Fast conformal bootstrap and constraints on 3d gravity
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Springer
Received: April 28, 2019 Accepted: May 5, 2019 Published: May 16, 2019
Nima Afkhami-Jeddi, Thomas Hartman and Amirhossein Tajdini Department of Physics, Cornell University, Ithaca, New York, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: The crossing equations of a conformal field theory can be systematically truncated to a finite, closed system of polynomial equations. In certain cases, solutions of the truncated equations place strict bounds on the space of all unitary CFTs. We describe the conditions under which this holds, and use the results to develop a fast algorithm for modular bootstrap in 2d CFT. We then apply it to compute spectral gaps to very high precision, find scaling dimensions for over a thousand operators, and extend the numerical bootstrap to the regime of large central charge, relevant to holography. This leads to new bounds on the spectrum of black holes in three-dimensional gravity. We provide numerical evidence that the asymptotic bound on the spectral gap from spinless modular bootstrap, at large central charge c, is ∆1 . c/9.1. Keywords: Conformal Field Theory, AdS-CFT Correspondence, Models of Quantum Gravity ArXiv ePrint: 1903.06272
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)087
JHEP05(2019)087
Fast conformal bootstrap and constraints on 3d gravity
Contents 1
2 Truncating the primal bootstrap 2.1 Setup 2.2 Truncation 2.3 Comments on monotonicity
3 3 4 5
3 Dual bootstrap 3.1 Review 3.2 Functionals parameterized by zeroes 3.3 Duality 3.4 From primal solutions to extremal functionals
5 6 6 8 9
4 Modular bootstrap algorithm 4.1 Setup 4.2 Algorithm
9 9 11
5 Modular bootstrap results 5.1 3d gravity and summary of existing bounds 5.2 Bound as a function of c 5.3 c = 12 and the modular j-function 5.4 Algorithm benchmarks
13 13 14 16 16
6 Discussion
17
A Direct derivation of optimization duality
19
B Generating guesses for Newton’s method
20
C High precision bounds with linear programming
21
1
Introduction
The conformal bootstrap is a rapidly growing array of techniques to solve strongly interacting quantum field theories. It had early success in two spacetime dimensions [1], and more recently has proved successful in higher dimensions [2]. An important impetus has been the development of numerical techniques to analyze the crossing equations, starting with the introduction of the functional bootstrap method in 2008 by Rattazzi, Rychkov,
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JHEP05(2019)087
1 Introduction
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Tonni, and Vichi [3]. This method uses linear (or semidefinite [4]) programming to constrain solutions to the crossing equations, thereby carving out the allowed parameter space of unitary CFTs. It has also been used to solve individual CFTs, including the 3d critical Ising model [5, 6]. The method relies on finding extremal functionals, often numerically, which bound the space of CFTs, and in certain cases encode the spectrum of the target theory [7, 8]. Despite this exciting progre
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