# Wavefunctions on $S^2$ with flux and branes

## TL;DR

This paper analyzes wavefunctions in a six-dimensional gauge theory compactified on a sphere with flux and branes, revealing unique localization, degeneracy, and CP violation features in the lowest KK modes.

## Contribution

It provides an analytical formulation of KK wavefunctions on $S^2$ with flux and branes, highlighting novel localization and degeneracy phenomena.

## Key findings

- Lowest KK modes can be sharply localized depending on brane configurations.
- Degeneracy of lowest KK modes can occur due to flux and brane effects.
- The setup allows for natural CP violation in Yukawa interactions.

## Abstract

We formulate a six dimensional $U(1)$ gauge theory compactified on a (two dimensional) sphere $S^2$ with flux and localized brane sources. Profiles of the lowest Kaluza-Klein (KK) wavefunctions and their masses are derived analytically. In contrast to ordinary sphere compactifications, the above setup can lead to the degeneracy of and the sharp localizations of the linearly independent lowest KK modes, depending on the number of branes and their tensions. Moreover, it can naturally accommodate CP violation in Yukawa interactions.