Higher depth quantum modular forms, multiple Eichler integrals, and ${\mathfrak{sl}}_3$ false theta functions

Abstract

We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose "companions" in the lower half-plane can be also realized both as double Eichler integrals and as non-holomorphic theta series having values of "double error" functions as coefficients. In particular, we prove that the false theta of ${\mathfrak{sl}}_3$, appearing in the character of the vertex algebra $W^0(p{)}_{A_2}$, can be written as the sum of two depth two quantum modular forms of positive integral weight.

39 pages

References in corpus (2)

• Logarithmic CFTs connected with simple Lie algebras
• Torus knots and quantum modular forms

Cited by in corpus (3)

• S-duality and refined BPS indices
• Vafa-Witten theory and iterated integrals of modular forms
• A $q$-series identity via the ${\mathfrak{sl}}_3$ colored Jones polynomials for the $(2,2m)$-torus link