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

TL;DR

This paper advances the understanding of Virasoro conformal blocks by analytically resumming logarithmic corrections in the $1/c$ expansion, revealing insights into chaos, high-order computations, and superconformal theories in AdS$_3$/CFT$_2$.

Contribution

It provides all-orders resummations of logarithmic factors, computes vacuum blocks to high order, and extends techniques to superconformal theories, enhancing the analytic toolkit for AdS$_3$/CFT$_2$.

Findings

Resummation of $rac{1}{c} ext{log} z$ terms in Lorentzian regime.

Demonstration that $1/c$ corrections shift Lyapunov exponents.

Explicit calculations of vacuum blocks up to order $1/c^3$.

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.