Crossing Symmetry in Alpha Space

TL;DR

Crossing Symmetry in Alpha Space develops a Sturm–Liouville based alpha-space framework for the 1D conformal bootstrap, recasting crossing as an eigenvalue problem for a crossing kernel built from SL$(2, ext{R})$ Casimir eigenfunctions. The work expresses four-point functions as alpha-space integrals with basis functions Ψ_α(z), relates alpha space to the Jacobi transform, and derives a closed-form crossing kernel in terms of Wilson functions, establishing its role as an intertwiner between s- and t-channel spaces. It also introduces a split kernel to manage analytic obstacles, analyzes asymptotics and OPE convergence, and explores the use of ξ_n bases to generate crossing-symmetric solutions, while outlining connections to representation theory and potential extensions to higher dimensions and more complex CFTs. The results provide analytic tools to constrain CFT data and illuminate the structure of crossing via integral transforms and kernel symmetries, potentially enriching the analytic bootstrap toolkit.

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 alpha. 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.

Figures (3)

• Figure 1: Plot of $z^{-1/2} \, \Psi_\alpha(z)$ for $\alpha = 2i$ (blue) and $\alpha = 20i$ (orange), as well as $k_h(z)$ for $h=3$ (dotted green). Both the oscillatory behaviour of the $\Psi_\alpha(z)$ near $z=0$ and their $O(\sqrt{z})$ growth are clearly visible.
• Figure 2: Choice of contour for a typical CFT correlator in the complex $\alpha$-plane. The blue dots, labeled by $\{1,2,3\}$, correspond to physical poles of $\widehat{f}(\alpha)$, whereas their mirrors (in red, with primed labels) are unphysical. The pole 1 has $\Re(\alpha) < 0$, hence it corresponds to an operator of dimension $h < 1/2$. The contour runs upwards along the imaginary axis, but in this case it must circle 1 in the positive and $1'$ in the negative direction, as indicated.
• Figure 3: Graphical representation of the crossing kernel $K(\alpha,\beta|h_1,h_2,h_3,h_4)$.