Bootstrapping non-unitary CFTs

Abstract

In this letter, we present an evolutionary algorithm-based approach to bootstrapping the spectrum of general conformal field theories (CFTs). Starting from a trial spectrum, we invert the crossing equations to extract the corresponding operator product expansion (OPE) coefficients. The statistical distribution of these coefficients, obtained by sampling over cross-ratios, provides a quantitative measure of how closely crossing symmetry is satisfied. We then define a reward function based on this statistic and employ a genetic algorithm to search for spectra that maximize the reward. A key advantage of our framework is that it does not rely on unitarity. We illustrate the method using Virasoro blocks with central charge $c<1$, where the optimal solutions align with the known minimal models. More broadly, this approach - solving systems of nonlinear constraints by exploiting the statistical properties of associated linear variables - provides a general strategy for broad class of bootstrap problems.

Figures (6)

• Figure 1: The reward distributions of minimal model processes inside the four regions where we perform the search.
• Figure 2: Search result in $c<1$ region with single internal operator. The blue dots are search results, while the black ones are analytic values from minimal models.
• Figure 3: Search results for non-unitary CFTs with $2$ and $3$ internal operators with similar $\left(h_{int,1}, c, h_{ext}\right)$ as Yang-Lee CFT.
• Figure 4: Search result of different truncated space where only processes with $5$ internal states are present.
• Figure 5: Search results around Yang-Lee CFT region with $2$ and $3$ internal operators in projection to ($O$, $h_{int,2}$) where $O\in\{h_{int,1}, h_{ext}, c\}$. The line of minimal model represents minimal models with $p,q\leq 200$.