# Nonabelian Hodge theory and vector valued modular forms

## TL;DR

This paper explores the connection between nonabelian Hodge theory and vector valued modular forms, proving new inequalities for certain nonunitary representations and proposing strategies to extend these results.

## Contribution

It introduces the concept of Higgs forms on the Dolbeault side to establish new instances of the three-term inequality for nonunitary representations.

## Key findings

- Proved new inequalities for nonunitary representations
- Established a link between Higgs forms and modular form inequalities
- Suggested a reduction strategy for general nilpotent Higgs bundles

## Abstract

We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, finite dimensional, irreducible representations of the modular group. This conjecture is known for irreducible unitary representations and for all irreducible representations of dimension at most 12. We prove new instances of the three-term inequality for certain nonunitary representations, corresponding to a class of maximally-decomposed variations of Hodge structure, by considering the same inequality with respect to a new type of modular form, called a "Higgs form", that arises naturally on the Dolbeault side of nonabelian Hodge theory. The paper concludes with a discussion of a strategy for reducing the general case of nilpotent Higgs bundles to the case under consideration in our main theorem.