Communications in Number Theory and Physics

Volume 4 (2010)

Number 1

A modern fareytail

Pages: 103 – 159

DOI: http://dx.doi.org/10.4310/CNTP.2010.v4.n1.a3

Authors

Jan Manschot (Institute for Theoretical Physics, University of Amsterdam, The Netherlands)

Gregory W. Moore (NHETC and Department of Physics and Astronomy, Rutgers University, New Jersey)

Abstract

We revisit the “fareytail expansions” of elliptic generawhich have been used in discussions of the AdS$_3$/CFT$_2$correspondence and the OSV conjecture. We show how to writesuch expansions without the use of the problematic“fareytail transform.” In particular, we show how towrite a general vector-valued modular form of non-positiveweight as a convergent sum over cosets ofSL$ (2,\mathbb{Z})$. This sum suggests a new regularizationof the gravity path integral in AdS$_3$, resolves thepuzzles associated with the “fareytail transform,” andleads to several new insights. We discuss constraints onthe polar coefficients of negative weight modular formsarising from modular invariance, showing how these arerelated to Fourier coefficients of positive weight cuspforms. In addition, we discuss the appearance ofholomorphic anomalies in the context of the fareytail.

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