Quantum Regge Trajectories and the Virasoro Analytic Bootstrap

TL;DR

<3-5 sentence high-level summary>We analyze irrational 2D CFTs with Virasoro symmetry by leveraging the Virasoro fusion kernel to perform a non-perturbative bootstrap that sums stress-tensor contributions exactly. This yields Virasoro Mean Field Theory (VMFT), producing a finite set of discrete quantum Regge trajectories plus a continuum, with twists $h_m=h_1+h_2+m+\delta h_m$ where $\delta h_m=-2(\alpha_1+mb)(\alpha_2+mb)+m(m+1)b^2$ and $c=1+6Q^2$, $Q=b+b^{-1}$. In the large-spin limit, unitary CFT$_2$ with $c>1$ and a twist gap exhibit universality toward VMFT data, while non-vacuum exchanges yield exponentially suppressed corrections in $\sqrt{\ell}$; the framework also yields cross-channel Virasoro blocks and late-time behavior with gravitational interpretations in AdS$_3$ as two-particle bound states and BTZ physics. These results provide non-perturbative insights into 3D gravity and offer a concrete, exact bootstrap handle on irrational CFT$_2$ dynamics at finite $c$.

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 CFT$_2$ analytic bootstrap, and extract several implications for AdS$_3$ 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 CFT$_2$ with $c > 1$ and a twist gap above the vacuum, the double-twist data approaches that of VMFT at large spin $\ell$. 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 AdS$_3$.

Figures (6)

• Figure 1: The twist spectrum of Virasoro double-twist operators arising from inversion of the holomorphic vacuum block, i.e. the spectrum of VMFT. The twists of the external operators are fixed while $c$ varies. For a given central charge $c$, there is a discrete number of operators, shown as the solid lines, below the continuum at $h>\frac{c-1}{24}$, shaded in grey. The exact formula for the twists is given in \ref{['anomdim']}. Each line, when combined with an anti-holomorphic component, forms a quantum Regge trajectory that is exactly linear in spin. At large $c$, one recovers the integer-spaced operators of MFT.
• Figure 2: In the semiclassical regime, the additivity rule \ref{['eq:VirasoroDoubleTwist']} for discrete momenta of large spin double-twist operators translates into additivity of conical defect angles in AdS$_3$, where the deficit angle is $\Delta\phi = \frac{4\pi}{Q} \alpha$. The leading-twist operator, whose dual conical defect is depicted as the sum of two constituent defects, has momentum $\alpha_1+\alpha_2$. We have suppressed the $\tfrac{4\pi}{Q}$ for clarity.
• Figure 3: For sufficiently heavy external pairwise identical operators (${\rm Re}(\alpha_1 + \alpha_2)>{Q\over 2}$), the fusion kernel has four semi-infinite lines of poles extending in either direction. In the case of vacuum exchange in the T-channel, these poles are simple, otherwise they are double poles. Here we show an example of this in the case that all external operators have weights in the continuum, with $0<b<1$. The dashed red curve denotes the contour of integration in the decomposition of the T-channel Virasoro block into S-channel blocks, while the blue and green crosses denote the poles of the fusion kernel.
• Figure 4: Here we plot the poles of the fusion kernel as a function of $\alpha_s$ in the case of pairwise identical external operators and $\operatorname{Re}(\alpha_1 + \alpha_2)<\frac{Q}{2}$, with $0<b<1$. In this case, the poles at $\alpha_s = \alpha_1 + \alpha_2 + mb< \frac{Q}{2}$ (and their reflections) cross the contour of integration and give discrete residue contributions to the T-channel Virasoro block. Note that despite the fact that the external operators have weights lying in the discrete range rather than the continuum we have given $\alpha_i$ small imaginary parts for the purpose of presentation.
• Figure 5: The VMFT spectrum, combining holomorphic and anti-holomorphic sectors.