Asian Journal of Mathematics

Volume 26 (2022)

Number 4

Parabolic Higgs bundles, $tt^\ast$ connections and opers

Pages: 455 – 506



Murad Alim (Fachbereich Mathematik, Universität Hamburg, Germany)

Florian Beck (Fachbereich Mathematik, Universität Hamburg, Germany)

Laura Fredrickson (Department of Mathematics, University of Oregon, Eugene, Or., U.S.A.)


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 $tt^\ast$-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.


parabolic Higgs bundles, $tt^\ast$ equations, opers, mirror symmetry, quasi-modular forms

2010 Mathematics Subject Classification

14D07, 14D21, 14J32, 53C07

Received 2 November 2021

Accepted 18 February 2022

Published 24 March 2023