Celebratio Mathematica

M. Kontsevich, A. Schwarz, and V. Vologodsky: “Integrality of instanton numbers and $p$ -adic B-model,” Phys. Lett. B 637 : 1–2 (2006), pp. 97–101. MR 2230876 Zbl 1247.14058 ArXiv hep-th/0603106 article

Vadim Aleksandrovich Vologodsky	Related
Maksim Lvovich Kontsevich	Related

A. Schwarz and V. Vologodsky: “Frobenius transformation, mirror map and instanton numbers,” Phys. Lett. B 660 : 4 (2008), pp. 422–427. MR 2396265 Zbl 1246.14053 ArXiv hep-th/0606151 article

A. Schwarz and V. Vologodsky: “Integrality theorems in the theory of topological strings,” Nuclear Phys. B 821 : 3 (2009), pp. 506–534. MR 2547214 Zbl 1203.81155 ArXiv 0807.1714 article

A. Schwarz, V. Vologodsky, and J. Walcher: “Framing the di-logarithm (over $Z$ ),” pp. 113–128 in String-Math 2012 (Bonn, Germany, 16–21 July 2012). Edited by R. Donagi, S. Katz, A. Klemm, and D. R. Morrison. Proceedings of Symposia in Pure Mathematics 90. American Mathematical Society (Providence, RI), 2015. MR 3409790 Zbl 1356.81196 ArXiv 1306.4298 incollection

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.

David Robert Morrison	Related
Sheldon H. Katz	Related
Vadim Aleksandrovich Vologodsky	Related
Albrecht Klemm	Related
Johannes Walcher	Related
Ron Yehuda Donagi	Related

A. Schwarz, V. Vologodsky, and J. Walcher: Integrality of framing and geometric origin of 2-functions. Preprint, February 2017. ArXiv 1702.07135 techreport

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.

Johannes Walcher	Related
Vadim Aleksandrovich Vologodsky	Related

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Albert S. Schwarz

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