Descending into the Modular Bootstrap

Abstract

In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization. The torus partition function of a 2d CFT is fixed by the spectrum of its primary operators and its chiral algebra, which we take to be the Virasoro algebra with $c>1$. We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra. Our approach involves two technical innovations that facilitate finding reliable candidate CFTs. The first is a strategy to estimate the uncertainty associated with truncating the spectrum to the lowest dimension operators. The second is the use of a new singular-value-based optimizer (Sven) that is more effective than gradient descent at navigating the hierarchical structure of the loss landscape. We numerically construct candidate truncated CFT partition functions with central charges between 1 and $\frac{8}{7}$, a range devoid of known examples, and argue that these candidates likely come from a continuous space of modular bootstrap solutions. We also provide evidence for a more stringent constraint on the spectral gap near $c = 1$ than the existing bound of $Δ_{\rm gap} \le \frac{c}{6} + \frac{1}{3}$.

Figures (23)

• Figure 1: Space of (a) established 2d CFTs compared to (b) the results of this study, in the $(c,\Delta_{\rm gap})$ plane of central charge versus spectral gap. The dashed black lines corresponds to the unitary bound $\Delta_{\rm gap} \ge 0$ and the known dual bound $\Delta_{\rm gap} \le \frac{c}{6}+\frac{1}{3}$Hellerman:2009buFriedan:2013cbaCollier:2016cls, while the yellow band indicates the refined exclusion from Ref. Fitzpatrick:2023lvh. For the established 2d CFTs, the purple dots represent compactified free boson theories at $c = 1$ which have spectral gaps that fill the entire $\Delta_{\rm gap} \in (0,\frac{1}{2})$ range. The orange dots correspond, from left to right, to the $\mathbb{Z}_5$ parafermion CFT with $(c,\Delta_{\rm gap}) = (\frac{8}{7}, \frac{4}{35})$Fateev:1985mm, the $N=1$ minimal model at $m=5$ with $(c,\Delta_{\rm gap}) = (\frac{81}{70}, \frac{3}{35})$Friedan:1984rv, and the Ising $\otimes$ tricritical Ising theory with $(c,\Delta_{\rm gap}) =(\frac{6}{5},\frac{1}{5})$. For our optimization results, which is a preview of Fig. \ref{['fig:survey_results']}, the blue dots correspond to numerical solutions of the primal modular bootstrap with a truncation scale of $\Delta_{\rm max} = 4$, with dark/medium/light blue corresponding to high/medium/low confidence in the solution. The red dots indicate poor optimization results, with dark/medium/light red corresponding to strong/medium/weak exclusion of that parameter point. Despite multiple attempts, we were unable to find good solutions in the upper left corner of this plot with $c \in (1.00,1.06)$ and $\Delta_{\rm gap} \in (0.3,0.5)$, suggestive of a primal/dual gap. We suspect that this gap might even grow with larger values of $\Delta_{\rm max}$.
• Figure 2: Spectrum of Virasoro primary operators below $\Delta_{\rm max} = 4$ for the $N=1$ minimal model (N1MM) with (a) $m = 5$, (b) $m=6$, (c) $m=7$, (d) $m=8$, and (e) $m=9$. Here and in other spectrum plots below, the degeneracy of each primary is proportional to the area of the corresponding dot, with the vacuum at $(\Delta,j) = (0,0)$ having degeneracy one, and every operator with positive spin has a negative spin counterpart. These five N1MM examples will be used to benchmark the loss value corresponding to CFT-like behavior, with $m = 5$ in Fig. \ref{['fig:n1mm_spectrum_m5']} used as our default. The losses $L$ come from Table \ref{['tab:n1mm_m_vs_r_table']} below, where order 1 values are considered CFT-like.
• Figure 3: Spectrum of Virasoro primary operators below $\Delta_{\rm max} = 6$ for free boson theories on the circle branch (FBCB) with radii (a) $R = 1$, (b) $R = \sqrt{2}$, (c) $R = \sqrt{3}$, and (d) $R = 2$. These four FBCB theories will used as baselines to estimate the uncertainty associated with truncating the spectrum at $\Delta_{\rm max}$. To estimate uncertainties for the N1MM models in Fig. \ref{['fig:n1mm_spectrum']} with $\Delta_{\rm max} = 4$, we take these FBCB baselines to have $\Delta_{\rm max} = 6$. The spectrum of the FBCB theory with $R = \sqrt{2}$ in Fig. \ref{['fig:fbcb_spectrum_rsprt2']} is most similar visually to those of the N1MM models, and we will use it for our final truncation uncertainty estimate.
• Figure 4: Values of $\tau = \tau_1 + i \, \tau_2$ in the fundamental domain used for (a) training, (b) testing, and (c) final reporting, along with their $S$-transformed counterparts. For numerical stability, we restrict $\tau_2 < \tau_2^{\rm max}$. Because we restrict to parity-preserving CFTs, we only need to consider $\tau_1 > 0$. During training, new $\tau$ values are chosen stochastically for each iteration. During testing and reporting, the same fixed grid of $\tau$ values are used, with higher fidelity during the reporting stage. Because the ML hyperparameters are fixed throughout the training process, we do not need a validation set of $\tau$ values.
• Figure 5: Distribution of individual terms in the loss for (a) the naive MSE-style loss in red and (b) the improved $\chi^2$-style loss in blue, for the default N1MM with $m=5$. For the naive loss, different values of $\tau$ yield vastly different loss terms, spanning 20 orders of magnitude in this case. For the improved loss, most values of $\tau$ yield order one loss values, with the small-side tails reasonably well modeled by a chi-squared distribution with 1 degree of freedom. The high-side depletion arises because the loss values are somewhat correlated, so that when one value of $\tau$ happens to yield a small loss, there are multiple nearby $\tau$ values that also have small losses.