Mirror symmetry in black holes, elementary particles, and mathematics

Abstract

Mirror symmetry is the study of two seemingly different objects that are conjectured to be the same. That is, we could think of them as mirror images of each other. For example, string theorists conjecture that there exists such a duality between black holes and elementary particles. This thesis discusses this example from string theory, and demonstrates some mathematical mirror symmetry examples involving polynomials and groups. Given a pair made up of a polynomial and a group, we create a vector space. The vector spaces created by the dual pair always turn out to be equivalent! Progress in this field has been made by breaking polynomials into smaller pieces (atomic-types), called Fermats, chains, and loops. In this thesis I will give a formula and complete description of the vector spaces associated to Fermat and loop polynomials with repeated exponents and using their maximal symmetry groups. Finally, explanations on general observations on the occurrence of vector spaces with various dimensions based on computations will be explored, and explanations for various methods and formulas obtained during my study of these objects will be given.