Radial Coordinates for Conformal Blocks

By Matthijs Hogervorst, S. Rychkov. 2013

271 citations recorded by the source corpus.

We develop the theory of conformal blocks in CFT_d expressing them as power series with Gegenbauer polynomial coefficients. Such series have a clear physical meaning when the conformal block is analyzed in radial quantization: individual terms describe contributions of descendants of a given spin. Convergence of these series can be optimized by a judicious choice of the radial quantization origin. We argue that the best choice is to insert the operators symmetrically. We analyze in detail the resulting "rho-series" and show that it converges much more rapidly than for the commonly used variable z. We discuss how these conformal block representations can be used in the conformal bootstrap. In particular, we use them to derive analytically some bootstrap bounds whose existence was previously found numerically.

Subfields: Applied Mathematics, Nuclear and High Energy Physics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/3eda60c489a7337eb40ce53b4c83997699a9de36

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In Nuclear and High Energy Physics

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Citations in the corpus

Projectors, shadows, and conformal blocks (2012, methodology): “to higher spin conformal blocks, perhaps along the lines of the methods recently developed in [25–27].”
Exact correlators of BPS Operators from the 3d superconformal bootstrap (2014, unknown): “(See, however, [3, 44–48].”
Looking for a bulk point (2015, methodology): “In radial quantization, this corresponds to the configuration shown in figure 6, where ρ = eτE+iφ is the radial coordinate introduced in [34, 39].”
Analyticity in spin in conformal theories (2017, methodology): “Subleading terms in zz̄ can be worked out systematically using the methods of [23]: one postulates an expansion in terms of powers times Gegenbauer polynomials,”
EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions (2016, background): “See our work [40] for an introduction to the issues arising for the external fields with spin, and [41] for the expansion in Gegenbauer polynomials.”
Weight shifting operators and conformal blocks (2017, unknown): “where r is the radial coordinate of [141, 142] and h∞,ρ is some relatively easily computable function, then we can write the conformal block as a sum over residues [14, 105].”
Bootstrapping 3D fermions (2015, methodology): “Computing derivatives of scalar blocks is straightforward, for example we use the pole expansion derived in [7] to compute the expansion in radial coordinates [47] to order ρ60.”
The ϵ ?> -expansion from conformal field theory (2015, background): “There is also a growing literature on the analytic expressions for the conformal correlation functions and conformal blocks, which is indispensable for the practical conformal bootstrap computations [54–73].”
A scaling theory for the long-range to short-range crossover and an infrared duality (2017, methodology): “Instead of z, it will be convenient to work with the radial coordinate ρ [23]”
All tree-level correlators in AdS5×S5 supergravity: hidden ten-dimensional conformal symmetry (2018, background): “J H E P 0 1 ( 2 0 1 9 ) 1 9 6 Comparing with the series expansion in powers of z, z̄ (see [63]) we find that the block indeed takes this form, with the function specifically chosen as”
Bootstrapping the O(N ) vector models (2013, methodology): “This can be done using either the Gegenbauer expansion and Casimir equation described in [41], or more efficiently using our recursion relation (3.”
A spacetime derivation of the Lorentzian OPE inversion formula (2017, methodology): “gion one can show that is large, of order 1=. To summarize, we have ˆ; 1 ˆ ; ;r2 ˘ 1 ; u;v˘1: (3.31) 9Note that this corresponds to starting with the standard Euclidean conguration described in [13,14] with ˆ= ˆ, and then applying a large Lorentzian relative boost between the points 1;2 and 3;4. Th”
Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$c-Minimization and Precise Critical Exponents (2012, methodology): “26 For the conformal blocks corresponding to identical external scalars used here, the expansion is actually in powers of r2 [46].”
Recursion relations for conformal blocks (2015, background): “g [37] ∆,l (u, v) = 4 g [1, 18] ∆,l (u, v) = (−2) ( d−2 2 ) l (d− 2)l g ∆,l (u, v) .”
TASI Lectures on the Conformal Bootstrap (2016, methodology): “Using conformal transformations, we can place four points on a plane in the configuration shown above (figure from [26]).”
Modular bootstrap revisited (2016, result): “ar functional. Presumably, the latter is more appropriately expressed as an integral transform, rather than derivatives taken at ˝= ˝ = i. Our preliminary attempt at a multi-point bootstrap approach [27,28] has yielded results consistent with mod, but it did not improve the numerical eciency due to ”
Bootstrapping mixed correlators in the 3D Ising model (2014, methodology): “where g̃1234 l (∆, r, η) is an entire function of ∆ and we describe the conformal cross ratios using radial coordinates [58].”
Defects in conformal field theory (2016, background): “We leave a thorough analysis for a future work,11 in which the choice of the most appropriate cross-ratio will be further discussed, along the lines of [49].”
Causality constraints in conformal field theory (2015, background): “To achieve this explicitly, following [50, 51], insert the operators as”
A proof of the conformal collider bounds (2016, methodology): “In fact, using radial coordinates [41], the region of convergence for the s-channel expansion can be expanded to the whole complex plane excluding [1,+∞).”