Symmetries and spectral statistics in chaotic conformal field theories

TL;DR

The work addresses how chaotic features emerge in two-dimensional CFTs with large central charge while respecting Virasoro symmetry and modular invariance. It introduces a modular-invariant decomposition to isolate the dense primary spectrum and uses harmonic analysis on the modular domain to identify spin-resolved chaotic data via Eisenstein series and Maass cusp forms. The authors show that random matrix universality near extremality can hold independently in each spin sector despite spectral determinacy, and they derive a Cardy-like transformation that maps near-extremal universality to eigenvalue repulsion in far-from-extremal, large-spin regimes. This connects holographic chaos, modular bootstrap, and spectral statistics, suggesting a sigma-model perspective for chaotic CFTs and guiding future explorations of discrete spectra and holographic duals.

Abstract

We discuss spectral correlations in coarse-grained chaotic two-dimensional CFTs with large central charge. We study a partition function describing the dense part of the spectrum of primary states in a way that disentangles the chaotic properties of the spectrum from those which are a consequence of Virasoro symmetry and modular invariance. We argue that random matrix universality in the near-extremal limit is an independent feature of each spin sector separately; this is a non-trivial statement because the exact spectrum is fully determined by only the spectrum of spin zero primaries and those of a single non-zero spin ("spectral determinacy"). We then describe an argument analogous to the one leading to Cardy's formula for the averaged density of states, but in our case applying it to spectral correlations: assuming statistical universalities in the near-extremal spectrum in all spin sectors, we find similar random matrix universality in a large spin regime far from extremality.

Figures (5)

• Figure 1: Cartoon of the spectrum of primary operators. The spectrum below the extremality bound (red dots outside the shaded region) is sparse and not chaotic. We remove it and all its modular images (red dots within the shaded region) from the primary partition function and focus on the statistics of the remaining dense spectrum after coarse-graining along fixed spin trajectories (blue). The first part of the paper (section \ref{['sec:ramp']}) demonstrates that chaos in different spin sectors near extremality (orange regions) contains independent information. The second part (section \ref{['sec:cardy']}) derives eigenvalue repulsion far from extremality by assuming random matrix statistics near extremality and performing a modular S-transformation (green regions).
• Figure 2: Plots of the non-convergent sum $\mathcal{S}_{\mathfrak{m}_1,\mathfrak{m}_2}(M)$ for $\mathfrak{m}_1 = \mathfrak{m}_2 - 1 \in \{10,10^3,10^5\}$ as a function of the cutoff $M$. We see that, although the limit $M\rightarrow \infty$ does not exist, the function oscillates within a bounded bandwidth. This is in stark contrast to the case $\mathfrak{m}_1 = \mathfrak{m}_2 \equiv \mathfrak{m}$, where ${\cal S}_{\mathfrak{m},\mathfrak{m}}(M)$ diverges. Note that for large $\mathfrak{m}_i$, the mean value around which the oscillations occur, tends to zero (dashed lines).
• Figure 3: The result of numerical evaluation of \ref{['eq:ZmZmInt']}: the contribution to the spin $(m_1,m_2)$ spectral form factor that originates from a spin 0 ramp through symmetries. We plot as a function of $y_1=y_2 \equiv y$. The plot confirms that this does not take the form of a ramp, but is rather a subleading correction to it, \ref{['eq:ZZresNum']}. Universal eigenvalue repulsion in the spin 0 sector hence doesn't imply the same in any other spin sector through symmetries (despite spectral determinacy in the exact partition function).
• Figure 4: The result of numerical evaluation of \ref{['eq:ZmZmInt']}, c.f., figure \ref{['fig:ZmZm1']}. In this plot we fix $y_1=20$ and plot as a function of $y_2 \equiv y$. Again, the result is well fitted by \ref{['eq:ZZresNum']}.
• Figure 5: The result of numerical evaluation of \ref{['eq:ZmZmspin1Imprint']}, i.e., the imprint of a ramp at spin 1 onto the spin $(m_1,m_2)$ spectrum, together with a linear fit (solid lines). The $y$-scaling is linear (as expected for a ramp), but the coefficient doesn't match the random matrix expectation for any $m_1=m_2>1$.