Distributions in CFT. Part I. Cross-ratio space
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Springer
Received: March 2, 2020 Accepted: May 8, 2020 Published: May 27, 2020
Distributions in CFT. Part I. Cross-ratio space
a
Institute for Advanced Study, Princeton, New Jersey 08540, U.S.A. b Laboratoire de Physique de l’Ecole normale sup´erieure, ENS, Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France c ´ Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France
E-mail: [email protected], [email protected] Abstract: We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations involving the boundary of the region of convergence, e.g. for deriving the dispersion relations. We establish our results by elementary methods, relying only on crossing symmetry and the standard convergence properties of the conformal block expansion. This is the first in a series of papers on distributional properties of correlation functions in conformal field theory. Keywords: Conformal and W Symmetry, Conformal Field Theory, Field Theories in Higher Dimensions, Field Theories in Lower Dimensions ArXiv ePrint: 2001.08778
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)137
JHEP05(2020)137
Petr Kravchuk,a Jiaxin Qiaob,c and Slava Rychkovb,c
Contents 1 Introduction
1
2 Conformal block expansion
3 6 7 10 11 13 14 16 19
4 Scalar four-point functions in higher dimensions 4.1 Conformal block expansion 4.2 Bounds on g(ρ, ρ) and partial sums of the conformal block expansion 4.3 Vladimirov’s theorem 4.4 Analytic functionals 4.5 Spinning operators 4.6 Single-variable dispersion relation for the four-point function in d > 2
22 23 24 25 27 28 28
5 Conclusions
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A Lorentzian 4pt correlator with no convergent OPE channel
32
B Proof of lemma 3.4
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C Comments on the proof of theorem 4.1
35
1
Introduction
Historically, distributions played a big role in axiomatic approaches to quantum field theory (QFT), via Wightman axioms [1] or Osterwalder-Schrader axioms [2, 3]. In particular, the language of tempered distributions allows clean treatment of correlation functions singularities at x2 = 0 in a UV-complete QFT, where x2 may be Euclidean or Lorentzian distance. In recent years, a new axiomatic approach — the conformal bootstrap — has emerged in the study of conformal field theories (CFTs) in dimension d > 2, i.e. quantum field theories invariant under the action of conformal group (see review [4]). This approach is both rigorous and calculable. On the numerical side, it has allowed precise determinatio
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