Bound on asymptotics of magnitude of three point coefficients in 2D CFT
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Springer
Received: November 1, 2019 Accepted: December 18, 2019 Published: January 7, 2020
Sridip Pal Department of Physics, University of California, San Diego La Jolla, CA 92093, U.S.A. School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, U.S.A.
E-mail: [email protected] Abstract: We use methods inspired from complex Tauberian theorems to make progress in understanding the asymptotic behavior of the magnitude of heavy-light-heavy three point coefficients rigorously. The conditions and the precise sense of averaging, which can lead to exponential suppression of such coefficients are investigated. We derive various bounds for the typical average value of the magnitude of heavy-light-heavy three point coefficients and verify them numerically. Keywords: Conformal and W Symmetry, Conformal Field Theory ArXiv ePrint: 1906.11223
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)023
JHEP01(2020)023
Bound on asymptotics of magnitude of three point coefficients in 2D CFT
Contents 1 The premise and the result
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3 Derivation of the result: warm up 3.1 A lemma 3.2 Main proof
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4 Allowing for power law growth 4.1 Modified HKS bounds 4.2 Estimation of three point coefficient
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5 Large central charge
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6 Extension and verification 6.1 Verification I: identity Module 6.2 Verification II: non identity module
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7 The scenario when ∆χ >
c 12
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8 Discussion and outlook
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A More on Tauberian theorems & upper bound
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B One more example!
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1
The premise and the result
The modular invariance plays a pivotal role in constraining the data of two dimensional conformal field theory. In two dimensions, a conformal field theory can be consistently defined on any Riemann surface, in particular, on a torus. The shape of the torus is determined by modular parameter τ ∈ H, where H is the upper half plane. The modular transformation acts on τ and maps it to another point in H, nonetheless the partition function of the conformal field theory defined on the torus remains invariant under such transformation. Physically, one cycle of the torus is interpreted as the spatial cycle while the other one is the thermal cycle. Modular transformation, for example, exchanges these cycles and thus can relate the low temperature behavior of a CFT with its high temperature behavior. This is how the universality in the low temperature behavior translates into a universal high temperature behavior, which is controlled by the asymptotic data of the
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2 Scheme of the proof
The aim of this paper is to make progress in understanding the asymptotic behavior of the magnitude of three point coefficients and investigating under which conditions one
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CFT. Hence, one can get insight to the asymptotic behavior of the CFT data utilizing the modular invariance [1–16]. Using AdS-CFT correspondence, the asymptotic data translates to a statement about gravity in AdS3 , in particular black holes [3–5,
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