Descending into the Modular Bootstrap

Nathan Benjamin; A. Liam Fitzpatrick; Wei Li; Jesse Thaler

hep-thApr 1, 2026

Descending into the Modular Bootstrap

Authors: Nathan Benjamin, A. Liam Fitzpatrick, Wei Li, Jesse Thaler

Organizations: Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA · Department of Physics, Boston University, Boston, MA 02215, USA · Center for Theoretical Physics – a Leinweber Institute, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Institut des Hautes ´Etudes Scientifiques, 91440 Bures-sur-Yvette, France ·

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 & tables

Panel of Figure 1 — 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 correspond to the unitarity bound $\Delta_{\rm gap}\geq 0$ and the known dual bound $\Delta_{\rm gap}\leq\frac{c}{6}+\frac{1}{3}$ Hellerman (2011) ; Friedan and Keller (2013) ; Collier et al. (2018) , while the yellow band indicates the refined exclusion from Ref. Fitzpatrick and Li (2024) . 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 and Zamolodchikov (1985) , the $N=1$ minimal model at $m=5$ with $(c,\Delta_{\rm gap})=(\frac{81}{70},\frac{3}{35})$ Friedan et al. (1985) , and the Ising $\otimes$ tricritical Ising theory with $(c,\Delta_{\rm gap})=(\frac{6}{5},\frac{1}{5})$ . For our optimization results, which are a preview of Fig. 13 , 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/intermediate/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 $\Delta_{\rm max}$ .

Panel of Figure 2 — 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. 2a used as our default. The losses $L$ come from Table 1 below, where order 1 values are considered CFT-like.

Panel of Figure 3 — 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 be 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. 2 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. 3b is most similar visually to those of the N1MM models, and we will use it for our final truncation uncertainty estimate.

Panel of Figure 4 — 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.

	$m=5$	$m=6$	$m=7$	$m=8$	$m=9$
$R=1$	$0.851$	$1.16$	$1.29$	$1.40$	$1.65$
$\boldsymbol{R=\sqrt{2}}$	$\mathbf{0.649}$	$\mathbf{0.904}$	$\mathbf{1.02}$	$\mathbf{1.12}$	$\mathbf{1.32}$
$R=\sqrt{3}$	$1.58$	$1.89$	$2.03$	$2.13$	$2.48$
$R=2$	$0.777$	$1.03$	$1.14$	$1.24$	$1.46$

Table 1: Loss values for the N1MM CFTs with

m\in\{5,6,7,8,9\}

, with truncation uncertainties estimated from the FBCB models with radii

R\in\{1,\sqrt{2},\sqrt{3},2\}

. Here, we take

\Delta^{\rm N1MM}_{\rm max}=4

\Delta^{\rm FBCB}_{\rm max}=6

, and

\tau_{2}^{\rm max}=6

as our defaults. All choices of

R

shown yield stable

\mathcal{O}(1)

loss values, but for concreteness we use

R=\sqrt{2}

from Fig. 3b when estimating truncation uncertainties.

Panel of Figure 5 — 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.

Panel of Figure 6 — Figure 6: Impact of different truncation choices on the distribution of loss terms, with the median loss shown as the solid line, the central 68% interval as the darker shading, and the central 95% interval as the lighter shading, for the default N1MM with $m=5$ . The top plots scan the truncation in operator dimension for $\Delta_{\rm max}\in\{2,3,\mathbf{4},5,6\}$ , while the bottom plots scan the truncation in the fundamental domain for $\tau_{2}^{\rm max}\in\{4,5,\mathbf{6},7,8\}$ , with the bold number indicating the choice made for the complementary scan. When estimating the truncation uncertainties, we always set $\Delta^{\rm FBCB}_{\rm max}=\Delta_{\rm max}+2$ . For the naive loss in the left (red) plots, there is a high degree of sensitivity to the truncation, with an exponential decrease in the median loss with increasing $\Delta_{\rm max}$ , and a nearly exponential increase in the upper edge of the 95% interval with increasing $\tau_{2}^{\rm max}$ . For the improved loss in the right (blue) plots, the distributions are much more stable across truncation choices, with less than an order of magnitude change in loss quantiles among the variations. The sharpest change in the improved loss is between $\Delta_{\rm max}=3$ and $\Delta_{\rm max}=4$ , due to the presence of multiple primary operators with $\Delta\approx 3$ in the default N1MM, as can be seen in Fig. 2a .

Figure 7: Fractional uncertainties on the central charges and primary dimensions for the default N1MM model with

m=5

from Fig. 2a , as estimated from the derivatives of the loss function. The uncertainties are highly correlated, so we show the maximum excursion on each parameter as 1, 2, 3, and 4 of them are varied simultaneously. When only 1 parameter is varied, the central charge and lowest dimension operators are constrained to better than 1 part in a million, but this neglects important correlations between the parameters.

