Renyi Entropies, the Analytic Bootstrap, and 3D Quantum Gravity at Higher Genus

Abstract

We compute the contribution of the vacuum Virasoro representation to the genus-two partition function of an arbitrary CFT with central charge $c>1$. This is the perturbative pure gravity partition function in three dimensions. We employ a sewing construction, in which the partition function is expressed as a sum of sphere four-point functions of Virasoro vacuum descendants. For this purpose, we develop techniques to efficiently compute correlation functions of holomorphic operators, which by crossing symmetry are determined exactly by a finite number of OPE coefficients; this is an analytic implementation of the conformal bootstrap. Expanding the results in $1/c$, corresponding to the semiclassical bulk gravity expansion, we find that---unlike at genus one---the result does not truncate at finite loop order. Our results also allow us to extend earlier work on multiple-interval Renyi entropies and on the partition function in the separating degeneration limit.

Figures (4)

• Figure 1: A depiction of the sewing construction as applied to $Z_{\rm vac}$, the contribution of the Virasoro vacuum representation to a genus-two CFT partition function. The coordinates $p_1$ and $p_2$ represent the widths of the two handles in a Schottky uniformization of the Riemann surface. The handles are replaced by a sum over pairwise operator insertions, where we include all Virasoro descendants of the identity, ${\cal O} \in {\cal H}_{\rm vac}$. This recasts $Z_{\rm vac}$ as a sum of sphere four-point functions, weighted by powers of $p_1$ and $p_2$. The operators ${\cal O}_i$ and ${\cal O}_j$ have holomorphic conformal weights $h_i$ and $h_j$, respectively. A detailed description of the sewing construction is presented in section \ref{['iv']} (see equations \ref{['Z2']} and \ref{['Ch1h2']}).
• Figure 2: The $n$-sheeted replica surface ${\mathscr R}_{2,n}$, which is the branched covering surface of the plane with two intervals and has genus $n-1$. On each sheet, there is a cycle separating the two intervals called the $A$-cycle, and another cycle encircling the two points $v_1$ and $u_2$, called the $B$-cycle. There are $n-1$ independent cycles of each type.
• Figure 3: An alternate depiction of the surface ${\mathscr R}_{2,n}$ in figure \ref{['fig-R2n']}. ${\mathscr R}_{2,n}$ can be visualized as two spheres connected by $n$ tubes. The two spheres, one for each interval, are made by cutting small holes around each pair of intervals on all $n$ sheets. The tubes connecting the holes on the two spheres represent the sheets. In this picture, the $A$-cycles wrap the $n$ tubes and the $B$-cycles run through two different tubes.
• Figure 4: A picture of the sewing approach to computing a genus-two partition function, $Z$. The mechanism was explained in figure \ref{['fig-sewing-vac']}. To compute $Z$ rather than $Z_{\rm vac}$, one simply lets the sum run over all operators in the CFT Hilbert space.