Conformal Partial Waves: Further Mathematical Results

By F. Dolan, H. Osborn. 2011

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Further results for conformal partial waves for four point functions for conformal primary scalar fields in conformally invariant theories are obtained. They are de- fined as eigenfunctions of the differential Casimir operators for the conformal group acting on two variable functions subject to appropriate boundary conditions. As well as the scale dimensionand spin l the conformal partial waves depend on two parameters a,b related to the dimensions of the operators in the four point function. Expressions for the Mellin transform of conformal partial waves are ob- tained in terms of polynomials of the Mellin transform variables given in terms of finite sums. Differential operators which change a,b by ±1, shift the dimension d by ±2 and also change �,l are found. Previous results for d = 2,4,6 are recovered. The trivial case of d = 1 and also d = 3 are also discussed. For d = 3 formulae for the conformal partial waves in some restricted cases as a single variable integral representation based on the Bateman transform are found.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f73fb1c5e3ce488f58d2e29419b49298c57339a1

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Citations in the corpus

Bootstrapping mixed correlators in the 3D Ising model (2014, background): “In the notation of [49], our blocks have a normalization coefficient of cl ≡ (−1) l 4 (ν)l (2ν)l .”
The analytic bootstrap and AdS superhorizon locality (2012, methodology): “On the right-hand side the blocks g(12,12) τ,` (v, u) again begin as v τ 2 at small v [34].”
Spinning conformal blocks (2011, background): “On the other hand, in odd dimensions, less explicit but perhaps still useful expressions exist, either in terms of simple integrals, or in terms of a double power series expansion in u and 1 − v for l = 0, combined with recursion relations for higher spins [14, 16].”
The lightcone bootstrap and the spectrum of the 3d Ising CFT (2016, background): “21 Using identities from [81], one can show that our G h,h (z, z) is symmetric under r ↔ s.”
Recursion relations for conformal blocks (2015, background): “In other cases one has to resort to more indirect methods like, integral representations [29–35], recursion relations that increase the spin of the exchanged operator [27, 36] or efficient series expansions [37, 38].”
Convexity and liberation at large spin (2012, background): “We find the following leading asymptotic form [42]”
The conformal bootstrap: Theory, numerical techniques, and applications (2018, background): “50 The quartic Casimir operator C4 = 12JABJBCJCDJDA has also proved useful in some conformal block studies (Dolan and Osborn, 2011; Hogervorst et al., 2013) .”
Asymptotic symmetries and celestial CFT (2020, methodology): “Here we used that the positive helicity Goldstone mode h̃zz;ww is related to the negative helicity Goldstone mode hzz;w̄w̄ by a shadow transform on the celestial sphere where orders of r are not mixed, and made use of the result of [58] that ∫ dz 2 ∏”
Universality of long-distance AdS physics from the CFT bootstrap (2014, background): “In the limit u 1, the minimal twist conformal blocks can be written as [59]”
Bootstrapping the 3d Ising twist defect (2013, background): “ = X O c 12Oc 34OG O (z); (4.3) where the sum runs over defect primaries, and G (z) is the 1d conformal block for equal external dimensions and internal dimension . The conformal blocks are given by [30] G (z) = z 2F 1( ;;2; z): (4.4) Colliding instead x 2 and x 3 and equating the two dierent repres”
A Mellin space approach to the conformal bootstrap (2016, methodology): “The conformal partial waves F (s) ν,` (u, v) are closely related [42, 30] to the conformal blocks being just linear combinations of a block of fictitious dimension ∆ = h+ν and its shadow with dimension d−∆ = h− ν.”
Quartic AdS interactions in higher-spin gravity from Conformal Field Theory (2015, background): “7) can be expressed in terms of conformal blocks corresponding to a given representation of the conformal group and its shadow [72,91].”