Journal of High Energy Physics

, 2014:156 | Cite as

Compact formulas for the completed mock modular forms

  • Tohru Eguchi
  • Yuji SugawaraEmail author
Open Access
Regular Article - Theoretical Physics


In this paper we present a new compact expression of the elliptic genus of SL(2)/U(1)-supercoset theory by making use of the ‘spectral flow method’ of the pathintegral evaluation. This new expression is written in a form like a Poincaré series with a non-holomorphic Gaussian damping factor, and manifestly shows the modular and spectral flow properties of a real analytic Jacobi form. As a related problem, we present similar compact formulas for the modular completions of various mock modular forms which appear in the representation theory of \( \mathcal{N}=2,\;4 \) superconformal algebras.

We further discuss the generalization to the cases of arbitrary spin-structures, that is, the world-sheet fermions with twisted boundary conditions parameterized by a continuous parameter. This parameter is naturally identified with the ‘u-variable’ in the Appell-Lerch sum.


Conformal Field Models in String Theory Superstring Vacua Black Holes in String Theory 


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This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Physics and Research Center for Mathematical PhysicsRikkyo UniversityTokyoJapan
  2. 2.Department of Physical Sciences, College of Science and EngineeringRitsumeikan UniversityShigaJapan

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