Framing the Di-Logarithm (over Z)

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.

22 pages, Contribution to Proceedings of String-Math 2012, Bonn

References in corpus (5)

• Integrality of instanton numbers and p-adic B-model
• Integrality theorems in the theory of topological strings
• Frobenius transformation, mirror map and instanton numbers
• Mahler Measure Variations, Eisenstein Series and Instanton Expansions
• On the Arithmetic of D-brane Superpotentials. Lines and Conics on the Mirror Quintic

Cited by in corpus (4)

• Knots-quivers correspondence
• Topological strings, strips and quivers
• BPS counting for knots and combinatorics on words
• Integrality of Framing and Geometric Origin of 2-functions