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We find formal and holomorphic normal forms for a class of meromorphic connections (the so called $(TE)$-structures) over the germ $\mathcal N_{2}$ at the origin of the $2$-dimensional globally nilpotent $F$-manifold. In order to obtain the holomorphic normal forms we prove that the restriction of any $(TE)$-structure $\nabla$ over $\mathcal N_{2}$ at the origin $0\in \mathcal N_{2}$ is either regular singular (in which case $\nabla$ is holomorphically isomorphic to its formal normal forms) or is holomorphically isomorphic to a Malgrange universal connection (in rank two, with pole of Poincar\'{e} rank one). We develop a careful treatment for such Malgrange universal connections. We find normal forms for Euler fields on $\mathcal N_{2}$ and we use them to answer the question when a given Euler field on $\mathcal N_{2}$ is induced by a $(TE)$-structure.

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Integrability of equations of topological-antitopological fusion (being proposed by Cecotti and Vafa) describing the ground state metric on a given 2D topological field theory (TFT) model, is proved. For massive TFT models these equations are reduced to a universal form (being independent on the given TFT model) by gauge transformations. For massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method. Also it is shown that the ground state metric together with some part of the underlined TFT structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms.

We consider the construction of Frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved. In particular, we prove that the underlying F-manifold is canonical. We then apply this construction to integrable systems of Hitchin type. Comment: 43 pages

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 OM–bilinear multiplication on the tangent sheaf TM, 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

We improve some earlier results [Part I, Prog. Math. 37, 353-379 (1983; Zbl 0528.32016)] on the existence of normal forms and universal deformations of integrable connections with polar singularities.

We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of F-manifold with compatible connection generalizing a structure introduced by Manin. Comment: LaTeX, 21 pages; Sections 5 and 6 completely rewritten

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