Automorphic forms in string theory : from moonshine to wall crossing

Abstract

Summary: This thesis is devoted to the study of applications of automorphic forms ((mock) modular forms and (mock) Jacobi forms to moonshine phenomena, and to BPS wall crossing of in N = 4, d = 4 string theory. Moonshine phenomena are deep mathematical relations between vertex operator algebras, (mock) modular forms and sporadic groups that is manifest as the encoding of the dimension of irreducible representations of a sporadic group in the coefficients of the Fourier/character expansion of a (mock) modular form. This thesis is devoted partly to understand the nature of Mathieu moonshine (when the elliptic genus of the K3 surface expanded in characters of an N = (4,4) superconformal algebra has expansion coefficients that capture the dimensions of the irreducible representations of the largest Mathieu group, M24). The second part of this thesis is devoted to the exact computation of single center 1⁄4 BPS black hole degeneracies in N = 4, d=4 string compactification from 1⁄2 BPS states . 1⁄4 BPS states in these theories suffer from wall crossing effects. We derive a formula that computes 1⁄2 BPS instanton degeneracies by keeping tract of all wall crossings precisely and use that to build the full indexed partition function of 1⁄4 BPS states.