Abstract
We demonstrate that Appell-Lerch sums with higher order poles as well as their modular covariant completions arise as partition functions in the cigar conformal field theory with worldsheet supersymmetry. The modular covariant derivatives of the elliptic genus of the cigar give rise to operator insertions corresponding to (powers of) right-moving momentum, left-moving fermion number, as well as a term corresponding to an ordinary zero mode partition sum. To show this, we demonstrate how the right-moving supersymmetric quantum mechanics (and in particular the Hamiltonian and spectral density) depend on the imaginary part of the chemical potential for angular momentum. A consequence of our analysis is that varying the imaginary part of the chemical potential for angular momentum on the cigar gives rise to a wall-crossing phenomenon in the bound state contribution to the elliptic genus, while the full elliptic genus is a continuous function of the chemical potential.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525 [INSPIRE].
J. Troost, The non-compact elliptic genus: mock or modular, JHEP 06 (2010) 104 [arXiv:1004.3649] [INSPIRE].
T. Eguchi and Y. Sugawara, Non-holomorphic Modular Forms and SL(2, ℝ)/U(1) Superconformal Field Theory, JHEP 03 (2011) 107 [arXiv:1012.5721] [INSPIRE].
S.K. Ashok and J. Troost, A Twisted Non-compact Elliptic Genus, JHEP 03 (2011) 067 [arXiv:1101.1059] [INSPIRE].
A. Hanany, N. Prezas and J. Troost, The Partition function of the two-dimensional black hole conformal field theory, JHEP 04 (2002) 014 [hep-th/0202129] [INSPIRE].
T. Eguchi and Y. Sugawara, SL(2, ℝ)/U(1) supercoset and elliptic genera of noncompact Calabi-Yau manifolds, JHEP 05 (2004) 014 [hep-th/0403193] [INSPIRE].
D. Israel, C. Kounnas, A. Pakman and J. Troost, The Partition function of the supersymmetric two-dimensional black hole and little string theory, JHEP 06 (2004) 033 [hep-th/0403237] [INSPIRE].
A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing and Mock Modular Forms, arXiv:1208.4074 [INSPIRE].
S. Zwegers, Mock Theta functions, PhD thesis, Utrecht University (2002).
D. Zagier, Ramanujan’s mock theta functions and their applications d’après Zwegers and Bringmann-Ono, Séminaire Bourbaki (2007) 986.
T. Eguchi and A. Taormina, Character Formulas for the N = 4 Superconformal Algebra, Phys. Lett. B 200 (1988) 315 [INSPIRE].
A.M. Semikhatov, A. Taormina and I.Y. Tipunin, Higher level Appell functions, modular transformations and characters, math/0311314 [INSPIRE].
T. Eguchi and K. Hikami, Superconformal Algebras and Mock Theta Functions, J. Phys. A 42 (2009) 304010 [arXiv:0812.1151] [INSPIRE].
J. Manschot, Stability and duality in N = 2 supergravity, Commun. Math. Phys. 299 (2010) 651 [arXiv:0906.1767] [INSPIRE].
A. Giveon, N. Itzhaki and J. Troost, The Black Hole Interior and a Curious Sum Rule, JHEP 03 (2014) 063 [arXiv:1311.5189] [INSPIRE].
A. Giveon, N. Itzhaki and J. Troost, Lessons on Black Holes from the Elliptic Genus, arXiv:1401.3104 [INSPIRE].
J.A. Harvey and S. Murthy, Moonshine in Fivebrane Spacetimes, JHEP 01 (2014) 146 [arXiv:1307.7717] [INSPIRE].
G. Lopes Cardoso, M. Cirafici, R. Jorge and S. Nampuri, Indefinite theta functions and black hole partition functions, JHEP 02 (2014) 019 [arXiv:1309.4428] [INSPIRE].
T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl. Phys. B 414 (1994) 191 [hep-th/9306096] [INSPIRE].
S. Murthy, A holomorphic anomaly in the elliptic genus, JHEP 06 (2014) 165 [arXiv:1311.0918] [INSPIRE].
S.K. Ashok, N. Doroud and J. Troost, Localization and real Jacobi forms, JHEP 04 (2014) 119 [arXiv:1311.1110] [INSPIRE].
R. Akhoury and A. Comtet, Anomalous Behavior of the Witten Index: Exactly Soluble Models, Nucl. Phys. B 246 (1984) 253 [INSPIRE].
S.K. Ashok, S. Nampuri and J. Troost, Counting Strings, Wound and Bound, JHEP 04 (2013) 096 [arXiv:1302.1045] [INSPIRE].
D. Israel, A. Pakman and J. Troost, D-branes in N = 2 Liouville theory and its mirror, Nucl. Phys. B 710 (2005) 529 [hep-th/0405259] [INSPIRE].
S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A New supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [INSPIRE].
B. Pioline, On the Witten index of N = 2 four dimensional gauge theories, talk at Mock-modular forms and physics, International Workshop at IMSc, Chennai, April 14-18 2014, based on work by S. Alexandrov, G. W. Moore, A. Neitzke and B. Pioline.
P. Kraus and F. Larsen, Partition functions and elliptic genera from supergravity, JHEP 01 (2007) 002 [hep-th/0607138] [INSPIRE].
S.K. Ashok and J. Troost, Elliptic genera and real Jacobi forms, JHEP 01 (2014) 082 [arXiv:1310.2124] [INSPIRE].
Open Access
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1404.7396
Unité Mixte du CNRS et de l’Ecole Normale Supérieure associée à l’université Pierre et Marie Curie 6, UMR, 8549
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), 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.
About this article
Cite this article
Ashok, S.K., Dell’Aquila, E. & Troost, J. Higher poles and crossing phenomena from twisted genera. J. High Energ. Phys. 2014, 87 (2014). https://doi.org/10.1007/JHEP08(2014)087
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2014)087