Abstract
We propose methods that efficiently impose integrality — i.e., the condition that the coefficients of characters in the partition function must be integers — into numerical modular bootstrap. We demonstrate the method with a number of examples where it can be used to strengthen modular bootstrap results. First, we show that, with a mild extra assumption, imposing integrality improves the bound on the maximal allowed gap in dimensions of operators in theories with a U(1)c symmetry at c = 3, and reduces it to the value saturated by the SU(4)1 WZW model point of c = 3 Narain lattices moduli space. Second, we show that our method can be used to eliminate all but a discrete set of points saturating the bound from previous Virasoro modular bootstrap results. Finally, when central charge is close to 1, we can slightly improve the upper bound on the scaling dimension gap.
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References
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [ar**v:0807.0004] [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [ar**v:0902.2790] [INSPIRE].
N. Benjamin and Y.-H. Lin, Lessons from the Ramond sector, SciPost Phys. 9 (2020) 065 [ar**v:2005.02394] [INSPIRE].
J. Kaidi and E. Perlmutter, Discreteness and integrality in Conformal Field Theory, JHEP 02 (2021) 064 [ar**v:2008.02190] [INSPIRE].
S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [ar**v:1608.06241] [INSPIRE].
T. Hartman, D. Mazáč and L. Rastelli, Sphere Packing and Quantum Gravity, JHEP 12 (2019) 048 [ar**v:1905.01319] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [ar**v:2006.04839] [INSPIRE].
S. El-Showk and M.F. Paulos, Bootstrap** Conformal Field Theories with the Extremal Functional Method, Phys. Rev. Lett. 111 (2013) 241601 [ar**v:1605.08087] [INSPIRE].
D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [ar**v:1307.6562] [INSPIRE].
D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [ar**v:1502.02033] [INSPIRE].
W. Landry and D. Simmons-Duffin, Scaling the semidefinite program solver SDPB, ar**v:1909.09745 [INSPIRE].
N. Afkhami-Jeddi, Conformal bootstrap deformations, JHEP 09 (2022) 225 [ar**v:2111.01799] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast Conformal Bootstrap and Constraints on 3d Gravity, JHEP 05 (2019) 087 [ar**v:1903.06272] [INSPIRE].
F. Caracciolo and V.S. Rychkov, Rigorous Limits on the Interaction Strength in Quantum Field Theory, Phys. Rev. D 81 (2010) 085037 [ar**v:1702.00423] [INSPIRE].
B.C. Rayhaun, Bosonic rational conformal field theories in small genera, chiral fermionization, and symmetry/subalgebra duality, J. Math. Phys. 65 (2024) 052301 [ar**v:2303.16921] [INSPIRE].
Acknowledgments
We thank David Simmons-Duffin and Yuan **n for helpful conversations, and Yuan **n for comments on the earlier draft. ALF and WL are supported by the US Department of Energy Office of Science under Award Number DE-SC0015845, and the Simons Collaboration on the Non-Perturbative Bootstrap. ALF thanks the Aspen Center of Physics for hospitality as this work was completed.
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ArXiv ePrint: 2308.08725
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Fitzpatrick, A.L., Li, W. Improving modular bootstrap bounds with integrality. J. High Energ. Phys. 2024, 58 (2024). https://doi.org/10.1007/JHEP07(2024)058
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DOI: https://doi.org/10.1007/JHEP07(2024)058