AdS$_3$/RMT$_2$ Duality

TL;DR

This work develops a trace-formula framework to quantify random-matrix universality in 2d CFT spectra and their AdS$_3$ gravity duals by decomposing torus partition functions into a chaotic piece $Z_{\rm spec}(\tau)$ via $SL(2,\mathbb{Z})$ spectral data. It introduces diagonal and Hecke projections to realize Berry-like coarse-graining, linking microscopic spectrum to a linear spectral-form-factor ramp with a precise $z=-1$ pole condition in the associated function $\mathcal{R}(z)$. In gravity, it interprets torus wormholes as diagonal projections (and their Hecke-symmetric refinements) of squared partition functions, culminating in the Cotler–Jensen wormhole saturating MaxRMT and enabling extraction of a microscopic $Z_{\rm RMT}(\tau)$ that captures heavy black-hole microstate structure. The results imply that pure AdS$_3$ gravity exhibits maximal random-matrix universality compatible with Virasoro symmetry and modular invariance, providing a concrete microscopic/CFT interpretation of wormholes and a path toward a consistent large-$c$ modular bootstrap incorporating spectral statistics. The framework also outlines how nonperturbative corrections must enter to maintain unitarity, potentially via Maxfield–Turiaci–type topologies, and suggests broad future directions for extending these ideas to higher dimensions and other observables.

Abstract

We introduce a framework for quantifying random matrix behavior of 2d CFTs and AdS$_3$ quantum gravity. We present a 2d CFT trace formula, precisely analogous to the Gutzwiller trace formula for chaotic quantum systems, which originates from the $SL(2,\mathbb{Z})$ spectral decomposition of the Virasoro primary density of states. An analogy to Berry's diagonal approximation allows us to extract spectral statistics of individual 2d CFTs by coarse-graining, and to identify signatures of chaos and random matrix universality. This leads to a necessary and sufficient condition for a 2d CFT to display a linear ramp in its coarse-grained spectral form factor. Turning to gravity, AdS$_3$ torus wormholes are cleanly interpreted as diagonal projections of squared partition functions of microscopic 2d CFTs. The projection makes use of Hecke operators. The Cotler-Jensen wormhole of AdS$_3$ pure gravity is shown to be extremal among wormhole amplitudes: it is the minimal completion of the random matrix theory correlator compatible with Virasoro symmetry and $SL(2,\mathbb{Z})$-invariance. We call this MaxRMT: the maximal realization of random matrix universality consistent with the necessary symmetries. Completeness of the $SL(2,\mathbb{Z})$ spectral decomposition as a trace formula allows us to factorize the Cotler-Jensen wormhole, extracting the microscopic object $Z_{\rm RMT}(τ)$ from the coarse-grained product. This captures details of the spectrum of BTZ black hole microstates. $Z_{\rm RMT}(τ)$ may be interpreted as an AdS$_3$ half-wormhole. We discuss its implications for the dual CFT and modular bootstrap at large central charge.

Figures (8)

• Figure 1: The density of heavy states $\rho(t)$ of a compact CFT, here approximated by the smooth black curve (as in a large $c$ limit) for illustrative purposes, is highly oscillatory on wavelengths of order $e^{-S_{\rm Cardy}(t)}$, the mean level spacing, while the modular completion of light states, $\widehat{\rho}_{L}(t)$, contains the smooth, universal Cardy-like growth (dashed line). Their difference, $\rho_{\rm spec}(t)$, captures the oscillations, encoding chaotic statistics. We have suppressed the spin index $j$.
• Figure 2: Off-shell wormhole amplitudes $Z_{\rm WH}(\tau_1,\tau_2)$ in semiclassical AdS$_3$ gravity are fixed by a single function $\mathcal{R}(z)$, where $y_i = \text{Im}(\tau_i)$ are inverse temperatures. The colored contours show Euclidean and Lorentzian kinematics in the low-temperature limit. The spectral form factor (SFF), for which $z$ lies on the unit circle, has a linear ramp at times $T \gg \beta$ if and only if $\mathcal{R}(z)$ has a simple pole at $z=-1$: see \ref{['ramppoleintro']}.
• Figure 3: The Cotler-Jensen wormhole $Z_{\rm CJ}(\tau_1,\tau_2)$ is generated by gluing two single boundary partition functions $Z_{\rm RMT}(\tau_1)$ and $Z_{\rm RMT}(\tau_2)$. Each factor may be interpreted as a half-wormhole in AdS$_3$ pure gravity. The gluing is performed by coarse-graining over the spectrum of the dual CFT: the (Hecke) projection $\mathcal{P}$ projects the factorized product onto the diagonal terms with respect to the $\sl$ spectral basis, analogously to trace formulas for non-disordered chaotic systems.
• Figure 4: Completeness of the $SL(2,\mathbb{Z})$ spectral basis permits factorization of the Cotler-Jensen wormhole. $Z_{\rm RMT}(\tau)$ encodes quantum substructure of the pure gravity black hole spectrum.
• Figure 5: $Z_{\rm diag}$ captures the ramp in the spectral form factor $K_\beta(T)$ of a 2d CFT.