On some aspects of mock and quantum modularity in theoretical physics

This thesis explores some aspects of mock and quantum modularity in two different topics arising from String Theory. The first topic is umbral moonshine, a series of surprising relations connecting representations of finite groups, mock modular forms, and K3 sigma models. Chapter 2 presents the construction of vertex algebra modules for some instances of umbral moonshine. This is achieved via the construction of cone vertex algebra modules whose graded traces can be expressed in terms of indefinite theta functions.
The second topic concerns a family of three-manifold invariants conjecturally related to Chern-Simons invariants. Chapter 3 explores the connection between mock theta functions and these invariants for Brieskorn spheres and gauge group SU(2). Using a relationship between the latter and mock modular forms obtained by indefinite theta functions, a regularisation procedure is proposed for the expression of the studied invariants under orientation reversal. Chapter 4 explores the connection with quantum modular forms of higher depth, and presents a conjectured recursive relation connecting the invariants for a gauge group of rank n to the invariants for gauge groups of lower ranks. This relation is proven for gauge group SU(3) for the invariants of Brieskorn spheres.