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Geometriae Dedicata

, Volume 167, Issue 1, pp 189–214 | Cite as

On adding a variable to a Frobenius manifold and generalizations

  • Liana David
Original Paper

Abstract

Let \(\pi :V\rightarrow M\) be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure \((\circ _{M},e_{M},g_{M})\) and typical fiber has the structure of a Frobenius algebra \((\circ _{V},e_{V},g_{V})\). Using a connection \(D\) on the bundle \(\pi : V{\,\rightarrow \,}M\) and a morphism \(\alpha :V\rightarrow TM\), we construct an almost Frobenius structure \((\circ , e_{V},g)\) on the manifold \(V\) and we study when it is Frobenius. In particular, we describe all (real) positive definite Frobenius structures on \(V\) obtained in this way, when \(M\) is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting, we add a real structure \(k_{M}\) on \(M\) and a real structure \(k_{V}\) on the bundle \(\pi : V \rightarrow M\). Using \(k_{M}\), \(k_{V}\) and \(D\) we define a real structure \(k\) on the manifold \(V\). We study when \(k\), together with an almost Frobenius structure \((\circ , e_{V}, g) \), satisfies the tt*- equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and \(tt^{*}\)-geometry.

Keywords

\(F\)-manifolds Frobenius manifolds \(tt^{*}\)-equations Saito bundles  Legendre transformations 

Mathematics Subject Classification

53D45 53B50 

Notes

Acknowledgments

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project no. PN-II-ID-PCE-2011-3-0362.

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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania

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