The non-compact elliptic genus: mock or modular

TL;DR

The paper investigates the elliptic genus of non-compact $N=2$ coset CFTs with $c>3$, focusing on the holomorphic part captured by an Appell-Lerch sum and its failure to form a modular object by itself. It develops three complementary analyses—the free-field (Section 2.1), the algebraic (Section 2.2), and the path-integral (Section 3)—to isolate the holomorphic piece and identify a non-holomorphic remainder arising from the continuous spectrum. The path-integral computation shows that $\chi = \chi_{hol} + \chi_{rem}$, with $\chi_{rem}$ precisely reproducing Zwegers' completion that converts the Appell-Lerch sum into a Jacobi form of weight $0$ and index $\frac{k(k+2)}{2}$, thereby achieving modular covariance. These results clarify modular properties in non-rational, non-compact CFTs and provide a framework for modular-invariant amplitudes in string theory, with potential extensions to higher supersymmetry and orbifolds and further study of holomorphic anomalies.

Abstract

We analyze various perspectives on the elliptic genus of non-compact supersymmetric coset conformal field theories with central charge larger than three. We calculate the holomorphic part of the elliptic genus via a free field description of the model, and show that it agrees with algebraic expectations. The holomorphic part of the elliptic genus is directly related to an Appell-Lerch sum and behaves anomalously under modular transformation properties. We analyze the origin of the anomaly by calculating the elliptic genus through a path integral in a coset conformal field theory. The path integral codes both the holomorphic part of the elliptic genus, and a non-holomorphic remainder that finds its origin in the continuous spectrum of the non-compact model. The remainder term can be shown to agree with a function that mathematicians introduced to parameterize the difference between mock theta functions and Jacobi forms. The holomorphic part of the elliptic genus thus has a path integral completion which renders it non-holomorphic and modular.