Abstract
In this work, we initiate a positive semi-definite numerical bootstrap program for multi-point correlators. Considering six-point functions of operators on a line, we reformulate the crossing symmetry equation for a pair of comb-channel expansions as a semi-definite programming problem. We provide two alternative formulations of this problem. At least one of them turns out to be amenable to numerical implementation. Through a combination of analytical and numerical techniques, we obtain rigorous bounds on CFT data in the triple-twist channel for several examples.
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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].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
S. El-Showk et al., Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. El-Showk et al., Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision Islands in the Ising and O(N) Models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
D. Poland, V. Prilepina and P. Tadić, The five-point bootstrap, JHEP 10 (2023) 153 [arXiv:2305.08914] [INSPIRE].
C. Bercini, V. Gonçalves and P. Vieira, Light-Cone Bootstrap of Higher Point Functions and Wilson Loop Duality, Phys. Rev. Lett. 126 (2021) 121603 [arXiv:2008.10407] [INSPIRE].
A. Antunes, M.S. Costa, V. Goncalves and J.V. Boas, Lightcone bootstrap at higher points, JHEP 03 (2022) 139 [arXiv:2111.05453] [INSPIRE].
A. Kaviraj, J.A. Mann, L. Quintavalle and V. Schomerus, Multipoint lightcone bootstrap from differential equations, JHEP 08 (2023) 011 [arXiv:2212.10578] [INSPIRE].
M.F. Paulos et al., The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
M. Hogervorst, H. Osborn and S. Rychkov, Diagonal Limit for Conformal Blocks in d Dimensions, JHEP 08 (2013) 014 [arXiv:1305.1321] [INSPIRE].
V. Rosenhaus, Multipoint Conformal Blocks in the Comb Channel, JHEP 02 (2019) 142 [arXiv:1810.03244] [INSPIRE].
A. Homrich et al., The S-matrix Bootstrap IV: Multiple Amplitudes, JHEP 11 (2019) 076 [arXiv:1905.06905] [INSPIRE].
V. Gonçalves, R. Pereira and X. Zhou, 20′ Five-Point Function from AdS5 × S5 Supergravity, JHEP 10 (2019) 247 [arXiv:1906.05305] [INSPIRE].
S. Parikh, A multipoint conformal block chain in d dimensions, JHEP 05 (2020) 120 [arXiv:1911.09190] [INSPIRE].
J.-F. Fortin, W. Ma and W. Skiba, Higher-Point Conformal Blocks in the Comb Channel, JHEP 07 (2020) 213 [arXiv:1911.11046] [INSPIRE].
D. Poland and V. Prilepina, Recursion relations for 5-point conformal blocks, JHEP 10 (2021) 160 [arXiv:2103.12092] [INSPIRE].
I. Buric et al., From Gaudin Integrable Models to d-dimensional Multipoint Conformal Blocks, Phys. Rev. Lett. 126 (2021) 021602 [arXiv:2009.11882] [INSPIRE].
I. Buric et al., Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D, JHEP 11 (2021) 182 [arXiv:2108.00023] [INSPIRE].
I. Buric et al., Gaudin models and multipoint conformal blocks III: comb channel coordinates and OPE factorisation, JHEP 06 (2022) 144 [arXiv:2112.10827] [INSPIRE].
I. Buric et al., Gaudin models and multipoint conformal blocks: general theory, JHEP 10 (2021) 139 [arXiv:2105.00021] [INSPIRE].
J.-F. Fortin et al., One- and two-dimensional higher-point conformal blocks as free-particle wavefunctions in \( {AdS}_3^{\otimes m} \), JHEP 01 (2024) 031 [arXiv:2310.08632] [INSPIRE].
M.F. Paulos and B. Zan, A functional approach to the numerical conformal bootstrap, JHEP 09 (2020) 006 [arXiv:1904.03193] [INSPIRE].
K. Ghosh and Z. Zheng, Numerical Conformal bootstrap with Analytic Functionals and Outer Approximation, arXiv:2307.11144 [INSPIRE].
A. Böttcher and S.M. Grudsky, Spectral Properties of Banded Toeplitz Matrices, Society for Industrial and Applied Mathematics (2005) [https://doi.org/10.1137/1.9780898717853].
I. Esterlis, A.L. Fitzpatrick and D. Ramirez, Closure of the Operator Product Expansion in the Non-Unitary Bootstrap, JHEP 11 (2016) 030 [arXiv:1606.07458] [INSPIRE].
