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arXiv:2512.07706

Bootstrapping non-unitary CFTs

Huang, Yu-tin, Lee, Shao-Cheng, Liao, Henry, Rumbutis, Justinas

Published Dec 8, 2025• 0 citations
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Abstract

We introduce a non-unitary-compatible numerical bootstrap strategy based on the statistical stability of OPE data inferred from crossing at multiple cross-ratios. For a trial spectrum, crossing determines OPE coefficients whose residual cross-ratio dependence directly measures the truncation error. This defines a scalar objective on the space of spectra, allowing bootstrap searches without imposing unitarity. Applied to two-dimensional Virasoro blocks, the method reproduces known A-series minimal models, including non-unitary examples, and yields candidate truncated solutions for c>1 with crossing violation comparable to that of minimal models. More generally, our framework provides a practical route to solving bootstrap constraints beyond the convex, unitary setting.

DOI
10.48550/arxiv.2512.07706

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We introduce a non-unitary-compatible numerical bootstrap strategy based on the statistical stability of OPE data inferred from crossing at multiple cross-ratios. For a trial spectrum, crossing determines OPE coefficients whose residual cross-ratio dependence directly measures the truncation error. This defines a scalar objective on the space of spectra, allowing bootstrap searches without imposing unitarity. Applied to two-dimensional Virasoro blocks, the method reproduces known A-series minimal models, including non-unitary examples, and yields candidate truncated solutions for c>1 with crossing violation comparable to that of minimal models. More generally, our framework provides a practical route to solving bootstrap constraints beyond the convex, unitary setting.
We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the “Gliozzi” bootstrap method, and provides a simpler setting in which to study technical challenges with the method. In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.
A bstract 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 87\frac{8}{7}78​ , 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 Δgap≤c6+13{\Delta}_{\text {gap}}\le \frac{c}{6}+\frac{1}{3}Δgap​≤6c​+31​ .
We introduce the use of reinforcement-learning (RL) techniques to the conformal-bootstrap program.We demonstrate that suitable soft Actor-Critic RL algorithms can perform efficient, relatively cheap highdimensional searches in the space of scaling dimensions and OPE-squared coefficients that produce sensible results for tens of CFT data from a single crossing equation.In this paper we test this approach in well-known 2D CFTs, with particular focus on the Ising and tricritical Ising models and the free compactified boson CFT.We present results of as high as 36-dimensional searches, whose sole input is the expected number of operators per spin in a truncation of the conformal-block decomposition of the crossing equations.Our study of 2D CFTs uses only the global soð2; 2Þ part of the conformal algebra, and our methods are equally applicable to higher-dimensional CFTs.When combined with other, already available, numerical and analytical methods, we expect our approach to yield an exciting new window into the nonperturbative structure of arbitrary (unitary or nonunitary) CFTs.
The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and operator product expansion (OPE) coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the extremal functional method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the two-dimensional Ising model based solely on the dimension of a single scalar quasiprimary--no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.
Here, we introduce spectral functions that capture the distribution of OPE coefficients and density of states in two-dimensional conformal field theories, and show that nontrivial upper and lower bounds on the spectral function can be obtained from semidefinite programming. We find substantial numerical evidence indicating that OPEs involving only scalar Virasoro primaries in a c > 1 CFT are necessarily governed by the structure constants of Liouville theory. Combining this with analytic results in modular bootstrap, we conjecture that Liouville theory is the unique unitary c > 1 CFT whose primaries have bounded spins. We also use the spectral function method to study modular constraints on CFT spectra, and discuss some implications of our results on CFTs of large c and large gap, in particular, to what extent the BTZ spectral density is universal.
In this Letter, we deploy for the first time reinforcement-learning algorithms in the context of the conformal-bootstrap program to obtain numerical solutions of conformal field theories (CFTs). As an illustration, we use a soft actor-critic algorithm and find approximate solutions to the truncated crossing equations of two-dimensional CFTs, successfully identifying well-known theories like the 2D Ising model and the 2D CFT of a compactified scalar. Our methods can perform efficient high-dimensional searches that can be used to study arbitrary (unitary or nonunitary) CFTs in any spacetime dimension.