构造双对数（在Z上）
Framing the Di-Logarithm (over Z)
摘要：
由于它们在拓扑弦论中的完整性和可积性的作用，我们引入了“s-函数”作为多函数的完整线性组合的一般数学概念。2-函数作为卡-丘D膜背景中的圆盘振幅出现，并形成最简单和最重要的特殊类。我们描述的Frobenius自同态正式幂级数的行动方面的s-功能，并使用这种描述来表征2-功能的代数K-理论的完整的幂级数环。这种表征导致了框架变换的完整性的一般证明，通过一定的正交关系在K-理论。我们评论各种可能的应用。在这里，我们只考虑幂级数与合理的系数，一般情况下，当系数属于一个任意的代数数域中处理的同伴文件。
英文摘要：
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.
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