Degenerate Operators and the $1/c$ Expansion: Lorentzian Resummations, High Order Computations, and Super-Virasoro Blocks

TL;DR

This work develops a robust framework for extracting Virasoro and super-Virasoro vacuum blocks in 2d CFTs by leveraging degenerate operators, enabling all-orders Lorentzian resummations and precise $1/c$ expansions up to $1/c^3$ for vacuum blocks. It combines differential-equation methods with the Coulomb gas formalism to determine the $f_{mn}( η_H,z)$ functions that encode heavy-light and all-light regimes, and demonstrates that leading Lorentzian corrections exponentiate into a corrected Lyapunov exponent while also enabling a Borel-resummed interpolation to the Ruelle regime. The results extend to $ ext{N}=1,2$ superconformal theories, showing that at leading order the heavy-light blocks are governed by pure Virasoro (with supersymmetric structure entering at subleading orders) and providing explicit expressions for super-Virasoro vacuum blocks via super-null-state equations. Together, these advances provide precise, analytically tractable control over high-order quantum corrections in AdS$_3$/CFT$_2$ and sharpen the understanding of chaos, information loss, and scrambling in holographic CFTs.

Abstract

One can obtain exact information about Virasoro conformal blocks by analytically continuing the correlators of degenerate operators. We argued in recent work that this technique can be used to explicitly resolve information loss problems in AdS$_3$/CFT$_2$. In this paper we use the technique to perform calculations in the small $1/c \propto G_N$ expansion: (1) we prove the all-orders resummation of logarithmic factors $\propto \frac{1}{c} \log z$ in the Lorentzian regime, demonstrating that $1/c$ corrections directly shift Lyapunov exponents associated with chaos, as claimed in prior work, (2) we perform another all-orders resummation in the limit of large $c$ with fixed $cz$, interpolating between the early onset of chaos and late time behavior, (3) we explicitly compute the Virasoro vacuum block to order $1/c^2$ and $1/c^3$, corresponding to $2$ and $3$ loop calculations in AdS$_3$, and (4) we derive the heavy-light vacuum blocks in theories with $\mathcal{N}=1,2$ superconformal symmetry.

Figures (2)

• Figure 1: Plot of the behavior of $1-F(t)$ as a function of time $t$ in the limit $c\rightarrow \infty$ with $c z$ fixed, with $h_L = h_H = \frac{1}{2}$. $F(t)$ is absolute value of the out-of-order correlator $\frac{\langle {\cal O}_L {\cal O}_H {\cal O}_L {\cal O}_H\rangle_\beta}{\langle{\cal O}_L {\cal O}_L\rangle \langle{\cal O}_H {\cal O}_H\rangle}$, and $t \equiv -\log( c z/6)$. The initial "Lyapunov" growth and the later "Ruelle" decay are labeled as in JoeRuelle. We have plotted only the contribution of an approximation to the vacuum Virasoro block, but the result has the qualitative features expected of the full correlator.
• Figure 2: Plots comparing the exact behavior from eq. (\ref{['eq:finalresumA']}) (black, dashed) for $1-F(t)$ in the limit $c\rightarrow \infty$ with $c z$ fixed, to the heavy-light approximation (\ref{['eq:heavylightlorentz']}) (red, solid). Left: $h_L = h_H = \frac{1}{2}$, Right: $h_L = \frac{3}{2}, h_H = \frac{3}{10}$. $F(t)$ and $t$ are as in figure \ref{['fig:LyapRuelle']}. Note that both curves only include contributions from the vacuum block, neglecting double-trace operators which could affect an AdS$_3$ calculation.