Framing the Di-Logarithm (over Z)

Understand

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.