero is most often C or R. Each manifold is endowed with the structure sheaf OM which is a sheaf of commutative K–algebras, and the tangent sheaf TM which is a locally free OM–module of (super)rank equal to the (super)dimension of M . TM acts on OM by derivations, and is a sheaf of Lie (super)algebras with an intrinsically defined Lie bracket [ , ]. There is a classical notion of Poisson structure on M which endows OM as well with a Lie bracket { , } satisfying a certain identity. Similarly, an F–structure on M endows TM with an extra operation: (super)commutative and associative OM–bilinear multiplication ◦ with identity e. In order to describe the structure identities imposed on these operations, we recall the notion of the Poisson tensor. Let generally A be a K–linear superspace (or a sheaf of superspaces) endowed with a K–bilinear multiplication and a K– bilinear Lie bracket [ , ]. Then for any a, b, c ∈ A put Pa(b, c) := [a, bc]− [a, b]c− (−1)b[a, c]. (1.1) (From here on, (−1)ab and similar notation refers to the sign occuring in superalgebra when two neighboring elements get permuted.) This tensor will be written for A = (OM , ·, { , }) in case of the Poisson structure, and for A = (TM , ◦, [ , ]) in case of an F–structure. Talk at the Conference dedicated to the memory of B. Segre, Inst. Mat. Guido Castelnuovo, Rome, June 2004. 1