Integrality of Framing and Geometric Origin of 2-functions

Abstract

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 $p$. One can prove that 2-functions with rational coefficients appear as building block of BPS generating functions in topological string theory. Using the Frobenius map we define 2-functions with coefficients in algebraic number fields. We establish two results pertaining to these functions. 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.

66 pages, no figures

References in corpus (4)

• Integrality of instanton numbers and p-adic B-model
• Integrality theorems in the theory of topological strings
• Frobenius transformation, mirror map and instanton numbers
• Supergeometry and Arithmetic Geometry