Figure 8: Initial spectrum of primary operators prior to optimization. This model has 60 free parameters, namely the dimensions

\{\Delta_{a}\}

of its 60 non-vacuum primaries. Note that the central charge is not a free parameter during the optimization, and while the

c=1.09

initialization is shown here for concreteness, the same initial spectrum is used for all

c

values, though with different starting loss

L

Panel of Figure 9 — Figure 9: Evolution of the test loss during training, with the gray bars at the bottom indicating the CFT-like thresholds from Eq. ( 74 ) with $n\in\{2,3,4\}$ . In the left panel, we compare different optimization strategies for the $c=1.09$ candidate. Whereas gradient descent only yields a modest decline in the loss, Sven is able to find orders of magnitude deeper minima as the number of computed singular values $k_{\rm SV}$ increases. For the default value of $k_{\rm SV}=16$ , the test loss crosses the CFT-like threshold in around 150 steps. In the right panel, we see that the training dynamics for the five candidate theories exhibit qualitatively similar behaviors. Note that we are only showing the test loss, but as explained in the text, the training loss faithfully tracks the test loss up to statistical fluctuations.

	$c=1.05$	$c=1.07$	$c=1.09$	$c=1.11$	$c=1.13$
$R=1$	$1.19$	$0.528$	$0.680$	$3.02$	$2.39$
$\boldsymbol{R=\sqrt{2}}$	$\mathbf{0.878}$	$\mathbf{0.395}$	$\mathbf{0.510}$	$\mathbf{2.24}$	$\mathbf{1.80}$
$R=\sqrt{3}$	$3.39$	$1.30$	$1.61$	$6.64$	$5.11$
$R=2$	$1.14$	$0.513$	$0.649$	$2.78$	$2.18$

Table 2: Same as Table 1 , but now for the five candidate CFTs found through optimization. We see the same stability as in the N1MM study as a function of

R

and the choice of model, though some of the loss values are a factor of 2 smaller or larger. While this is some evidence for overfitting (for smaller losses) and incomplete optimization (for larger losses), we do not expect this to impact the qualitative features of the identified spectra.

Panel of Figure 10 — Figure 10: Final spectrum of Virasoro primary operators after optimization, where we have merged operators with $|\Delta_{a}-\Delta_{b}|<0.15$ for ease of visualization. The fact that these spectra are qualitatively similar to the N1MM models in Fig. 2 is additional evidence for their CFT-like structure. The presence of non-vacuum primaries with $\Delta\approx|j|$ suggests that these candidates could be in the vicinity of rational CFTs.

Figure 11: Fractional uncertainties on the central charges and primary dimensions for the

c=1.09

candidate, as estimated from the derivatives of the loss function. While the central charge was fixed during training, we allow it to float when reporting uncertainties. Even if there is evidence for operator degeneracy, we vary their dimensions independently. Like for the N1MM benchmark in Fig. 7 , the central charge and lowest primary dimensions have less than 1 part per million uncertainties when varied individually, but there are sizable correlated uncertainties when multiple parameters are varied simultaneously. The uncertainties for the other four benchmark models (see Figs. 20 and 21 ) are qualitatively similar.

Figure 12: Same initial spectrum prior to optimization as Fig. 8 , but now imposing a non-trivial spectral gap

\Delta_{\rm gap}

. This model will evolve into the strong candidate CFT shown in Fig. 15a .

Panel of Figure 13 — Figure 13: Results of the $(c,\Delta_{\rm gap})$ survey in the plane of central charge versus spectral gap, separating the preview plot from Fig. 1b . In the left panel, we show candidate CFTs identified in our survey. The blue dots indicate the degree of confidence in the solution: darkest blue for loss $L<4$ , medium blue for $L\in(4,9)$ , and the lightest blue for $L\in(9,16)$ . In the right panel, we show poor optimization results indicative of a modular bootstrap obstruction. The red dots indicate the degree of deficiency in the optimization: darkest red for loss $L>10^{7}$ , medium red for $L\in(10^{3},10^{7})$ , and lightest red for $L\in(16,10^{3})$ . The dashed black lines correspond to the unitarity bound $\Delta_{\rm gap}\geq 0$ and HCLY bound $\Delta_{\rm gap}\leq\frac{c}{6}+\frac{1}{3}$ Collier et al. (2018) .

Figure 14: Distribution of loss values found in the

(c,\Delta_{\rm gap})

survey. The losses for the CFT candidates are peaked near 1 (logarithm peaked near 0), as expected for chi-squared-like behavior, albeit with an asymmetric skew towards higher loss values. While the boundary between CFT candidates and weakly-excluded spectra is fluid, losses corresponding to intermediate/strong exclusion are well separated from those of good solutions.

