The non-compact elliptic genus: mock or modular

TL;DR

This paper investigates the elliptic genus of non-compact supersymmetric coset conformal field theories, revealing its connection to mock modular forms and providing a path integral completion that restores modularity.

Contribution

It offers a detailed analysis of the holomorphic and non-holomorphic parts of the elliptic genus, linking physical models to mathematical structures like Appell-Lerch sums and mock theta functions.

Findings

Holomorphic part matches algebraic expectations.

Path integral captures both holomorphic and non-holomorphic components.

Non-holomorphic remainder aligns with mathematical functions describing mock modularity.

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.