Abstract
We initiate the study of the conformal bootstrap using Sturm-Liouville theory, specializing to four-point functions in one-dimensional CFTs. We do so by decomposing conformal correlators using a basis of eigenfunctions of the Casimir which are labeled by a complex number α. This leads to a systematic method for computing conformal block decompositions. Analyzing bootstrap equations in alpha space turns crossing symmetry into an eigenvalue problem for an integral operator K. The operator K is closely related to the Wilson transform, and some of its eigenfunctions can be found in closed form.
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Hogervorst, M., van Rees, B.C. Crossing symmetry in alpha space. J. High Energ. Phys. 2017, 193 (2017). https://doi.org/10.1007/JHEP11(2017)193
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DOI: https://doi.org/10.1007/JHEP11(2017)193