Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2007-12-04
Commun.Num.Theor.Phys. 4, 103-159, 2010
Physics
High Energy Physics
High Energy Physics - Theory
56 pages, published version
Scientific paper
We revisit the "fareytail expansions" of elliptic genera which have been used in discussions of the AdS_3/CFT_2 correspondence and the OSV conjecture. We show how to write such expansions without the use of the problematic "fareytail transform." In particular, we show how to write a general vector-valued modular form of non-positive weight as a convergent sum over cosets of SL(2,Z). This sum suggests a new regularization of the gravity path integral in AdS_3, resolves the puzzles associated with the "fareytail transform," and leads to several new insights. We discuss constraints on the polar coefficients of negative weight modular forms arising from modular invariance, showing how these are related to Fourier coefficients of positive weight cusp forms. In addition, we discuss the appearance of holomorphic anomalies in the context of the fareytail.
Manschot Jan
Moore Gregory W.
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