\(tt^*\) geometry, Frobenius manifolds, their connections, and the construction for singularities.

*(English)*Zbl 1040.53095This paper constructs and studies a natural generalization of variations of Hodge structures called \(tt^*\)-geometry, as defined by S. Cecotti and C. Vafa [Topological-antitopological fusion, Nucl. Phys. B 367, 359–461 (1991), Commun. Math. Phys. 158, 569–644 (1993; Zbl 0787.58049)] together with Frobenius manifold structures on the base space of a semi-universal unfolding of a hypersurface singularity (not necessarily quasi-homogeneous) in a unified way. Both, Frobenius structures and \(tt^*\)-structures are first generalized and studied on abstract holomorphic vector bundles. These are related by a careful analysis of associated flat connections with poles and flat pairings on the pull-back bundle \(\pi^*K\), where \(\pi\) is the projection \(\pi:{P}^1 \times M \to M\). Bundles with such a (generalized) \(tt^*\)-structure are special cases of C. T. Simpson’s “harmonic bundles” [J. Am. Math. Soc. 3, 713–770 (1990; Zbl 0713.58012)].

In the second half of the paper these investigations are specialized to the case of the tangent bundle \(TM\). Using oscillating integrals the author then reconstructs the Saito Frobenius manifold structure on the semi-universal unfolding of a hypersurface singularity, previously done using the Gauss-Manin connection [C. Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts Math. 151, Cambridge University Press (2002; Zbl 1023.14018)], and he shows that it has a canonical compatible \(tt^*\)-geometry – or rather, CDV-structure (Cecotti-Dubrovin-Vafa) – outside of a real analytic subvariety \(R\subset M\), invariant under the flow of \(E-\overline{E}\), where \(E\) is the Euler vector field of the Frobenius manifold.

Finally, the author shows that starting at suitable semi-simple or nilpotent points and going sufficiently far along the flow of the real vector field \(E+\overline{E}\) one no longer meets this bad locus \(R\); the Hermitian form \(h:=g(\cdot,\kappa \cdot)\) is positive definite, where \(g\) is the metric for the Frobenius manifold, and \(\kappa\) is the anti-involution of the \(tt^*\)-structure; and the endomorphism \(\mathcal Q\) given by the \(tt^*\)-structure has eigenvalues tending toward a known set.

In the second half of the paper these investigations are specialized to the case of the tangent bundle \(TM\). Using oscillating integrals the author then reconstructs the Saito Frobenius manifold structure on the semi-universal unfolding of a hypersurface singularity, previously done using the Gauss-Manin connection [C. Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts Math. 151, Cambridge University Press (2002; Zbl 1023.14018)], and he shows that it has a canonical compatible \(tt^*\)-geometry – or rather, CDV-structure (Cecotti-Dubrovin-Vafa) – outside of a real analytic subvariety \(R\subset M\), invariant under the flow of \(E-\overline{E}\), where \(E\) is the Euler vector field of the Frobenius manifold.

Finally, the author shows that starting at suitable semi-simple or nilpotent points and going sufficiently far along the flow of the real vector field \(E+\overline{E}\) one no longer meets this bad locus \(R\); the Hermitian form \(h:=g(\cdot,\kappa \cdot)\) is positive definite, where \(g\) is the metric for the Frobenius manifold, and \(\kappa\) is the anti-involution of the \(tt^*\)-structure; and the endomorphism \(\mathcal Q\) given by the \(tt^*\)-structure has eigenvalues tending toward a known set.

Reviewer: Tyler J. Jarvis (Provo)

##### MSC:

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |

32S30 | Deformations of complex singularities; vanishing cycles |

32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |

53C28 | Twistor methods in differential geometry |

##### Keywords:

Frobenius manifold; \(tt^*\)-structure; TERP; CDV; hypersurface singularity; harmonic bundle; unfolding; oscillating integrals##### References:

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