Abstract
Every conformal field theory (CFT) above two dimensions contains an infinite set of Regge trajectories of local operators which, at large spin, asymptote to “double-twist” composites with vanishing anomalous dimension. In two dimensions, due to the existence of local conformal symmetry, this and other central results of the conformal bootstrap do not apply. We incorporate exact stress tensor dynamics into the CFT2 analytic bootstrap, and extract several implications for AdS3 quantum gravity. Our main tool is the Virasoro fusion kernel, which we newly analyze and interpret in the bootstrap context. The contribution to double-twist data from the Virasoro vacuum module defines a “Virasoro Mean Field Theory” (VMFT); its spectrum includes a finite number of discrete Regge trajectories, whose dimensions obey a simple formula exact in the central charge c and external operator dimensions. We then show that VMFT provides a baseline for large spin universality in two dimensions: in every unitary compact CFT2 with c > 1 and a twist gap above the vacuum, the double-twist data approaches that of VMFT at large spin ℓ. Corrections to the large spin spectrum from individual non-vacuum primaries are exponentially small in sqrt{ell } for fixed c. We analyze our results in various large c limits. Further applications include a derivation of the late-time behavior of Virasoro blocks at generic c; a refined understanding and new derivation of heavy-light blocks; and the determination of the cross-channel limit of generic Virasoro blocks. We deduce non-perturbative results about the bound state spectrum and dynamics of quantum gravity in AdS3.
Highlights
Introduction and summaryRecent years have seen enormous progress in understanding generic conformal field theories (CFTs), largely through conformal bootstrap methods
We show that Virasoro Mean Field Theory” (VMFT) provides a baseline for large spin universality in two dimensions: in every unitary compact CFT2 with c > 1 and a twist gap above the vacuum, the double-twist data approaches that of VMFT at large spin
Analysis in the lightcone limit implies that the spectrum of any d > 2 CFT approaches that of Mean Field Theory (MFT) at large spin, with anomalous dimensions suppressed by inverse powers of spin −2ht, where ht is the twist (2ht = ∆t − t) of some T-channel operator
Summary
Recent years have seen enormous progress in understanding generic conformal field theories (CFTs), largely through conformal bootstrap methods. Some of the most powerful results exist at large N or in an expansion in large spin, where analytic methods reveal features of the operator product expansion (OPE) that are universal to all CFTs. In two dimensions, the enhancement of spacetime symmetry to the infinite-dimensional Virasoro algebra would seem to aid efforts to analytically explore the space of CFTs and, via holography, the properties of AdS3 quantum gravity. We will combine some of the maxims of recent analytic bootstrap studies in higher dimensions with the power of Virasoro symmetry to uncover universal properties of irrational CFTs at finite central charge, and their implications for AdS3 quantum gravity These results represent the complete, exact summation of the stress tensor contributions to certain OPE data. We will leverage the power of an underexploited tool, the Virasoro fusion kernel
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