Conformal partial waves and the operator product expansion

By F. Dolan, H. Osborn. 2003

603 citations recorded by the source corpus.

Abstract By solving the two variable differential equations which arise from finding the eigenfunctions for the Casimir operator for O ( d ,2) succinct expressions are found for the functions, conformal partial waves, representing the contribution of an operator of arbitrary scale dimension Δ and spin l together with its descendants to conformal four point functions for d =4, recovering old results, and also for d =6. The results are expressed in terms of ordinary hypergeometric functions of variables x , z which are simply related to the usual conformal invariants. An expression for the conformal partial wave amplitude valid for any dimension is also found in terms of a sum over two variable symmetric Jack polynomials which is used to derive relations for the conformal partial waves.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8d25b10849793de448474f063b49d29757cbaa38

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Citations in the corpus

Projectors, shadows, and conformal blocks (2012, methodology): “For example, Dolan and Osborn derived scalar blocks by solving an eigenvalue equation for the quadratic-Casimir of the conformal group, which takes the form of a single second-order PDE [22].”
Recursion relations for conformal blocks (2015, methodology): “Equation (40) is a second order differential equation (since we are using the quadratic Casimir) that can be used to compute the full conformal block G∆l [28].”
All tree-level correlators in AdS5×S5 supergravity: hidden ten-dimensional conformal symmetry (2018, neutral): “up to an overall prefactor conventionally written as follows [43]: 〈 O1(x1, y1) · · · O4(x4, y4) 〉 ≡ ( y2 12 x(2)12 ) p1+p2 2 ( y2 34 x(2)34 ) p3+p4 2 ( x(2)14y 2 24 x(2)24y 2 14 ) p2−p1 2 ( x(2)14y 2 13 x(2)13y 2 14 ) p3−p4 2”
Central charge bounds in 4D conformal field theory (2010, background): “Now, it has been shown by Dolan and Osborn [14] that every primary spin l, dimension ∆ operator O∆,l appearing in the φ×φ OPE with a coefficient c∆,l gives a contribution to the 4-point 1If φ is not Hermitean, we can consider its real and imaginary parts. function of φ of the following form: 〈φ(x1)φ”
The lightcone bootstrap and the spectrum of the 3d Ising CFT (2016, methodology): “However, because we also have small z, we can replace each conformal block by its expansion in small z [73, 74], g∆,`(z, 1− z) = zhk2h(1− z) +O(z), (3.”
Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions (2006, methodology): “1 The d = 2 Case Explicit expressions for the partial waves Th,h̄ and Sh,h̄ exist for d even [9, 10], and are particularly simple in d = 2 where the problem factorizes in left/right equations for z and z̄.”
Eikonal approximation in AdS/CFT: resumming the gravitational loop expansion (2007, background): “The four-point amplitude A is just a function of two cross-ratios z, z̄ which we define, following [13, 14], in terms of the kinematical invariants as3”
Spinning conformal blocks (2011, background): “(2.4) satisﬁes the Casimir diﬀerential equation , h i This diﬀerential equation can then be rewritten 4 as an equation for G O ( u, v ), and solved in closed form when d is an even integer [15].”
Holography from Conformal Field Theory (2009, background): “(5.7) The conformal partial wave expansion of this correlator can be found using a series representation of the D-function [24, 32].”
Witten diagrams revisited: the AdS geometry of conformal blocks (2015, methodology): “It is not obvious by inspection that these results agree with those in [5], but we will verify this in various cases to assuage any doubts that our general arguments are valid.”
Virasoro conformal blocks and thermality from classical background fields (2015, neutral): “and G is the global conformal block [42–44]”
Bootstrapping 3D fermions (2015, methodology): “There are many approaches to computing conformal blocks in D > 2, including direct summation [2, 31–35], solving the Casimir differential equation [5, 36–39], pole expansions [7, 9], and evaluating monodromyprojected conformal integrals [27, 40, 41].”
The conformal bootstrap: Theory, numerical techniques, and applications (2018, methodology): “We conclude that the CPW, and hence the conformal block, satisfies a second-order partial differential equation.51 The actual form of this “Casimir equation” is most conveniently found using the embedding formalism (Dolan and Osborn, 2004).”
Looking for a bulk point (2015, background): “Conformal blocks in d-dimensions are eigenfunctions of the Casimir operator for the conformal group SO(d, 2) [35].”
Eikonal approximation in AdS/CFT: from shock waves to four-point functions (2006, neutral): “The four–point amplitude A is just a function of two cross–ratios z, z̄ which we define, following [25, 26], in terms of the kinematical invariants pij as 3”
EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions (2016, background): “For the conformal blocks in D > 3, start with the very influential papers of Dolan and Osborn [38, 39].”
Rigorous limits on the interaction strength in quantum field theory (2009, background): “It would be also interesting to derive analogous bounds in two spacetime dimensions, where explicit expressions for conformal blocks are also known [13].”
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].”
Bounding the space of holographic CFTs with chaos (2016, positive): “Indeed, the four-point function of J0 in the d-dimensional free O(N) model [103], 〈J0(x1)J0(x2)J0(x3)J0(x4)〉〈J0(x1)J0(x2)〉〈J0(x3)J0(x4)〉 = 1 + ud−2 + (u v )d−2”