Abstract
We analyse Virasoro blocks in the regime of heavy intermediate exchange (hp → ∞). For the 1-point block on the torus and the 4-point block on the sphere, we show that each order in the large-hp expansion can be written in closed form as polynomials in the Eisenstein series. The appearance of this structure is explained using the fusion kernel and, more markedly, by invoking the modular anomaly equations via the 2d/4d correspondence. The existence of these constraints allows us to develop a faster algorithm to recursively construct the blocks in this regime. We then apply our results to find corrections to averaged heavy-heavy-light OPE coefficients.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
M. Beşken, S. Datta and P. Kraus, Semi-classical Virasoro blocks: proof of exponentiation, JHEP 01 (2020) 109 [arXiv:1910.04169] [INSPIRE].
A.B. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Sov. Phys. JETP 63 (1986) 1061.
A. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensions: An explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].
M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, JHEP 04 (2019) 018 [arXiv:1703.09805] [INSPIRE].
P. Kraus and A. Maloney, A cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].
D. Das, S. Datta and S. Pal, Charged structure constants from modularity, JHEP 11 (2017) 183 [arXiv:1706.04612] [INSPIRE].
E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].
A. Romero-Bermúdez, P. Sabella-Garnier and K. Schalm, A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled, JHEP 09 (2018) 005 [arXiv:1804.08899] [INSPIRE].
Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].
D. Das, S. Datta and S. Pal, Universal asymptotics of three-point coefficients from elliptic representation of Virasoro blocks, Phys. Rev. D 98 (2018) 101901 [arXiv:1712.01842] [INSPIRE].
B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U(q)(SL(2, ℤ)), Commun. Math. Phys. 224 (2001) 613 [math.QA/0007097] [INSPIRE].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP 05 (2019) 212 [arXiv:1811.05710] [INSPIRE].
E.M. Brehm and D. Das, Aspects of the S transformation Bootstrap, J. Stat. Mech. 2005 (2020) 053103 [arXiv:1911.02309] [INSPIRE].
S. Collier, A. Maloney, H. Maxfield and I. Tsiares, Universal dynamics of heavy operators in CFT2 , JHEP 07 (2020) 074 [arXiv:1912.00222] [INSPIRE].
M.C.N. Cheng, T. Gannon and G. Lockhart, Modular Exercises for Four-Point Blocks — I, arXiv:2002.11125 [INSPIRE].
C. Cardona, Virasoro blocks at large exchange dimension, arXiv:2006.01237 [INSPIRE].
A.-K. Kashani-Poor and J. Troost, The toroidal block and the genus expansion, JHEP 03 (2013) 133 [arXiv:1212.0722] [INSPIRE].
A.-K. Kashani-Poor and J. Troost, Transformations of Spherical Blocks, JHEP 10 (2013) 009 [arXiv:1305.7408] [INSPIRE].
A.-K. Kashani-Poor and J. Troost, Quantum geometry from the toroidal block, JHEP 08 (2014) 117 [arXiv:1404.7378] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
J.A. Minahans, D. Nemeschansky and N.P. Warner, Instanton expansions for mass deformed N = 4 superYang-Mills theories, Nucl. Phys. B 528 (1998) 109 [hep-th/9710146] [INSPIRE].
M. Billó, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential in \( \mathcal{N} \) = 2★ theories (I): the ADE algebras, JHEP 11 (2015) 024 [arXiv:1507.07709] [INSPIRE].
S.K. Ashok, E. Dell’Aquila, A. Lerda and M. Raman, S-duality, triangle groups and modular anomalies in \( \mathcal{N} \) = 2 SQCD, JHEP 04 (2016) 118 [arXiv:1601.01827] [INSPIRE].
A. Kanazawa and J. Zhou, Lectures on BCOV holomorphic anomaly equations, in Fields Institute Monographs 34, Springer, New York NY U.S.A. (2015), pp. 445–473 [arXiv:1409.4105] [INSPIRE].
V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].
M.-x. Huang, On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit, JHEP 06 (2012) 152 [arXiv:1205.3652] [INSPIRE].
M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Deformed N = 2 theories, generalized recursion relations and S-duality, JHEP 04 (2013) 039 [arXiv:1302.0686] [INSPIRE].
M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Modular anomaly equation, heat kernel and S-duality in N = 2 theories, JHEP 11 (2013) 123 [arXiv:1307.6648] [INSPIRE].
