Fermion Wavefunctions in Magnetized branes: Theta identities and Yukawa couplings

TL;DR

This work develops explicit fermion wavefunctions on toroidal compactifications with general constant fluxes, including oblique fluxes that do not commute, and derives Yukawa couplings from wavefunction overlaps. It introduces generalized Riemann theta identities to treat theta-function products with matrix-valued modular parameters, enabling closed-form Yukawa expressions for both factorized and non-factorized tori. The authors construct negative-chirality wavefunctions and establish mappings to positive-chirality bases, ensuring the equations of motion map consistently across chiralities and complex structures. The framework is applied to mass-generation scenarios for non-chiral states in an SU(5) GUT model, and the results have broad implications for moduli-stabilized magnetized brane constructions and their string-theoretic duals and extensions.

Abstract

Computation of Yukawa couplings, determining superpotentials as well as the Kähler metric, with oblique (non-commuting) fluxes in magnetized brane constructions is an interesting unresolved issue, in view of the importance of such fluxes for obtaining phenomenologically viable models. In order to perform this task, fermion (scalar) wavefunctions on toroidally compactified spaces are presented for general fluxes, parameterized by Hermitian matrices with eigenvalues of arbitrary signatures. We also give explicit mappings among fermion wavefunctions, of different internal chiralities on the tori, which interchange the role of the flux components with the complex structure of the torus. By evaluating the overlap integral of the wavefunctions, we give the expressions for Yukawa couplings among chiral multiplets arising from an arbitrary set of branes (or their orientifold images). The method is based on constructing certain mathematical identities for general Riemann theta functions with matrix valued modular parameter. We briefly discuss an application of the result, for the mass generation of non-chiral fermions, in the SU(5) GUT model presented by us in arXiv:0709.2799.