An update on semisimple quantum cohomology and F-manifolds

By C. Hertling, Yuri I. Manin, C. Teleman. 2008

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AbstractIn the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative$$ \mathcal{O}_M $$-bilinear multiplication on its tangent sheaf$$ \mathcal{T}_M $$is an F-manifold in the sense of Hertling-Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle TM*, is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau-Ginzburg models for Fano varieties.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/52b467be246d4ad660293758aaed2df2bbd5abcf

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Citations in the corpus

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, background): “On the negative side, it was observed by Hertling, Manin and Teleman [28] that a projective algebraic manifold M cannot have generically semisimple quantum homology if there are any nonzero Hodge numbers h.”
The structure of 2D semi-simple field theories (2007, background): “(More is true: it turns out that semi-simplicity of the even part H of the quantum cohomology ring forces the vanishing of odd cohomology [16].”