3D Gravity and Chaos in CFTs with Fermions

Abstract

Pure 3d gravity in AdS is believed to admit a holographic description in terms of 2d CFT. We introduce a theory of fermionic 3d gravity where we sum over geometries equipped with spin structure, and propose it is holographically described by fermionic 2d CFT data. We evaluate the leading contributions to the gravity path integral with one and two torus boundaries, extracting both the spectrum and its spectral statistics from the torus wormhole. Strikingly, the theory has fermionic black hole microstates, even in the absence of bulk fermionic matter. We then incorporate subtle bulk topological field theories, classified by appropriate cobordism groups, and evaluate the one and two-boundary torus partition functions. The spectral statistics we derive from gravity are shown, in all cases, to be consistent with the pattern of anomalies expected from classifications of fermionic 2d CFT. We also define a version of RMT$_2$, a random-matrix framework compatible with the symmetries of 2d CFTs, which naturally accommodates fermionic spectra and reproduces our gravitational results across all cases we analyze.

Figures (10)

• Figure 1: The $D-1$ boundary $Y_{D-1}$ is labeled by its topology, metric, and spin structure $\mathfrak{s}$. In the gravitational path integral we integrate over all manifolds $X_D$ with a bulk spin structure compatible with that of $Y_{D-1}$.
• Figure 2: Conventions for the torus spin structures. R and NS Hilbert spaces correspond to $\nu=0$ and $\nu=1$ respectively. $\mu=0$ corresponds to a $(-1)^{\sf F}$ insertion.
• Figure 3: Example of the fact that the gravitational path integral with boundary 2d spin surfaces $Y=Y_1 \cup Y_2 \cup Y_3 \cup \ldots$ vanishes if $\sum_i \zeta_i = 1~\text{mod}~2$. When $\zeta \neq 0$ no 3d spin surface exists with such a boundary. This applies to a union of tori but also more general surfaces.
• Figure 4: (a) Spectrum of pure fermionic gravity in $\mathcal{H}_\mathrm{R}$ according to the gravitational path integral on a solid torus. The vertical black lines denote black hole states and appear for $E\geq |j|$ and $j\in \mathbb{Z}$. Each comes in two identical copies of bosonic and fermionic states. (b) Same result for $\mathcal{H}_{\mathrm{NS}}$. Bosonic black holes with $E\geq |j|$ and $j\in \mathbb{Z}$ are denoted by the black lines and fermionic $j\in \mathbb{Z}+1/2$ by the blue lines. In (a) and (b) negativities are denoted by the red segments, but all states have a $1/\sqrt{E-|j|}$ edge.
• Figure 5: In this section we compute the two-boundary torus wormhole with boundary moduli $(\tau_1,\bar{\tau}_1)$ and $(\tau_2,\bar{\tau}_2)$ as well as spin structure $\mathfrak{s}_1$ and $\mathfrak{s}_2$.