A family of vector-valued quantum modular forms of depth two

Abstract

We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta functions with coefficients given by double error functions. Further, we view these Eichler integrals in a modular setting as parts of certain weight two indefinite theta series.

A short note on higher Mordell integrals

“…The relation (2.4) extends the definition of M 2 (u) to include u 2 = 0 or u 1 = κu 2 -note however that M 2 is discontinuous across these loci. Further, it is shown in the proof of Lemma 7.1 of [8] that for u = u(n…”

“…This short note serves to show that techniques of Bringmann, Kaszian, and Milas of relating their double Eichler integral to higher Mordell integrals in [3] immediately carry over to the more general setting of [8]. In a similar fashion to [3] we define…”

“…In [8] a family of functions is given as a generalisation of the function F 1 . Each function F in this more general family from [8] is of the shape (up to addition by one-dimensional partial theta functions) α∈S ε(α)…”

“…where D := 4a 1 a 3 − a 2 2 > 0, and θ 1 , θ 2 are given explicitly in Section 3. It is shown in [8] that using Shimura theta functions we may rewrite this in the form (1.3).…”

Higher depth mock theta functions and $q$-hypergeometric series

“…Recently, such forms were connected to black holes by Alexandrov, Pioline [1], to the Gromov-Witten theory of elliptic orbifolds [10], and to indefinite theta functions on arbitrary lattices of signature (r, n − r) -see [2] for the r = 2 case and [25] for general r, each of which are generalizations of Zwegers' groundbreaking thesis [32] where r = 1. There are also further applications after relaxing to the notion of higher depth quantum modular forms [8,9,24]. The q-hypergeometric structure of examples of mock theta functions is also crucial to applications in geometry and topology.…”