Abstract
We construct real Jacobi forms with matrix index using path integrals. The path integral expressions represent elliptic genera of two-dimensional \( \mathcal{N} \) = (2, 2) supersymmetric theories. They arise in a family labeled by two integers N and k which determine the central charge of the infrared fixed point through the formula c = 3N (1 + 2N/k). We decompose the real Jacobi form into a mock modular form and a term arising from the continuous spectrum of the conformal field theory. For a given N and k we argue that the Jacobi form represents the elliptic genus of a theory defined on a 2N dimensional linear dilaton background with U(N) isometry, an asymptotic circle of radius \( \sqrt{{k\alpha \prime }} \) and linear dilaton slope \( N\sqrt{{{2 \left/ {k} \right.}}} \). We also present formulas for the elliptic genera of their orbifolds.
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References
A. Schellekens and N. Warner, Anomalies and modular invariance in string theory, Phys. Lett. B 177 (1986) 317 [INSPIRE].
E. Witten, Elliptic genera and quantum field theory, Commun. Math. Phys. 109 (1987) 525 [INSPIRE].
T. Eguchi, H. Ooguri, A. Taormina and S.-K. Yang, Superconformal algebras and string compactification on manifolds with SU(N) holonomy, Nucl. Phys. B 315 (1989) 193 [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].
E. Witten, On the Landau-Ginzburg description of N = 2 minimal models, Int. J. Mod. Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].
J. Troost, The non-compact elliptic genus: mock or modular, JHEP 06 (2010) 104 [arXiv:1004.3649] [INSPIRE].
S. Zwegers, Mock theta functions, Ph.D. thesis, Utrecht University, Utrecht The Netherlands (2002).
D. Zagier, Ramanujan’s mock theta functions and their applications d’après Zwegers and Bringmann-Ono, Séminaire Bourbaki 986, Astérisque France (2007).
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].
S.K. Ashok, S. Nampuri and J. Troost, Counting strings, wound and bound, JHEP 04 (2013) 096 [arXiv:1302.1045] [INSPIRE].
E. Kiritsis, C. Kounnas and D. Lüst, A large class of new gravitational and axionic backgrounds for four-dimensional superstrings, Int. J. Mod. Phys. A 9 (1994) 1361 [hep-th/9308124] [INSPIRE].
K. Hori and A. Kapustin, Duality of the fermionic 2D black hole and N = 2 Liouville theory as mirror symmetry, JHEP 08 (2001) 045 [hep-th/0104202] [INSPIRE].
K. Hori and A. Kapustin, World sheet descriptions of wrapped NS five-branes, JHEP 11 (2002) 038 [hep-th/0203147] [INSPIRE].
C. Ziegler, Jacobi forms of higher degree, Abh. Math. Semi. Univ. Hamburg 59 (1989) 191.
N.-P. Skoruppa, Jacobi forms of critical weight and Weil representations, arXiv:0707.0718.
A. Semikhatov, A. Taormina and I.Y. Tipunin, Higher level Appell functions, modular transformations and characters, math.QA/0311314 [INSPIRE].
S.K. Ashok and J. Troost, Elliptic genera of non-compact Gepner models and mirror symmetry, JHEP 07 (2012) 005 [arXiv:1204.3802] [INSPIRE].
K. Miki, The representation theory of the SO(3) invariant superconformal algebra, Int. J. Mod. Phys. A 5 (1990) 1293 [INSPIRE].
T. Eguchi and Y. Sugawara, Modular bootstrap for boundary N = 2 Liouville theory, JHEP 01 (2004) 025 [hep-th/0311141] [INSPIRE].
D. Israel, A. Pakman and J. Troost, Extended SL(2,R)/U(1) characters, or modular properties of a simple nonrational conformal field theory, JHEP 04 (2004) 043 [hep-th/0402085] [INSPIRE].
S. Elitzur, A. Forge and E. Rabinovici, Some global aspects of string compactifications, Nucl. Phys. B 359 (1991) 581 [INSPIRE].
G. Mandal, A.M. Sengupta and S.R. Wadia, Classical solutions of two-dimensional string theory, Mod. Phys. Lett. A 6 (1991) 1685 [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, arXiv:1308.4896 [INSPIRE].
A. Gadde and S. Gukov, 2d index and surface operators, arXiv:1305.0266 [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozcaz, G. Lockhart and C. Vafa, M-strings, arXiv:1305.6322 [INSPIRE].
V.G. Kac and M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, hep-th/9407057 [INSPIRE].
M. Stern and P. Yi, Counting Yang-Mills dyons with index theorems, Phys. Rev. D 62 (2000) 125006 [hep-th/0005275] [INSPIRE].
A. Dabholkar, S. Murthy and D. Zagier, Quantum black holes, wall crossing and mock modular forms, arXiv:1208.4074 [INSPIRE].
T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 surface and the Mathieu group M 24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].
J.A. Harvey and S. Murthy, Moonshine in fivebrane spacetimes, arXiv:1307.7717 [INSPIRE].
B. Haghighat, J. Manschot and S. Vandoren, A 5d/2d/4d correspondence, JHEP 03 (2013) 157 [arXiv:1211.0513] [INSPIRE].
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ArXiv ePrint: 1310.2124
Unité Mixte du CNRS et de l’Ecole Normale Supérieure associée à l’université Pierre et Marie Curie 6, UMR 8549.
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Ashok, S.K., Troost, J. Elliptic genera and real Jacobi forms. J. High Energ. Phys. 2014, 82 (2014). https://doi.org/10.1007/JHEP01(2014)082
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DOI: https://doi.org/10.1007/JHEP01(2014)082