M-Theory on G₂ Manifolds and Its Connection to Number Theory

In this project, we focus on M-Theory in its low-energy limit, supergravity, on G₂ manifolds. A G₂ manfiold is a 7-dimensional, Ricci-flat, Riemannian manifold with a G₂ holonomy group, where G₂ is an exceptional simple Lie group that is a proper subgroup of SO(7), preserving a spinor in the 8-dimensional spinor representation. Compactifying 11-dimesnional supergravity on the G₂ manifold results in a effective 4-dimensional N =1 theory which is phenomenologically important since there exists a family of N=1 supersymmetric extensions to the Standard Model that are realistic and breaking supersymmetry at an appropriate energy scale. We aim to understand the interesting mathematical structures, such as mock modular forms, of the 11-dimensional supergravity on G₂ manifold, offering a potential bridge between M-Theory and number theory.