Indefinite Theta Functions and Higher Depth Mock Modular Forms

This thesis consists of research articles on indefinite theta functions, higher depth mock modular forms, higher depth quantum modular forms, and iterated Eichler integrals. First, we study an explicit indefinite theta function of signature (1,3) that occurs in the Gromov-Witten theory of elliptic orbifolds and determine its modular completion, showing that it is a mock modular form of depth 3 using methods of Zwegers and Alexandrov, Banerjee, Manschot, and Pioline. We continue by studying higher rank false theta functions and show that they have the same asymptotic behavior near rationals as iterated Eichler integrals of theta functions. The modular behavior of the Eichler integrals implies that the higher rank false theta functions are quantum modular forms of higher depth. Additionally, one of the occurring Eichler integrals is essentially the purely non-holomorphic part of an indefinite theta function of signature (2,2). Furthermore, we show that these false theta-functions actually satisfy higher depth vector-valued quantum modular behavior. We also generalize a connection between Eichler integrals and Mordell integrals to a two-dimensional case. Finally, we suggest $q$-series that should contribute the other components of the higher depth vector-valued quantum modular forms and show that they span a space that is essentially closed under modular transformation in a special case