A. Antunes et al., Towards bootstrapping RG flows: sine-Gordon in AdS, JHEP 12 (2021) 094 [arXiv:2109.13261] [INSPIRE].
L. Córdova, Y. He and M.F. Paulos, From conformal correlators to analytic S-matrices: CFT1/QFT2, JHEP 08 (2022) 186 [arXiv:2203.10840] [INSPIRE].
M. Yamashita, K. Fujisawa, M. Fukuda, K. Nakata and M. Nakata, A high-performance software package for semidefinite programs: Sdpa 7, Research Report B-463, Dept. of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, Japan (2010).
M. Yamashita, K. Fujisawa and M. Kojima, Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0), Optim. Meth. Software 18 (2003) 491.
M. Nakata, A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD., in the proceedings of the 2010 IEEE International Symposium on Computer-Aided Control System Design, Yokohama, Japan, September 08–10 (2010) [https://doi.org/10.1109/cacsd.2010.5612693].
D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
K. Fujisawa, M. Kojima, K. Nakata and M. Yamashita, Sdpa (semidefinite programming algorithm) user’s manual — version 6.00, Math. Comp. Sci. Series B: Oper. Res. (2002).
M. Dür, Copositive programming — a survey, in Recent Advances in Optimization and its Applications in Engineering, M. Diehl, F. Glineur, E. Jarlebring and W. Michiels eds., Springer Berlin Heidelberg (2010), p. 3–20 [https://doi.org/10.1007/978-3-642-12598-0_1].
D. Mazac, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP 02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
K. Ghosh, A. Kaviraj and M.F. Paulos, Charging up the functional bootstrap, JHEP 10 (2021) 116 [arXiv:2107.00041] [INSPIRE].
K. Ghosh, A. Kaviraj and M.F. Paulos, Polyakov blocks for the 1D conformal field theory mixed-correlator bootstrap, Phys. Rev. D 109 (2024) L061703 [arXiv:2307.01257] [INSPIRE].
M.F. Paulos, Dispersion relations and exact bounds on CFT correlators, JHEP 08 (2021) 166 [arXiv:2012.10454] [INSPIRE].
P. Ferrero, K. Ghosh, A. Sinha and A. Zahed, Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs, JHEP 07 (2020) 170 [arXiv:1911.12388] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Dispersive CFT Sum Rules, JHEP 05 (2021) 243 [arXiv:2008.04931] [INSPIRE].
A. Adams et al., Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, AdS bulk locality from sharp CFT bounds, JHEP 11 (2021) 164 [arXiv:2106.10274] [INSPIRE].
W. Knop and D. Mazac, Dispersive sum rules in AdS2, JHEP 10 (2022) 038 [arXiv:2203.11170] [INSPIRE].
F. Serra and L.G. Trombetta, Five-Point Superluminality Bounds, arXiv:2312.06759 [INSPIRE].
M.F. Paulos and Z. Zheng, Bounding 3d CFT correlators, JHEP 04 (2022) 102 [arXiv:2107.01215] [INSPIRE].
S. El-Showk and M.F. Paulos, Bootstrapping Conformal Field Theories with the Extremal Functional Method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].
S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, JHEP 03 (2018) 148 [arXiv:1605.08087] [INSPIRE].
N. Drukker, D.J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].
J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].
N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].
P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N} \) = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping the half-BPS line defect, JHEP 10 (2018) 077 [arXiv:1806.01862] [INSPIRE].
A. Cavaglià, N. Gromov, J. Julius and M. Preti, Integrability and conformal bootstrap: One dimensional defect conformal field theory, Phys. Rev. D 105 (2022) L021902 [arXiv:2107.08510] [INSPIRE].
S. Giombi and S. Komatsu, Exact Correlators on the Wilson Loop in \( \mathcal{N} \) = 4 SYM: Localization, Defect CFT, and Integrability, JHEP 05 (2018) 109 [Erratum ibid. 11 (2018) 123] [arXiv:1802.05201] [INSPIRE].
J. Barrat, A. Gimenez-Grau and P. Liendo, Bootstrapping holographic defect correlators in \( \mathcal{N} \) = 4 super Yang-Mills, JHEP 04 (2022) 093 [arXiv:2108.13432] [INSPIRE].
