# L-functions for meromorphic modular forms and sum rules in conformal field theory

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## Abstract

We define L-functions for meromorphic modular forms that are regular at cusps, and use them to: (i) find new relationships between Hurwitz class numbers and traces of singular moduli, (ii) establish predictions from the physics of T-reflection, and (iii) express central charges in two-dimensional conformal field theories (2d CFT) as a literal sum over the states in the CFTs spectrum. When a modular form has an order-*p* pole away from cusps, its *q*-series coefficients grow as *n*^{p−1}*e*^{2πnt} for \( t\ge \frac{\sqrt{3}}{2} \). Its L-function must be regularized. We define such L-functions by a deformed Mellin transform. We study the L-functions of logarithmic derivatives of modular forms. L-functions of logarithmic derivatives of Borcherds products reveal a new relationship between Hurwitz class numbers and traces of singular moduli. If we can write 2d CFT path integrals as infinite products, our L-functions confirm T-reflection predictions and relate central charges to regularized sums over the states in a CFTs spectrum. Equating central charges, which are a proxy for the number of degrees of freedom in a theory, directly to a sum over states in these CFTs is new and relies on our regularization of such sums that generally exhibit exponential (Hagedorn) divergences.

## Keywords

Conformal Field Theory Anomalies in Field and String Theories Space-Time Symmetries## Notes

