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Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. I · 1953
Earlier work this paper cites.
McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I · 1954
Earlier work this paper cites.
Amer. Math. Soc., Providence, R.I., 1965
A. Selberg, “On the estimation of Fourier coefficients of modular forms,” in Proc. Sympos. Pure Math., Vol. VIII · 1965
Earlier work this paper cites.
Tata Inst. Fundamental Res., Bombay, 1981
D. Zagier, “Eisenstein series and the Riemann zeta function,” in Automorphic forms, representation theory and arithmetic (Bombay, 1979) · 1981
Earlier work this paper cites.
D. Goldfeld and P. Sarnak, “Sums of Kloosterman sums,” Invent. Math
1983
Earlier work this paper cites.
J. L. Cardy, “Operator Content of Two-Dimensional Conformally Invariant Theories,” Nucl.Phys
1986
Earlier work this paper cites.
A. Castro, M. R. Gaberdiel, T. Hartman, A. Maloney, and R. Volpato, “The Gravity Dual of the Ising Model,” Phys.Rev
Original
1987
Earlier work this paper cites.

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2014

Keller, Christoph A., Maloney, Alexander

Modular invariance strongly constrains the spectrum of states of two dimensional conformal field theories.

By summing over the images of the modular group, we construct candidate CFT partition functions that are modular invariant and have positive spectrum.
This allows us to efficiently extract the constraints on the CFT spectrum imposed by modular invariance, giving information on the spectrum that goes beyond the Cardy growth of the asymptotic density of states.
Some of the candidate modular invariant partition functions we construct have gaps of size (c-1)/12, proving that gaps of this size and smaller are consistent with modular invariance.

Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. I · 1953
Earlier work this paper cites.
McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I · 1954
Earlier work this paper cites.
Amer. Math. Soc., Providence, R.I., 1965
A. Selberg, “On the estimation of Fourier coefficients of modular forms,” in Proc. Sympos. Pure Math., Vol. VIII · 1965
Earlier work this paper cites.
Tata Inst. Fundamental Res., Bombay, 1981
D. Zagier, “Eisenstein series and the Riemann zeta function,” in Automorphic forms, representation theory and arithmetic (Bombay, 1979) · 1981
Earlier work this paper cites.
D. Goldfeld and P. Sarnak, “Sums of Kloosterman sums,” Invent. Math
1983
Earlier work this paper cites.
J. L. Cardy, “Operator Content of Two-Dimensional Conformally Invariant Theories,” Nucl.Phys
1986
Earlier work this paper cites.
A. Castro, M. R. Gaberdiel, T. Hartman, A. Maloney, and R. Volpato, “The Gravity Dual of the Ising Model,” Phys.Rev
Original
1987
Earlier work this paper cites.

M. Banados, C. Teitelboim, and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys.Rev.Lett
1992
Cited alongside, same era.
J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv.Theor.Math.Phys
1998
Cited alongside, same era.
J. M. Maldacena and A. Strominger, “AdS(3) black holes and a stringy exclusion principle,” JHEP
1998
Cited alongside, same era.
S. Carlip, “Logarithmic corrections to black hole entropy from the Cardy formula,” Class.Quant.Grav
2000
Cited alongside, same era.
W. d. A. Pribitkin, “A generalization of the Goldfeld-Sarnak estimate on Selberg’s Kloosterman zeta-function,” Forum Math
Original
2000
Cited alongside, same era.