Parabolic Higgs bundles, $tt^*$ connections and opers

TL;DR

This paper explores the relationship between non-abelian Hodge theory and mirror symmetry in the context of Calabi-Yau families, revealing new connections between Higgs bundles, opers, and special functions.

Contribution

It demonstrates that the $tt^*$-connection is gauge equivalent to an oper derived from Picard-Fuchs equations, linking differential geometry with special functions.

Findings

Parabolic degrees are expressed via Picard-Fuchs exponents.

The $tt^*$-connection is gauge equivalent to an oper.

New differential relations generalize Ramanujan's relations.

Abstract

The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi-Yau manifolds. This setup enables us to apply techniques from mirror symmetry. For example, the corresponding Higgs bundles extend to parabolic Higgs bundles to the compactification of the base of the families. We determine the parabolic degrees of the underlying parabolic bundles in terms of the exponents of the Picard-Fuchs equations obtained from the variations of Hodge structure. Moreover, we prove in this setup that the flat non-abelian Hodge or $tt^*$-connection is gauge equivalent to an oper which is determined by the corresponding Picard-Fuchs equations. This gauge equivalence puts forward a new derivation of non-linear differential relations between special functions on the moduli space which generalize Ramanujan's relations for the differential ring of quasi-modular forms.