Conformal blocks are the central ingredient of the conformal bootstrap programme. We elaborate on our recent observation that uncovered a relation with wave functions of an integrable Calogero-Sutherland Hamiltonian in order to develop a systematic theory of conformal blocks. Our main goal here is to review central ingredients of the Heckman-Opdam theory for scattering states of Calogero-Sutherland models with special emphasis to the relation with scalar 4-point blocks. We will also discuss a number of direct consequences for conformal blocks, including a new series expansion for blocks of arbitrary complex spin and a complete analysis of their poles and residues. Applications to the Froissart-Gribov formula for conformal field theory, as well as extensions to spinning blocks and defects are briefly discussed before we conclude with an outlook on forthcoming work concerning algebraic consequences of integrability.

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Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory

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Integrability of Conformal Blocks I: Calogero-Sutherland Scattering Theory

Wavelet packets and wavelet frame packets on local fields of positive characteristic

The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24. [2] , [3] ]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8] ], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most ln 2 ln 3 order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1] ] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1. There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511. [9] , [10] , [11] , [12] ] is based on random Cantor set which takes the golden mean dimension 5 - 1 2 as shown by El Naschie.

3-Adic Cantor function on local fields and its p-adic derivative

The concept of derivatives of functions plays a key role in the study of local fields. Such a definition was given by virtue of pseudo-differential operators by Su in 1992 [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China (series A) 1992;35(7A):826–36; Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5–6):805–24]. In this paper, a kind of Weierstrass-like functions in the p -series local fields are found, these Weierstrass-like functions [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: John Wiley & Sons, Inc.; 1990. [1] ] are continuous, and m order differentiable with m

Weierstrass-like functions on local fields and their p-adic derivatives

In this note, we present expressions of characters on a local field Kq, including p-series fields, p-adic number fields and their algebraic extensions. Then we define (p)-type derivative for a function fon Kq and evaluate the pointwise derivative of a character χ of K
 q
 +
 .

A note to the concept of derivatives on local fields

Let f∈Lp(R+, Δ(t)dt), we consider conditions under which the space spanned by generalized translates of f is dense in Lp(R+, Δ(t)dt) in terms of Fourier-Jacobi trans form of f. This generalizes and improves the earlier results of A. Sitaram on semi-simple Lie groups of rank one.

A Wiener-Tauberian theorem for Fourier-Jacobi transform

LetG=H p
 (H
 ) be the (2n+1)-dimensional Heisenberg group over local fieldK. In this paper we prove some theorems about convolution operators onH
 p
 (G) and vector-valued Hardy spaces. As an example, the distribution
 
$$\int {_{k^* } {{\varphi dt} \mathord{\left/ {\vphantom {{\varphi dt} {|t|}}} \right. \kern-\nulldelimiterspace} {|t|}}} $$

 for some φ∈S(G), ξ φ=0 is a ramified 0-type kernel. These results can be applied to characterizeH
 p
 (G) spaces by square functions.

Convolution operators onH p (H k n )

For any a,b∈R let ϕa,b(x)=ax+b(x∈R). Suppose 0<a<1. Let Ca,b be the generalized a-Cantor set with generating iterated function systme {ϕa,0, ϕa,b; ϕa,l}. Then we prove the Hausdorff dimension of$C_{a,c^2 } $ is$\frac{{ln(3 - \sqrt 5 - ln2}}{{lna}}$ when 0<a≤2 cos 80°.

On the hausdorff dimensions of certain attractors

Let G -= H~ be the (2r~ + 1)-dimensional Heisenberg group over local field K. In this paper we prove some theorems about convolution operators on HP(G) and vector-valued Hardy spaces. As an example, the distribution fk* Tdt/(t( for some ~ E S(G), f ~ = 0 is a ramified 0-type kernel. These results can be applied to characterize HP(G) spaces by square functions.

Zheng Weixing

Papers

A note to the concept of derivatives on local fields

A Wiener-Tauberian theorem for Fourier-Jacobi transform

Convolution operators onH p (H k n )

On the hausdorff dimensions of certain attractors

Convolution Operators on HP(H )