# Modular Inflation at Higher Level $N$

## TL;DR

This paper extends modular inflation models to higher levels, analyzing their modular properties and proposing new models based on hauptmodul functions, with implications for CMB observations and quantum gravity conjectures.

## Contribution

It generalizes modular inflation to higher levels and introduces a new class of models using hauptmodul functions, linking them to observable CMB signatures.

## Key findings

- Modular structure ensures almost holomorphicity of inflation parameters.
- Hauptmodul-based models provide testable CMB predictions.
- Discussion of quantum gravity conjectures in the context of higher-level modular inflation.

## Abstract

We introduce the framework of modular inflation with level structure, generalizing the level one theory considered previously to higher levels. We analyze the modular structure of CMB observables in this framework and show that the nontrivial geometry of the target space suffices to ensure the almost holomorphic modularity of the relevant parameters. We further introduce a concrete class of models based on hauptmodul functions that provide generators of the corresponding inflationary potentials at level $N>1$. The phenomenology of this class of models provides targets for ground-based CMB experiments in the immediate future. In the framework of our models we also discuss the status of the quantum gravity conjectures that have been formulated in the context of the swampland.