# Modularity from Monodromy

## TL;DR

This paper presents a method to compute how a specific Fourier-Mukai transformation acts on brane charges in elliptically fibered Calabi-Yau threefolds, revealing a modular group action in certain cases.

## Contribution

It introduces a technique to determine the Fourier-Mukai transformation's effect on brane charges, highlighting the modular symmetry in the presence of a section.

## Key findings

- Fourier-Mukai kernel is the ideal sheaf of the relative diagonal.
- In fibrations with a section, the kernel is essentially the Poincaré sheaf.
- The transformation induces a modular group action on 2-brane charges.

## Abstract

In this note we describe a method to calculate the action of a particular Fourier-Mukai transformation on a basis of brane charges on elliptically fibered Calabi-Yau threefolds with and without a section. The Fourier-Mukai kernel is the ideal sheaf of the relative diagonal and for fibrations that admit a section this is essentially the Poincar\'e sheaf. We find that in this case it induces an action of the modular group on the charges of 2-branes.

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Source: https://tomesphere.com/paper/1902.08215