2 1 O ct 1 99 8 WEAK FROBENIUS MANIFOLDS

We establish a new universal relation between the Lie bracket and • –multiplication of tangent fields on any Frobenius (super)manifold. We use this identity in order to introduce the notion of " weak Frobenius manifold " which does not involve metric as part of structure. As another application, we show that the powers of an Euler field generate (a half of) the Virasoro algebra on an arbitrary, not necessarily semi–simple, Frobenius supermanifold. 0. Introduction. B. Dubrovin introduced and thoroughly studied in [D] the notion of Frobenius manifold. By definition, it is a structure (M, g, •) where M is a manifold, • is an associative, commutative and O M –bilinear multiplication on the tangent sheaf T M , and g is a flat metric on M, invariant with respect to •. The main axiom is the local existence of a function Φ (Frobenius potential) such that the structure constants of • in the basis ∂ a of flat local fields are given by the tensor of third derivatives A ab c = Φ ab c with one index raised with the help of g. We start with establishing a new universal identity (1) between the • –multiplica-tion and the Lie bracket. It follows formally from the Poisson (or Leibniz) identity, but is strictly weaker, and the algebra of tangent fields on a Frobenius manifold is never Poisson. For further comments see section 5. We show that this identity encodes an essential part of the potentiality property, at least in the semisimple case. We then use it in order to introduce in section 3 " weak Frobenius manifolds " that is, Frobenius manifolds without a fixed flat metric. We explain the relation of this notion to Dubrovin's notion of twisted Frobenius manifolds ([D], Appendix B.) The importance of weak Frobenius manifolds is related to the fact that in the constructions of K. Saito and Barannikov–Kontsevich the metric is the part of the structure that comes last, and (at least in the theory of singularities) requires considerable additional work. Finally, in section 6 we extend the construction of the Virasoro algebra from the Euler field, previously known only in the semisimple case, to the general situation. As a general reference on the basics of the theory of Frobenius manifolds we use [M] (summarized in [MM].) In particular, our manifolds are supermanifolds (say, in the complex analytic category). The multiplication • is called …