Panel of Figure 15 — Figure 15: Example primary spectra found in the $(c,\Delta_{\rm gap})$ survey, corresponding to CFT candidates with (a) high confidence, (b) medium confidence, and (c) low confidence, and poor optimization results with (d) weak exclusion, (e) intermediate exclusion, and (f) strong exclusion. These examples have $c=1.08$ and $\Delta_{\rm gap}\in\{0.32,0.36,0.40,0.44,0.48,0.52\}$ , showing the transition from CFT-like to non-CFT-like behavior as the spectral gap increases for fixed central charge.

Figure 16: Scrutinizing the robustness of the

\Delta_{\rm max}=4

gap by doing a fresh optimization where operators are allowed to have non-integer degeneracy. Since there are explicit constructions that saturate the HCLY bound without integrality, we expect (and observe) the gap to close completely.

Panel of Figure 17 — Figure 17: Scrutinizing the robustness of the $\Delta_{\rm max}=4$ gap by relaxing the truncation scale to $\Delta_{\rm max}=3$ . In the left panel, we perform a fresh optimization and focus just on the gap region. In the right panel, we reinterpret the spectra already identified from the $\Delta_{\rm max}=4$ survey from Fig. 13 . In both cases, the gap closes significantly (though not completely), implying that the feature seen in the $\Delta_{\rm max}=4$ case corresponds to constraints imposed by integrality from multiple $j=2$ and $j=3$ operators. The agreement between the two panels is strong evidence that the poor optimization results we found at $\Delta_{\rm max}=4$ nevertheless have the right $\Delta_{\rm max}=3$ structures.

Panel of Figure 18 — Figure 18: Scrutinizing the robustness of the $\Delta_{\rm max}=4$ gap by increasing the truncation threshold to $\Delta_{\rm max}=5$ . Because of computational limitations, we have to do a reduced scan in the $(c,\Delta_{\rm gap})$ plane. In the left panel, we show the results of a fresh optimization with $\Delta_{\rm max}=5$ . We find that the gap region persists in this $\Delta_{\rm max}=5$ survey, though new poor optimization regions appear close to the HCLY bound and at $(c,\Delta_{\rm gap})=(1.02,1.04)$ . In the right panel, we reinterpret these spectra using the $\Delta_{\rm max}=4$ criteria, finding the qualitatively same behavior as in the original $\Delta_{\rm max}=4$ survey from Fig. 13 . We therefore conclude that the apparent contraction of the space of candidate CFTs is due to new constraints appearing from integrality at $\Delta_{\rm max}=5$ .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

	$m=5$	$m=6$	$m=7$	$m=8$	$m=9$
$R=1$	$0.424$	$0.449$	$0.431$	$0.428$	$0.469$
$\boldsymbol{R=\sqrt{2}}$	$\mathbf{0.371}$	$\mathbf{0.399}$	$\mathbf{0.388}$	$\mathbf{0.389}$	$\mathbf{0.429}$
$R=\sqrt{3}$	$1.14$	$1.04$	$0.982$	$0.957$	$1.05$
$R=2$	$0.422$	$0.453$	$0.449$	$0.455$	$0.506$

Appendix

Table 3: Loss values for the N1MM CFTs using an alternative strategy to estimate truncation uncertainties. The loss values are around a factor of 2–3 smaller than for the default strategy in Table 1 , but otherwise quite similar in their qualitative patterns.

	$c=1.05$	$c=1.07$	$c=1.09$	$c=1.11$	$c=1.13$
$R=1$	$0.870$	$0.358$	$0.416$	$1.73$	$1.31$
$\boldsymbol{R=\sqrt{2}}$	$\mathbf{0.725}$	$\mathbf{0.305}$	$\mathbf{0.360}$	$\mathbf{1.50}$	$\mathbf{1.15}$
$R=\sqrt{3}$	$3.42$	$1.27$	$1.43$	$5.71$	$4.46$
$R=2$	$0.866$	$0.364$	$0.421$	$1.71$	$1.31$

Appendix

Table 4: Loss values for the five candidate CFTs found in Sec. 6 using an alternative strategy to estimate truncation uncertainties. Here, the loss values are around the same size as for the default strategy in Table 2 , which makes sense since we did not specifically optimize using this alternative uncertainty strategy.

Panel of Figure 19 — Figure 19: Primary spectrum at the end of training, when using (a) gradient descent and (b–f) Sven with different numbers of singular values. The plots here correspond to the plateaus in Fig. 9a , and Fig. 19f is a duplicate of Fig. 10c for ease of comparison. To probe the spectrum up to $\Delta_{\rm max}$ , it seems that roughly $2^{\Delta_{\rm max}}$ singular values are needed.

Panel of Figure 20 — Figure 20: Same as Fig. 11 but for the (a) $c=1.05$ and (b) $c=1.07$ candidates.

Panel of Figure 21 — Figure 21: Same as Fig. 11 but for the (a) $c=1.11$ and (b) $c=1.13$ candidates.

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