Lightcone expansions of conformal blocks in closed form
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Springer
Received: February 27, 2020 Accepted: May 29, 2020 Published: June 16, 2020
Wenliang Li Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan
E-mail: [email protected] Abstract: We present new closed-form expressions for 4-point scalar conformal blocks in the s- and t-channel lightcone expansions. Our formulae apply to intermediate operators of arbitrary spin in general dimensions. For physical spin `, they are composed of at most (` + 1) Gaussian hypergeometric functions at each order. Keywords: Conformal Field Theory, Conformal and W Symmetry, Nonperturbative Effects ArXiv ePrint: 1912.01168
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)105
JHEP06(2020)105
Lightcone expansions of conformal blocks in closed form
Contents 1 Introduction
1
2 Lightcone limits of conformal blocks
5 7 7 10
4 Conclusion
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A Alternative s-channel lightcone expansion
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B Sums of t-channel conformal blocks
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1
Introduction
Conformal blocks are fundamental to the conformal bootstrap program [1, 2], which aims to classify and solve conformal field theories (CFTs) by general consistency requirements, such as associativity of operator product expansion (OPE). In two dimensions, the conformal symmetry is infinite-dimensional, and the conformal bootstrap program can be carried out rather successfully [3]. The well-known examples include the minimal models, which describe the critical phenomena of 2d statistical models, such as the Lee-Yang, Ising, and Potts models [4]. On the other hand, the d > 2 conformal bootstrap is considerably more challenging as the conformal symmetry is usually finite-dimensional. When performing operator product expansions in conformal field theories, the contributions of a primary and its descendants are related by conformal symmetry. A conformal block is defined by the contributions of a full conformal multiplet, which includes a primary and infinitely many descendants. The study of conformal blocks has a long history [5–9], which dates back to the 1970’s when the conformal bootstrap proposal was just formulated. The understanding of conformal blocks was significantly advanced by the works of Dolan and Osborn [10–12], in which they found explicit analytic expressions in d = 2, 4, 6 dimensions, recursion relations and Casimir differential equations in general dimensions. These results had paved the road for the revival of the d > 2 conformal bootstrap [13], where a new numerical conformal bootstrap method was proposed. Let us mention here the precise determinations of 3d Ising critical exponents [14–17], but refer to [18] for a comprehensive review on many other impressive results.1 1
See also [19–22] for useful lecture notes.
–1–
JHEP06(2020)105
3 Lightcone expansions of conformal blocks 3.1 The t-channel lightcone expansion 3.2 The s-channel lightcone expansion
2
Other active analytic approaches include the Polyakov-Mellin bootstrap [52–56], analytic functionals [57–61]
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