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.
Similar content being viewed by others
References
Cortes, V.: Topological anti-topological fusion equations. In: Kowalski, O., Musso, E., Perone, D. (eds.) Complex Contact and Symmetric manifolds, Progress in Mathematics 234. Birkhauser, Basel (2005)
Dubrovin, B.: Geometry and integrability of topological-antitopological fusion. Commun. Math. Phys. 152, 539–564 (1992)
Dubrovin, B.: Geometry of \(2D\)-topological field theory. In: Francaviglia, M., Marcolli, M. (eds.) Integrable Systems and Quantum Groups Francaviglia, Aspects of Mathematics, vol. E36. Vieweg, New York (2004)
Hertling, C., Manin, Y.: Weak Frobenius manifolds. Int. Math. Res. Notices 6, 277–286 (1999)
Hertling, C.: \(tt^{*}\)-geometry, Frobenius manifolds, their connections and the construction for singularities. J. Reine Angew. Math. 555, 77–161 (2003)
Hertling, C.: Frobenius manifolds and moduli spaces for singularities. Cambridge University Press, Cambridge (2002)
Hitchin, N.: Frobenius manifolds. In: Calderbank, D., Hurtubise, J., Lalonde, F. (eds.) Gauge Theory and Symplectic Geometry, pp. 69–112. Kluwer Academic Publishers, Dordrecht (1997)
Lin, J.: Some constraints on Frobenius manifolds with a \(tt^{*}\)-structure. Math. Z. vol. 267(1–2), 81–108 (2011)
Manin, Y.: Frobenius Manifolds, Quantum Cohomology and Moduli Spaces vol. 47. American Mathematical Society, Colloquim Publications, Providence (1999)
Sabbah, C.: Isomonodromic Deformations and Frobenius Manifolds—An Introduction. Springer and EDP Sciences, Berlin (2007)
Sabbah, C.: Universal unfolding of Laurent polynomials and \(tt^{*}\) structures. From Hodge theory to integrability and TQFT: \(tt^{*}\)-geometry. In: Proceedings of Symposia in Pure Math., vol. 78, pp. 1–29 (2008)
Saito K. (1983) The higher residue pairings \(K_{F}^{(k)} \) for a family of hypersurfaces singular points, Singularities. In: Proceedings of Symposia in Pure Math., vol. 40. American Mathematical Society, pp. 441–463.
Saito, K.: Period mapping associated to a primitive form. Publ. RIMS Kyoto Univ. 19, 1231–1264 (1983)
Schäfer, L.: Harmonic bundles, topological-antitopological fusion and the related pluriharmonic maps. J. Geom. Phys. 26, 830–842 (2006)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
David, L. On adding a variable to a Frobenius manifold and generalizations. Geom Dedicata 167, 189–214 (2013). https://doi.org/10.1007/s10711-012-9809-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-012-9809-y