The Most Irrational Rational Theories

TL;DR

The paper constructs a two-parameter family of modular-invariant 2d CFT partition functions, dual to pure AdS3 gravity, controlled by the central charge c and the dimension of an SL(2,ℤ) representation via Bantay-Gannon vector-valued modular forms. It demonstrates that, at large c and with large representations, the spectral form factor exhibits chaotic-like dip-ramp-plateau behavior, while minimal models at large m show analogous SFF features, linking RCFTs to gravity-like dynamics. It also analyzes unitarity/holomorphicity constraints, discusses the possibility of larger gaps beyond c/24, and outlines future bootstrap-related avenues to realize such theories. Overall, the work provides a structured route to realize non-holomorphic, large-c partition functions with gravity-like spectral features, bridging rational models and semiclassical gravity through modular representation theory.

Abstract

We propose a two-parameter family of modular invariant partition functions of two-dimensional conformal field theories (CFTs) holographically dual to pure three-dimensional gravity in anti de Sitter space. Our two parameters control the central charge, and the representation of $SL(2,\mathbb{Z})$. At large central charge, the partition function has a gap to the first nontrivial primary state of $\frac{c}{24}$. As the $SL(2,\mathbb{Z})$ representation dimension gets large, the partition function exhibits some of the qualitative features of an irrational CFT. This, for instance, is captured in the behavior of the spectral form factor. As part of these analyses, we find similar behavior in the minimal model spectral form factor as $c$ approaches $1$.

Figures (10)

• Figure 1: Left: SFF's at continuous, early $t$, of different models. Right: SFF's at continuous, late $t$, of the same models. By eye one can see that the behavior is more irregular at late times. The dotted spikes are from the $J$-invariant, which is strictly periodic with period equal to the mini-recurrence time. The dashed and solid curves are from $m=10$ and $m=100$ minimal models respectively. The minimal model SFF's are not exactly periodic, but clearly the SFF peaks at every $t_n=2\pi n$.
• Figure 2: SFF of the $m=50$ minimal model at $\beta=\frac{3\pi}{2}$ without any averaging. The dip and ramp behave as predicted by (\ref{['eq:dipramppred']}).
• Figure 3: Upper left: SFF of the $m=10^4$ minimal model at $\beta = \pi/5$ without any averaging. The plot depicts about one-third of the full recurrence time. This unaveraged SFF has a clear dip and the late time behavior is obscured by oscillation. Upper right: The same SFF only averaged over different $m$ with a window of $\pm\delta m = 50$. Lower left: The same SFF only averaged over a gaussian time window of standard deviation $\delta t = 5$, for integral time greater than 100. Lower right: The same SFF first averaged over different $m$, then averaged over $t$, and zoomed in on the dip and plateau. The averaged behavior shows more clearly the ramp and plateau.
• Figure 4: Upper: Level statistics of the absolute energy levels of the $m=10^3$ minimal model. Lower left: Level statistics of the $m=10^3$ minimal model with energy levels taken modulo 1 and degeneracies ignored. Lower right: Same as left graph but averaging minimal models between $m=10^3-50$ to $m=10^3+50$.
• Figure 5: The smooth curve from averaging the level statistics of the $m=10^3\pm50$ minimal models with moded energy levels, compared with the Gaussian Ensembles and the Poisson distribution. A curve with $m=3 \times 10^3$ looks identical to the $m=10^3$ curve. Left: Distribution as a function of log of the unfolded energy level $r$. Right: Distribution as a function of the unfolded energy level $r$.