About adding a variable to a Frobenius manifold and generalizations

Details About adding a variable to a Frobenius manifold and generalizations About adding a variable to a Frobenius manifold and generalizations

2012-01-04

arxiv.org/abs/1201.0948v1

Mathematics

Differential Geometry

29 pages

Scientific paper

Let \pi : V \rightarrow M be a vector bundle whose base is a Frobenius manifold and typical fiber a Frobenius algebra. Using a connection D on the bundle V and a morphism \alpha : V \rightarrow TM, we construct an almost Frobenius structure on the manifold V and we study when it is Frobenius. When M is semisimple this reduces to a single equation on the map \alpha. We describe all Frobenius structures on V, obtained by this method, when M is semisimple with non-vanishing rotation coefficients \gamma_{ij} (i\neq j) and rank(V)=2. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and tt*-geometry.

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