Finite operator product expansions in two-dimensional CFT

While the conformal bootstrap method has recently enjoyed the wide popularity that it deserves, its applications have been mostly restricted to unitary conformal field theories. (By definition, in a unitary theory, there is a positive definite scalar product on the space of states, such that the dilatation operator is self-adjoint.) Unitarity brings the technical advantage that three-point structure constants are real, so squared structure constants are positive, leading to bounds on allowed conformal dimensions. However, dealing with non-unitary theories using similar methods is surely possible, at the expense of having the signs of squared structure constants as extra discrete variables. And unitarity is sometimes assumed even in cases where it brings no discernible technical benefit, such as in studies of torus partition functions, where multiplicities are positive integers whether the theory is unitary or not.

So it is refreshing that, in their recent article, Esterlis, Fitzpatrick and Ramirez apply the conformal bootstrap method to non-unitary theories.
Their main ideas are to look for OPEs that involve only finitely many primary fields, and to focus on two-dimensional theories with local conformal symmetry (i.e. a Virasoro symmetry algebra, rather than just \(s\ell_2\)). The focus on two-dimensional theories allow them to interpret their results in terms of minimal models and/or Coulomb gas integrals. In other words, they validate their numerical bootstrap results by comparing them with the analytic bootstrap results that have been accumulating since the 1984 article by Belavin, Polyakov and Zamolodchikov. The hope is of course that their numerical bootstrap method might be useful in cases where no analytic results are available.

The methods

 

The OPE \(\phi\times \phi\) is constrained by numerically requiring crossing symmetry of the four-point function \(\left<\phi\phi\phi\phi\right>\). In the Gliozzi bootstrap method, this implies that a certain matrix has a vanishing kernel. But Esterlis, Fitzpatrick and Ramirez convincingly argue that it is better in practice, although equivalent in principle, to require that matrix to have a vanishing singular value. This is why they have nice plots of how singular values depend on \(\Delta_\phi\).

They follow the good practice of including some of their code in the arXiv files, and this code has already been used in another article. Of course, it would be even better to include all the necessary code, and to avoid using a closed software such as Mathematica.

There is also a brave attempt at analytically solving crossing symmetry using explicit formulas for the fusing matrix. Since this is limited to cases that involve degenerate fields, this approach is not yet very convincing, given that the results can more easily be deduced from the fusion rules, as I will argue below.

The results

 

Let me summarize the results of Esterlis, Fitzpatrick and Ramirez. The idea is to look for finite OPEs of the types \[\phi \times \phi = 1 + \phi \quad , \quad \phi\times \phi = 1 \quad , \quad \phi\times \phi = 1 + \epsilon \quad , \quad \phi \times \phi = \epsilon\ ,\] where \(\phi,\epsilon\) are spinless Virasoro primary fields of arbitrary dimensions \(\Delta_\phi,\Delta_\epsilon\), and \(1\) is a field of dimension zero. In order to write the results, it is convenient to introduce the dimensions \(\Delta_{r,s}=2h_{r,s}\) with \[h_{r, s} = \frac14\left[ (\beta - \beta^{-1})^2 - (\beta r - \beta^{-1}s)^2\right] \quad \text{with} \quad c = 1 - 6(\beta -\beta^{-1})^2\ .\] The corresponding primary fields \(\phi_{r,s}\) are degenerate if \(r,s\in\{1,2,3,\dots\}\). Here \(c\) is the central charge, and minimal models correspond to the values \[c_{p, p'} = 1-6\frac{(p-p')^2}{pp'} \quad \text{with} \quad p,p'\in \{2,3,4,\dots\} \ , \quad \text{i.e.} \quad \beta^{2} =\frac{p}{p'}\ .\] Up to the symmetry \(\phi_{r,s} = \phi_{p'-r,p-s}\), the only solutions are found to be \[\renewcommand{\arraystretch}{1.3}  \begin{array}{l|l|l} \text{Case} & \text{Conditions on } \phi,\epsilon & \text{Condition on } c \\ \hline \phi \times \phi = 1 + \phi & \phi = \phi_{1,3} & c = c_{5,p'} \\ \phi \times \phi = 1 & \phi = \phi_{1, 1} & \\ \phi \times \phi = 1 & \phi = \phi_{1, p-1} & c = c_{p, p'} \\ \phi \times \phi = 1 + \epsilon & \phi = \phi_{1, 2} ,\ \epsilon = \phi_{1,3} & \\ \phi \times \phi = 1 + \epsilon & \phi = \phi_{1, p-2} ,\ \epsilon = \phi_{1,3} & c = c_{p, p'} \\ \phi \times \phi = \epsilon & \phi = \phi_{\frac12, \frac12},\ \epsilon = \phi_{0,0} & \end{array}\] In the first five series of solutions, all fields are degenerate, and the solutions could likely be deduced from the degenerate field fusion rules, \[\phi_{r_1,s_1} \times \phi_{r_2,s_2} = \sum_{r=|r_1-r_2|+1}^{r_1+r_2} \sum_{s=|s_1-s_2|+1}^{s_1+s_2} \phi_{r,s}\ ,\] where the sums run in increments of \(2\), and if \(c=c_{p,p'}\) the symmetry \(\phi_{r,s} = \phi_{p'-r,p-s}\) should be taken into account, resulting in fewer fields in fusion products. For example, in the case \(\phi \times \phi = 1 + \phi\), it is natural to try \(\phi = \phi_{1,3}\), whose fusion product with itself is in general \(\phi_{1,3}\times \phi_{1, 3} = 1 + \phi_{1, 3} + \phi_{1, 5}\). We then need an extra null vector that would kill the field \(\phi_{1,5}\). The existence of an extra null vector implies \(c = c_{p,p'}\). Then, for the extra null vector of \(\phi_{1, 3} = \phi_{p'-1,p-3}\) to kill \(\phi_{1, 5}\) but not \(\phi_{1,3}\), we need \(4 < p<6\) i.e. \(p=5\).

It is likely that, with similar arguments, we can derive the first five series of solutions under the assumption that all fields are degenerate. But the case \(\phi \times \phi = \epsilon\) is different, and does not include any degenerate fields. As the authors point out, it can however be explained by a Coulomb gas construction of the corresponding four-point function. I would add that the presence of fields with half-integer indices reminds me of the recently proposed OPE \(\phi_{0,\frac12}\times \phi_{0,\frac12} = \sum_{m,n\in\mathbb{Z}} \phi_{2m,n+\frac12}\) in the Potts model (where however fields are non-diagonal unless $m=0$).

It is very possible that exploring further cases would yield genuinely new results, that could well be inaccessible to other known methods. An obvious next case would be \(\phi\times \phi = \epsilon +\epsilon'\). The presence of non-diagonal fields could also be allowed. The existence of consistent CFTs with finite OPEs but no degenerate fields would obviously be very interesting.