Abstract
The partition function of 2d conformal field theory is a modular invariant function. It is known that the partition function of a holomorphic CFT whose central charge is a multiple of 24 is a polynomial in the Klein function. In this paper, by using the medium temperature expansion we show that every modular invariant partition function can be mapped to a holomorphic partition function whose structure can be determined similarly. We use this map to study partition function of CFTs with half-integer left and right conformal weights. We show that the corresponding left and right central charges are necessarily multiples of 4. Furthermore, the degree of degeneracy of high-energy levels can be uniquely determined in terms of the degeneracy in the low energy states.
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References
A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 538] [INSPIRE].
A.A. Migdal, Conformal invariance and bootstrap, Phys. Lett. B 37 (1971) 386 [INSPIRE].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D conformal field theories with global symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].
A. Vichi, Improved bounds for CFT’s with global symmetries, JHEP 01 (2012) 162 [arXiv:1106.4037] [INSPIRE].
F. Caracciolo and V.S. Rychkov, Rigorous limits on the interaction strength in quantum field theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Central charge bounds in 4D conformal field theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
F. Loran, M.M. Sheikh-Jabbari and M. Vincon, Beyond logarithmic corrections to Cardy formula, JHEP 01 (2011) 110 [arXiv:1010.3561] [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal spectrum of 2D conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
G. Höhn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Math. Schr. 286 (1996) 1 [arXiv:0706.0236].
G. Höhn, Conformal designs based on vertex operator algebras, math/0701626.
A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
A.N. Schellekens, Meromorphic C = 24 conformal field theories, Commun. Math. Phys. 153 (1993) 159 [hep-th/9205072] [INSPIRE].
I.B. Frenkel, J. Lepowsky and A. Meurman, A natural representation of the Fischer-Griess monster with the modular function J as character, Proc. Natl. Acad. Sci. U.S.A. 81 (1984) 3256.
T. Apostol, Modular functions and Dirichlet series in number theory, Springer, Germany (1990).
M.R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [INSPIRE].
S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
D. Friedan and C.A. Keller, Constraints on 2D CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
S. Hellerman and C. Schmidt-Colinet, Bounds for state degeneracies in 2D conformal field theory, JHEP 08 (2011) 127 [arXiv:1007.0756] [INSPIRE].
C.A. Keller, Modularity, Calabi-Yau geometry and 2d CFTs, Proc. Symp. Pure Math. 88 (2014) 307 [arXiv:1312.7313] [INSPIRE].
J.D. Qualls and A.D. Shapere, Bounds on operator dimensions in 2D conformal field theories, JHEP 05 (2014) 091 [arXiv:1312.0038] [INSPIRE].
J.D. Qualls, Universal bounds on operator dimensions in general 2D conformal field theories, arXiv:1508.00548 [INSPIRE].
C.A. Keller and H. Ooguri, Modular constraints on Calabi-Yau compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE].
N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, Universal bounds on charged states in 2d CFT and 3d gravity, JHEP 08 (2016) 041 [arXiv:1603.09745] [INSPIRE].
P. di Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer, Germany (1997).
C.A. Keller, Phase transitions in symmetric orbifold CFTs and universality, JHEP 03 (2011) 114 [arXiv:1101.4937] [INSPIRE].
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ArXiv ePrint: 1607.08516
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Ashrafi, M., Loran, F. Non-chiral 2d CFT with integer energy levels. J. High Energ. Phys. 2016, 121 (2016). https://doi.org/10.1007/JHEP09(2016)121
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DOI: https://doi.org/10.1007/JHEP09(2016)121