Closure of the Operator Product Expansion in the Non-Unitary Bootstrap

TL;DR

Problem and scope: identify finite, closed sub-algebras of the operator product expansion in two-dimensional CFTs without assuming unitarity. Approach: perform numerical 2D bootstrap using the Gliozzi method to search for solutions with $[\phi]\times [\phi] = [1]$ or $[1]+[\phi]$, analyzing both Virasoro and global-block truncations. Findings: solutions with the vacuum in the algebra reproduce minimal-model data; additional lines are accessible via the Coulomb-gas formalism and include degenerate-operator families, with a notable line $c=32 h_\phi + 1$ and a log-CFT at $h=1,c=-2$; non-unitary features emerge in these loci. Significance: clarifies how degeneracy controls OPE closure in non-unitary bootstrap, links to the crossing-matrix structure, and provides a practical analytic-toolkit including a Mathematica notebook for crossing matrices and OPE coefficients.

Abstract

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.

Figures (11)

• Figure 1: A plot of the zero contours of the first, third, fifth, and seventh derivatives of the vacuum block. Black dots indicate minimal model operators with OPE closure $[\phi]\times[\phi] = [1]$. For reference, the values are $(c,h)=(0,2),(1/2,1/2),(7/10,3/2)$. Vertical lines are $c=0,1/2,7/10$.
• Figure 2: Same as figure \ref{['fig:vac+closure0c1']}, except in the region $c<0$, and showing the zero contours of the zeroth, second, and fourth derivatives of the vacuum block. Values are $(c,h)=(-3/5,3/4),(-25/7,5/4)$. Vertical lines are $c=-3/5,-27/5$.
• Figure 3: Zero contours of the functions \ref{['eq:vir+gli+eqn']} in two sub-regions of region $\mathcal{R}$. Squares are the first few minimal models with OPE closure $[\phi] \times [\phi] = [1]+[\phi]$. Triangles are minimal models with OPE closure $[\phi]\times[\phi]=[1]$. Dashed, black lines are null curves passing through these minimal model values. Points at which all contours intersect are putative solutions to the crossing equation \ref{['eq:vir+gli+eqn']}. The only intersections we find correspond precisely to minimal models, plus the $h=1, c=-2$ point. As explained in the text, the logarithmic CFT at $h=1, c=-2$ looks effectively to the numerics like a $[\phi]\times[\phi] = [1]+[\phi]$ operator algebra.
• Figure 4: The (log of the) smallest singular value of matrix $f^{(m,n)}_{\Delta_\phi,\Delta,L}$. Sharp dips, where this singular value vanishes, correspond to solutions to \ref{['eq:glob+gli+eqn']}. Filled circles are minimal models with OPE closure $[\phi] \times [\phi] = [1] + [\phi]$, along with the $(h=1,c=-2)$ log CFT. Open circles are minimal models with OPE closure $[\phi] \times [\phi] = [1]$. In this plot we take $N =111, ~M =112$.
• Figure 5: The (log of the) smallest singular value of the matrix $f^{(m,n)}_{\Delta_\phi,\Delta,L}$ for the OPE $[\phi]\times[\phi] = [1]$. Filled circles are minimal models with OPE closure $[\phi] \times [\phi] = [1]$. In this case the minimal model with $\Delta_\phi = 3/2$ is clearly found while the model with $\Delta_\phi = 5/2$ is found to within $\sim 10 \%$. In this plot we take $N = M = 36$. The reason we include a smaller number of operators than in figure \ref{['fig:single_op_global']} is that we find the matrix $f^{(m,n)}_{\Delta_\phi,\Delta,L}$ becomes numerically unstable more quickly as more operators are included than with the OPE $[\phi] \times [\phi] = [1] + [\phi]$.