Integrality of framing and geometric origin of 2-functions

We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. Such 2-functions appear as building blocks of BPS generating functions in open topological string theory. Using the Frobenius map we define 2-functions with coefficients in algebraic number fields, and establish two results: First, we show that the class of 2-functions is closed under the so-called framing operation (related to compositional inverse of power series). Second, we show that 2-functions arise naturally in geometry as $q$-expansion of the truncated normal function associated with an algebraic cycle extending a degenerating family of Calabi-Yau 3-folds. This is joint work with Albert Schwarz and Vadim Vologodsky.