A Modern Fareytail

TL;DR

This paper refines the mathematical framework for elliptic genera in AdS_3/CFT_2, removing problematic transforms and proposing a new regularization method that enhances understanding of modular forms and gravity path integrals.

Contribution

It introduces a new way to express vector-valued modular forms without the fareytail transform, offering a regularization for the AdS_3 gravity path integral and clarifying modular invariance constraints.

Findings

A convergent sum representation for non-positive weight modular forms.

A new regularization scheme for the AdS_3 gravity path integral.

Insights into the relation between polar coefficients and cusp form coefficients.

Abstract

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.