Abstract
We initiate the analytical functional bootstrap study of conformal field theories with large N limits. In this first paper we particularly focus on the 1D O(N) vector bootstrap. We obtain a remarkably simple bootstrap equation from the O(N) vector crossing equations in the large N limit. The numerical conformal bootstrap bound is saturated by the generalized free field theories, while its extremal functional actions do not converge to any non-vanishing limit. We study the analytical extremal functionals of this crossing equation, for which the total positivity of the SL(2, ℝ) conformal block plays a critical role. We prove the SL(2, ℝ) conformal block is totally positive in the limits with large ∆ or small 1 − z and show that the total positivity is violated below a critical value \( {\Delta }_{\textrm{TP}}^{\ast } \) ≈ 0.32315626. The SL(2, ℝ) conformal block forms a surprisingly sophisticated mathematical structure, which for instance can violate total positivity at the order 10−5654 for a normal value ∆ = 0.1627! We construct a series of analytical functionals {αM} which satisfy the bootstrap positive conditions up to a range ∆ ⩽ ΛM. The functionals {αM} have a trivial large M limit. However, due to total positivity, they can approach the large M limit in a way consistent with the bootstrap positive conditions for arbitrarily high ΛM. Moreover, in the region ∆ ⩽ ΛM, the analytical functional actions are consistent with the numerical bootstrap results, therefore it clarifies the positive structure in the crossing equation analytically. Our result provides a concrete example to illustrate how the analytical properties of the conformal block lead to nontrivial bootstrap bounds. We expect this work paves the way for large N analytical functional bootstrap in higher dimensions.
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Acknowledgments
The author would like to thank Nima Arkani-Hamed, Greg Blekherman, Miguel Paulos and David Poland for discussions. The author is grateful to David Poland for the valuable support. The author thanks the organizers of the conferences “Bootstrapping Nature: Nonperturbative Approaches to Critical Phenomena” at Galileo Galilei Institute, “Positivity” at Princeton Center for Theoretical Science and Simons Collaboration on the Nonpertur- bative Bootstrap Annual Meeting for creating stimulating environments. This research was supported by Shing-Tung Yau Center and Physics Department at Southeast University, the Startup Funding 4007022314 of the Southeast University, the Simons Foundation grant 488651 (Simons Collaboration on the Nonperturbative Bootstrap) and DOE grant DE-SC0017660. The bootstrap computations were carried out on the Yale Grace computing cluster, supported by the facilities and staff of the Yale University Faculty of Sciences High Performance Computing Center.
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Li, Z. Large N analytical functional bootstrap. Part I. 1D CFTs and total positivity. J. High Energ. Phys. 2023, 167 (2023). https://doi.org/10.1007/JHEP07(2023)167
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DOI: https://doi.org/10.1007/JHEP07(2023)167