Framing the Di-Logarithm (over Z)

TL;DR

This paper introduces the concept of s-functions, a class of poly-logarithmic series relevant to topological string theory, and proves their integrality properties through algebraic K-theory and Frobenius actions.

Contribution

It defines s-functions and characterizes 2-functions using algebraic K-theory, providing a proof of the integrality of the framing transformation.

Findings

s-functions are characterized via Frobenius endomorphism

2-functions relate to disk amplitudes in Calabi-Yau backgrounds

The framing transformation's integrality is proven using K-theory orthogonality

Abstract

Motivated by their role for integrality and integrability in topological string theory, we introduce the general mathematical notion of "s-functions" as integral linear combinations of poly-logarithms. 2-functions arise as disk amplitudes in Calabi-Yau D-brane backgrounds and form the simplest and most important special class. We describe s-functions in terms of the action of the Frobenius endomorphism on formal power series and use this description to characterize 2-functions in terms of algebraic K-theory of the completed power series ring. This characterization leads to a general proof of integrality of the framing transformation, via a certain orthogonality relation in K-theory. We comment on a variety of possible applications. We here consider only power series with rational coefficients; the general situation when the coefficients belong to an arbitrary algebraic number field is treated in a companion paper.