M. Beccaria and G. Macorini, Exact partition functions for the Ω-deformed \( \mathcal{N} \) = 2∗ SU(2) gauge theory, JHEP 07 (2016) 066 [arXiv:1606.00179] [INSPIRE].
N. Nemkov, On new exact conformal blocks and Nekrasov functions, JHEP 12 (2016) 017 [arXiv:1606.05324] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
E. Perlmutter, Virasoro conformal blocks in closed form, JHEP 08 (2015) 088 [arXiv:1502.07742] [INSPIRE].
H. Chen, C. Hussong, J. Kaplan and D. Li, A Numerical Approach to Virasoro Blocks and the Information Paradox, JHEP 09 (2017) 102 [arXiv:1703.09727] [INSPIRE].
Y. Kusuki, Large c Virasoro Blocks from Monodromy Method beyond Known Limits, JHEP 08 (2018) 161 [arXiv:1806.04352] [INSPIRE].
Y. Kusuki, New Properties of Large-c Conformal Blocks from Recursion Relation, JHEP 07 (2018) 010 [arXiv:1804.06171] [INSPIRE].
M. Beşken, S. Datta and P. Kraus, Quantum thermalization and Virasoro symmetry, J. Stat. Mech. 2006 (2020) 063104 [arXiv:1907.06661] [INSPIRE].
N. Nemkov, On modular transformations of toric conformal blocks, JHEP 10 (2015) 039 [arXiv:1504.04360] [INSPIRE].
D. Galakhov, A. Mironov and A. Morozov, S-duality as a beta-deformed Fourier transform, JHEP 08 (2012) 067 [arXiv:1205.4998] [INSPIRE].
N. Nemkov, S-duality as Fourier transform for arbitrary 𝜖1, 𝜖2 , J. Phys. A 47 (2014) 105401 [arXiv:1307.0773] [INSPIRE].
F.W.J. Olver et al. eds., Power Series, in NIST Digital Library of Mathematical Functions, section 20.6, (2020) https://dlmf.nist.gov/20.6.
A. Iqbal, C. Kozcaz and C. Vafa, The Refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
M. Billó, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential of \( \mathcal{N} \) = 2★ theories (II): the non-simply laced algebras, JHEP 11 (2015) 026 [arXiv:1507.08027] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, AMS/IP Stud. Adv. Math. 1 (1996) 655 [hep-th/9302103] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 hep-th/9309140] [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].
C. Liu and D.A. Lowe, Notes on Scrambling in Conformal Field Theory, Phys. Rev. D 98 (2018) 126013 [arXiv:1808.09886] [INSPIRE].
H.R. Hampapura, A. Rolph and B. Stoica, Scrambling in Two-Dimensional Conformal Field Theories with Light and Smeared Operators, Phys. Rev. D 99 (2019) 106010 [arXiv:1809.09651] [INSPIRE].
C.-M. Chang, D.M. Ramirez and M. Rangamani, Spinning constraints on chaotic large c CFTs, JHEP 03 (2019) 068 [arXiv:1812.05585] [INSPIRE].
P. Kraus, A. Sivaramakrishnan and R. Snively, Late time Wilson lines, JHEP 04 (2019) 026 [arXiv:1810.01439] [INSPIRE].
Y. Kusuki, Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity, JHEP 01 (2019) 025 [arXiv:1810.01335] [INSPIRE].
N. Wyllard, AN −1 conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
R. Poghossian, Recurrence relations for the \( {\mathcal{W}}_3 \) conformal blocks and \( \mathcal{N} \) = 2 SYM partition functions, JHEP 11 (2017) 053 [Erratum JHEP 01 (2018) 088] [arXiv:1705.00629] [INSPIRE].
M. Beccaria, A. Fachechi and G. Macorini, Virasoro vacuum block at next-to-leading order in the heavy-light limit, JHEP 02 (2016) 072 [arXiv:1511.05452] [INSPIRE].
A. Bombini, S. Giusto and R. Russo, A note on the Virasoro blocks at order 1/c, Eur. Phys. J. C 79 (2019) 3 [arXiv:1807.07886] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, A Quantum Correction To Chaos, JHEP 05 (2016) 070 [arXiv:1601.06164] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2007.10998
Rights and permissions
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.
About this article
Cite this article
Das, D., Datta, S. & Raman, M. Virasoro blocks and quasimodular forms. J. High Energ. Phys. 2020, 10 (2020). https://doi.org/10.1007/JHEP11(2020)010
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2020)010