Wormholes and Spectral Statistics in the Narain Ensemble

TL;DR

The paper studies spectral statistics of primary operators in the Narain ensemble of free boson CFTs by averaging over Narain moduli with the Siegel-Weil framework, giving exact two-point functions of the density of states via degree-2 Eisenstein series. It interprets these correlators as sums over Euclidean wormholes in a bulk U(1) gravity theory, with detailed analyses of rank-1 and rank-2 contributions and their bulk-topology interpretations. The authors compute the spectral form factor and demonstrate a late-time plateau consistent with spectral discreteness, while finding no linear ramp characteristic of random-matrix theory. This work provides a concrete, solvable setting where ensemble averaging yields controlled gravitational-like amplitudes and enables comparisons with pure Einstein gravity in AdS3. It also lays groundwork for higher-genus and multi-boundary observables in an explicitly averaged holographic dual.

Abstract

We study the spectral statistics of primary operators in the recently formulated ensemble average of Narain's family of free boson conformal field theories, which provides an explicit (though exotic) example of an averaged holographic duality. In particular we study moments of the partition function by explicit computation of higher-degree Eisenstein series. This describes the analog of wormhole contributions coming from a sum of over geometries in the dual theory of "U(1) gravity" in AdS$_3$. We give an exact formula for the two-point correlation function of the density of primary states. We compute the spectral form factor and show that the wormhole sum reproduces precisely the late time plateau behaviour related to the discreteness of the spectrum. The spectral form factor does not exhibit a linear ramp.

Figures (5)

• Figure 1: The degeneration of a genus-two handlebody into two solid tori. The $A$-cycles of the tori are contractible in the bulk, corresponding to the manifold $M_{0,\mathbb{1}}$.
• Figure 2: Here we illustrate how to construct the manifold $M_{C_0,D_0}$ from the pinching limit of a quotient of hyperbolic three-space. The statement that the cycles given by (\ref{['eq:torusWormholeContractibleCycles']}) are contractible in the bulk can be implemented (up to an orientation reversal that flips the sign of one of the $B$ cycles) by "filling in" the three-manifold by rotating the top half through the bulk around the $x$-axis by an angle of $\pi$ as indicated by the bold arrow so that $A_1$ is identified with $A_2$ and $B_1$ is identified with $B_2$. The manifold $M_{C_0,D_0}$ is then obtained by pinching the cycle $A_1\,B_1\,A_1^{-1}\,B_1^{-1}$, which is labelled in green. The resulting topology is that of a torus times an interval. One can even endow this with a constant negative curvature metric by replacing the straight line segments with circles that are identified via Mobius transformations which become hyperbolic isometries in the bulk. Strictly speaking, this gives a constant negative curvature metric whose boundary is two copies of a punctured torus, rather than just two copies of the torus; this had to be the case, of course, since there is no smooth constant negative curvature manifold whose conformal boundary is the disconnected union of two tori.
• Figure 3: A plot of the normalized spectral form factor $G(\beta,T) = g(\beta,T)/g(\beta,0)$ as a function of time in the averaged Narain CFT, for $s=100,10,5$ and with $\beta = {\pi\over 1000}$. The horizontal gridline corresponds to the plateau ${\langle Z^p(2\beta)\rangle \over \langle Z^p(\beta)^2\rangle}$. At smaller values of $\beta$, the rank-two contributions to the two-point function dominate at early times and decay monotonically, until a crossover time when they are surpassed by the plateau coming from the rank-one contributions.
• Figure 4: A plot of the normalized spectral form factor as a function of time in the averaged Narain CFT, for $s=100,10,5$ and with $\beta = {5\pi}$. At large values of $\beta$, the rank-two contributions to the two-point function are essentially never important, and the spectral form factor approaches the plateau in an oscillatory manner due to the rank-one contributions. The horizontal gridline corresponds to the plateau ${\langle Z^p(2\beta)\rangle \over \langle Z^p(\beta)^2\rangle}$.
• Figure 5: A finely-tuned example of the normalized spectral form factor at a moderate value of the temperature $\beta = {\pi\over 5}$ with $s=100$. In the second plot on the right we zoom in to the region where the rank-one contributions overtake the rank-two contributions, demonstrating an apparent approach to the plateau from below. In the third plot at the bottom we show the separate rank-one and rank-two contributions to the spectral form factor.