Magnetized Riemann Surface of Higher Genus and Eta Quotients of Semiprime Level

TL;DR

This paper investigates zero mode solutions of a Dirac operator on higher genus magnetized Riemann surfaces, introducing a method to construct cusp form bases and exploring applications like Yukawa couplings.

Contribution

It presents a new method to construct cusp form bases for higher genus magnetized Riemann surfaces and applies it to semiprime levels, with implications for physics.

Findings

Successful construction of cusp form bases for selected levels

Demonstration of zero mode solutions for lower weights

Discussion of applications to Yukawa couplings and matrix regularization

Abstract

We study the zero mode solutions of a Dirac operator on a magnetized Riemann surface of higher genus. In this paper, we define a Riemann surface of higher genus as a quotient manifold of the Poincar$\acute{\text{e}}$ upper half-plane by a congruence subgroup, especially $\Gamma_{0}(N)$. We present a method to construct basis of cusp forms since the zero mode solutions should be cusp forms. To confirm our method, we select a congruence subgroup of semiprime level and show the demonstration to some lower weights. In addition, we discuss Yukawa couplings and matrix regularization as applications.