S. Giombi, S. Komatsu, B. Offertaler and J. Shan, Boundary reparametrizations and six-point functions on the AdS2 string, arXiv:2308.10775 [INSPIRE].
A. Cavaglià, N. Gromov, J. Julius and M. Preti, Bootstrability in defect CFT: integrated correlators and sharper bounds, JHEP 05 (2022) 164 [arXiv:2203.09556] [INSPIRE].
P. Ferrero and C. Meneghelli, Bootstrapping the half-BPS line defect CFT in N = 4 supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. D 104 (2021) L081703 [arXiv:2103.10440] [INSPIRE].
J. Barrat, P. Liendo, G. Peveri and J. Plefka, Multipoint correlators on the supersymmetric Wilson line defect CFT, JHEP 08 (2022) 067 [arXiv:2112.10780] [INSPIRE].
J. Barrat, P. Liendo and G. Peveri, Multipoint correlators on the supersymmetric Wilson line defect CFT. Part II. Unprotected operators, JHEP 08 (2023) 198 [arXiv:2210.14916] [INSPIRE].
M.F. Paulos, S. Rychkov, B.C. van Rees and B. Zan, Conformal Invariance in the Long-Range Ising Model, Nucl. Phys. B 902 (2016) 246 [arXiv:1509.00008] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, Long-range critical exponents near the short-range crossover, Phys. Rev. Lett. 118 (2017) 241601 [arXiv:1703.03430] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, A scaling theory for the long-range to short-range crossover and an infrared duality, J. Phys. A 50 (2017) 354002 [arXiv:1703.05325] [INSPIRE].
C. Behan, Bootstrapping the long-range Ising model in three dimensions, J. Phys. A 52 (2019) 075401 [arXiv:1810.07199] [INSPIRE].
C. Behan, E. Lauria, M. Nocchi and P. van Vliet, Analytic and numerical bootstrap for the long-range Ising model, JHEP 03 (2024) 136 [arXiv:2311.02742] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFTd, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
M. Isachenkov, P. Liendo, Y. Linke and V. Schomerus, Calogero-Sutherland Approach to Defect Blocks, JHEP 10 (2018) 204 [arXiv:1806.09703] [INSPIRE].
M. Lemos, P. Liendo, M. Meineri and S. Sarkar, Universality at large transverse spin in defect CFT, JHEP 09 (2018) 091 [arXiv:1712.08185] [INSPIRE].
P. Liendo, Y. Linke and V. Schomerus, A Lorentzian inversion formula for defect CFT, JHEP 08 (2020) 163 [arXiv:1903.05222] [INSPIRE].
I. Burić and V. Schomerus, Defect Conformal Blocks from Appell Functions, JHEP 05 (2021) 007 [arXiv:2012.12489] [INSPIRE].
E. Lauria, P. Liendo, B.C. Van Rees and X. Zhao, Line and surface defects for the free scalar field, JHEP 01 (2021) 060 [arXiv:2005.02413] [INSPIRE].
I. Buric and V. Schomerus, Universal spinning Casimir equations and their solutions, JHEP 03 (2023) 133 [arXiv:2211.14340] [INSPIRE].
A.A. Ahmadi et al., Improving efficiency and scalability of sum of squares optimization: Recent advances and limitations, in the proceedings of the 2017 IEEE 56th Annual Conference on Decision and Control (CDC), Melbourne, Australia, December 12–15 (2017) [https://doi.org/10.1109/cdc.2017.8263706].
P. Kraus, S. Megas and A. Sivaramakrishnan, Anomalous dimensions from thermal AdS partition functions, JHEP 10 (2020) 149 [arXiv:2004.08635] [INSPIRE].
Acknowledgments
We thank Till Bargheer, Julien Barrat, Carlos Bercini, Ilija Buric, Pedro Liendo, Jeremy Mann, Junchen Rong, Francesco Russo, Slava Rychkov, Alessandro Vichi and Sasha Zhiboedov for useful comments and discussions. This project received funding from the German Research Foundation DFG under Germany’s Excellence Strategy - EXC 2121 Quantum Universe - 390833306. SH is further supported by the Studienstiftung des Deutschen Volkes.
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Antunes, A., Harris, S., Kaviraj, A. et al. Lining up a positive semi-definite six-point bootstrap. J. High Energ. Phys. 2024, 58 (2024). https://doi.org/10.1007/JHEP06(2024)058
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DOI: https://doi.org/10.1007/JHEP06(2024)058