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## References

- [1]T.M. Apostol,
*Modular functions and Dirichlet series in number theory*, Graduate Texts in Mathematics, London, Springer (1997).Google Scholar - [2]D. Zagier,
*Traces of singular moduli*, in*Motives, Polylogarithms, and Hodge Theory*F. Bogomolov and L. Katzarkov eds., Lect. Ser. 3 Intl. Press, Somerville (2002), pp. 209–244.Google Scholar - [3]J.H. Bruinier, J. Funke and O. Imamoglu,
*Regularized theta liftings and periods of modular functions*,*J. reine angew. Math.***703**(2015) 43 [arXiv:1112.3444].MathSciNetzbMATHGoogle Scholar - [4]K. Bringmann, K.H. Fricke and Z.A. Kent,
*Special L-values and periods of weakly holomorphic modular forms*,*Proc. Am. Math. Soc.***142**(2014) 3425.MathSciNetCrossRefzbMATHGoogle Scholar - [5]K. Bringmann, N. Diamantis and S. Ehlen,
*Regularized Inner Products and Errors of Modularity*,*Int. Math. Res. Not.***2017**(2017) 7420 [arXiv:1603.03056].MathSciNetGoogle Scholar - [6]D.A. McGady,
*Temperature-reflection I: field theory, ensembles and interactions*, arXiv:1711.07536 [INSPIRE]. - [7]D.A. McGady,
*Temperature-reflection II: Modular Invariance and T-reflection*, arXiv:1806.09873 [INSPIRE]. - [8]P. Di Francesco, P. Mathieu and D. Senechal,
*Conformal field theory*, New York, U.S.A., Springer (1997) [INSPIRE].CrossRefzbMATHGoogle Scholar - [9]O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk,
*The Hagedorn-deconfinement phase transition in weakly coupled large N gauge theories*,*Adv. Theor. Math. Phys.***8**(2004) 603 [hep-th/0310285] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar - [10]G. Bașar, A. Cherman, K.R. Dienes and D.A. McGady, 4
*D-*2*D equivalence for large-N Yang-Mills theory*,*Phys. Rev.***D 92**(2015) 105029 [arXiv:1507.08666] [INSPIRE].ADSMathSciNetGoogle Scholar - [11]G. Bașar, A. Cherman, K.R. Dienes and D.A. McGady,
*Modularity and*4*D-*2*D spectral equivalences for large-N gauge theories with adjoint matter*,*JHEP***06**(2016) 148 [arXiv:1512.07918] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [12]R. Hagedorn,
*Statistical thermodynamics of strong interactions at high-energies*,*Nuovo Cim. Suppl.***3**(1965) 147 [INSPIRE].Google Scholar - [13]T.D. Cohen,
*QCD and the Hagedorn spectrum*,*JHEP***06**(2010) 098 [arXiv:0901.0494] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [14]G. Basar, A. Cherman, D.A. McGady and M. Yamazaki,
*Temperature-reflection symmetry*,*Phys. Rev.***D 91**(2015) 106004 [arXiv:1406.6329] [INSPIRE].ADSGoogle Scholar - [15]G.H. Hardy and S. Ramanujan,
*On the coefficients in the expansions of certain modular functions*,*Proc. Roy. Soc. Lond.***A 95**(1918) 144.ADSCrossRefzbMATHGoogle Scholar - [16]B. Berndt, P. Bialek and A. Yee,
*Formulas of Ramanujan for the power series coefficients of certain quotients of Eisenstein series*,*Int. Math. Res. Not.***2002**(2002) 1077.MathSciNetCrossRefzbMATHGoogle Scholar - [17]P. Bialek,
*Ramanujan’s formulas for the coefficients in the power series expansions of certain modular forms*, Ph.D. Thesis, University of Illinois at Urbana-Champaign (1995).Google Scholar - [18]B. Berndt and P. Bialek
*On the Power Series Coefficients of Certain Quotients of Eisenstein Series*,*Trans. Am. Math. Soc.***357**(2005) 4379.Google Scholar - [19]K. Bringmann and B. Kane,
*Ramanujan and coefficients of meromorphic modular forms*,*J. Math. Pures Appl.***107**(2017) 100 [arXiv:1603.07079].MathSciNetCrossRefzbMATHGoogle Scholar - [20]K. Bringmann and B. Kane,
*Ramanujan-like formulas for Fourier coefficients of all meromorphic cusp forms*, arXiv:1603.09250. - [21]K. Bringmann, B. Kane, S. Lobrich, K. Ono and L. Rolen,
*On Divisors of Modular Forms*,*Adv. Math.***329**(2018) 541 [arXiv:1609.08100].MathSciNetCrossRefzbMATHGoogle Scholar - [22]J.H. Bruinier, W. Kohnen and K. Ono,
*The arithmetic of the values of modular functions and the divisors of modular forms*,*Compos. Math.***140**(2004) 552.MathSciNetCrossRefzbMATHGoogle Scholar - [23]
- [24]A. Dabholkar, S. Murthy and D. Zagier,
*Quantum Black Holes, Wall Crossing and Mock Modular Forms*, arXiv:1208.4074 [INSPIRE]. - [25]R.E. Borcherds,
*Automorphic forms on O*_{s}+ 2*,*2(*R*)*and infinite products*,*Invent. Math.***120**(1995) 161.ADSMathSciNetCrossRefzbMATHGoogle Scholar - [26]G. Bol,
*Invarianten linearer Differentialgleichungen*,*Abh. Mat. Sem. Univ. Hamburg***16**(1949) 1.MathSciNetCrossRefzbMATHGoogle Scholar - [27]J. Lewis and D. Zagier,
*Period functions for Maass wave forms I*,*Annals Math.***153**(2001) 191. [math/0101270]. - [28]I.B. Frenkel, J. Lepowsky and A. Meurman,
*A Natural Representation of the Fischer-Griess Monster With the Modular Function J As Character*,*Proc. Nat. Acad. Sci. U.S.A.***81**(1984) 3256.ADSMathSciNetCrossRefzbMATHGoogle Scholar - [29]
- [30]
- [31]E. Witten,
*Global gravitational anomalies*,*Commun. Math. Phys.***100**(1985) 197 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [32]D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm,
*Heterotic String Theory. 2. The Interacting Heterotic String*,*Nucl. Phys.***B 267**(1986) 75 [INSPIRE]. - [33]J. Polchinski,
*String theory. Vol. 2: Superstring theory and beyond*, Cambridge University Press (1998) [INSPIRE]. - [34]W. Duke,
*Modular functions and the uniform distribution of CM points*,*Math. Ann.***334**(2006) 241.MathSciNetCrossRefzbMATHGoogle Scholar - [35]S. Lobrich,
*Niebur-Poincare Series and Traces of Singular Moduli*, arXiv:1704.08147. - [36]D. Kutasov and N. Seiberg,
*Number of degrees of freedom, density of states and tachyons in string theory and CFT*,*Nucl. Phys.***B 358**(1991) 600 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [37]K.R. Dienes,
*Modular invariance, finiteness and misaligned supersymmetry: New constraints on the numbers of physical string states*,*Nucl. Phys.***B 429**(1994) 533 [hep-th/9402006] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [38]G. Basar, A. Cherman, D. Dorigoni and M. Ünsal,
*Volume Independence in the Large N Limit and an Emergent Fermionic Symmetry*,*Phys. Rev. Lett.***111**(2013) 121601 [arXiv:1306.2960] [INSPIRE].ADSCrossRefGoogle Scholar - [39]G. Basar, A. Cherman and D.A. McGady,
*Bose-Fermi Degeneracies in Large N Adjoint QCD*,*JHEP***07**(2015) 016 [arXiv:1409.1617] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar