The lightcone bootstrap and the spectrum of the 3d Ising CFT

Abstract

Abstract:We compute numerically the dimensions and OPE coefficients of several operators in the 3d Ising CFT, and then try to reverse-engineer the solution to crossing symmetry analytically. Our key tool is a set of new techniques for computing infinite sums of SL(2, R) conformal blocks. Using these techniques, we solve the lightcone bootstrap to all orders in an asymptotic expansion in large spin, and suggest a strategy for going beyond the large spin limit. We carry out the first steps of this strategy for the 3d Ising CFT, deriving analytic approximations for the dimensions and OPE coefficients of several infinite families of operators in terms of the initial data {∆ σ , ∆ , f σσ , f , c T }. The analytic results agree with numerics to high precision for about 100 low-twist operators (correctly accounting for O(1) mixing effects between large-spin families). Plugging these results back into the crossing equations, we obtain approximate analytic constraints on the initial data.

More on supersymmetric and 2d analogs of the SYK model

“…However, the number of operators and the behavior of their anomalous dimensions is more reminiscent of the behavior of double-twist families that were discussed in [37][38][39]; see also [40] for data in the 3d Ising model.…”

“…In this appendix, we give a direct explanation of this identity, which actually is an example of a general relationship between bound state wavefunctions and scattering data for Schrodinger-like operators. 40 The eigenvalue equation for the wavefunction Ψ h (χ) can be written…”

Long multiplet bootstrap

“…Parity odd blocks, anti-symmetric under this exchange, were considered in [78]. 10 Note that due to the explicit ∆ex factors in the crossing equations, and derivatives of blocks, one must be careful to consistently approximate all terms in the crossing equation to a polynomial of the same degree in ∆ex.…”

“…The bootstrap approach, by relying only on symmetries, combined with a few spectral assumptions, allows one to obtain complete non-perturbative answers, without reference to any type of perturbative description. The most striking example of this is the three-dimensional Ising model, where the best determination of the critical exponents comes from the bootstrap [5][6][7][8][9][10]. A tremendous amount of work has been done studying superconformal field theories (SCFTs) in various dimensions and with various amounts of supersymmetry .…”

AdS4/CFT3 for unprotected operators

“…Secondly, unlike in even dimensions, there is no closed form for the 3d conformal blocks, which makes it hard to extract CFT data from Mellin amplitudes for arbitrary twist and spin, as was done in even dimension [25,38,39]. In this work, we develop an efficient algorithm for extracting CFT data order by order in the twist, based on the expansion of 3d conformal blocks into lightcone blocks initiated in [40,41]. Lastly, we expect there to be n + 1 unprotected operators with the nth lowest twist, so for all but the lowest twist there will be mixing that cannot be resolved from studying the stress tensor four-point function alone.…”