A universal inequality for CFT and quantum gravity

By S. Hellerman. 2009

232 citations recorded by the source corpus.

We prove that every unitary two-dimensional conformal field theory (with no extended chiral algebra, and with c, $ \tilde{c} > 1 $) contains a primary operator with dimension ∆1 that satisfies $ 0 < {\Delta_1} < \frac{{c + \tilde{c}}}{{12}} + 0.473695 $. Translated into gravitational language using the AdS3/CFT2 dictionary, our result proves rigorously that the lightest massive excitation in any theory of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than $ {{1} \left/ {{\left( {4{G_N}} \right)}} \right.} + o\left( {\sqrt {{ - \Lambda }} } \right) $. In the flat-space approximation, this limiting mass is twice that of the lightest BTZ black hole. The derivation applies at finite central charge for the boundary CFT, and does not rely on an asymptotic expansion at large central charge. Neither does our proof rely on any special property of the CFT such as supersymmetry or holomorphic factorization, nor on any bulk interpretation in terms of string theory or semiclassical gravity. Our only assumptions are unitarity and modular invariance of the dual CFT. Our proof demonstrates for the first time that there exists a universal center-of-mass energy beyond which a theory of ”pure” quantum gravity can never consistently be extended.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/e377f83493eb0a19b226e52a60e1ae12e4da0bdd

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Citations in the corpus

Spinning conformal blocks (2011, background): “It is plausible that studying crossing symmetry of stress tensor four point functions could lead to interesting constraints on central charges in CFTs, perhaps not unrelated to the bounds of [34] or [35].”
Universal constraints on conformal operator dimensions (2009, result): “A similar in spirit general prediction of string gravity, although in a different context and by using different methods, was obtained recently in [14].”
The Swampland: Introduction and Review (2019, unknown): “For early work on the Weak Gravity Conjecture and AdS/CFT see [65,66].”
Unitarity and the holographic S-Matrix (2011, methodology): “This means that one can obtain robust results about quantum gravity by modeling the correlators of low-dimension operators using bulk effective field theory, and then using the bootstrap approach [17, 19, 35, 36, 25, 20, 37, 38, 39, 26] to constrain correlators that involve operators of larger dimen”
Virasoro conformal blocks and thermality from classical background fields (2015, unknown): “There has also been signi(cid:12)cant recent progress [38{40] in using modular invariance to understand the spectrum of CFT 2 at large central charge.”
Comments on holographic entanglement entropy and RG flows (2012, background): “12) must be constrained in some way, along the lines of various bounds found in [56–64] or [108–112].”
Universal spectrum of 2d conformal field theory in the large c limit (2014, methodology): “In [11], this technique was used to lowest order to prove that every CFT has a state with scaling dimension ∆1 = EL + ER + c 12 ≤ c 6 + 0.”
The bootstrap program for boundary CFTd (2012, background): “6 One could in fact also go ahead (5)Analogous “sum rule” techniques can also be used to obtain non-trivial bounds from modular invariant partition functions, see [14, 15, 16, 17, 18].”
Looking for a bulk point (2015, background): “Formulated in terms of g(q, q̄), the four-point function bootstrap [55–59] is almost identical to the modular bootstrap [60, 61].”
Holographic Hydrodynamics with a Chemical Potential (2009, background): “Recently, there has been some discussion of this conjecture in the context of the AdS/CFT correspondence and the implications for the spectrum of the CFT [54, 55] — see also footnote 4 in [19].”
The Weak Gravity Conjecture in three dimensions (2016, methodology): “10) is well suited for the modular bootstrap approach pioneered in this context in [73].”
Central charge bounds in 4D conformal field theory (2010, background): “We would like to mention a parallel line of development in 2D CFT, where Hellerman [5] and others [6],[7] have also studied constraints on the gap in the spectrum, in terms of the central charge.”
Emergent spacetime and holographic CFTs (2011, background): “Before we close this section let us mention that the constraints from modular invariance of the boundary CFT, in the context of the AdS3/CFT2 correspondence, have been recently discussed in [76], [77], [78], [79], [80], [81].”
Bounds on 4D conformal and superconformal field theories (2010, background): “Finally, we note that a somewhat similar bound on the central charge was derived in [ 36 ] in the context of 2D conformal field theories.”
A cardy formula for three-point coefficients or how the black hole got its spots (2016, background): “Many of the more precise generalizations of Cardy’s formula, such as [13, 14, 15], rely on the fact that in Cardy’s formula the partition function is a sum of positive definite terms.”
Analyticity and the holographic S-matrix (2011, methodology): “In any case, our work suggests that there is an intimate connection between the Bootstrap Program for large N CFTs [27, 21, 20, 22, 23, 24, 25, 26, 50] and the S-Matrix Program, and that solving general CFTs via the Bootstrap will be at least as difficult as finding consistent S-Matrices using the c”