Uniqueness of the $2d$ critical Ising model

This post is motivated by a request from JHEP to review a recent article by Anton de la Fuente. I am grateful to the author for stimulating correspondence.

 

The conformal bootstrap: analytic vs numerical

 

The critical Ising model is described by a unitary conformal field theory. In two dimensions, that theory is part of a family called minimal models, which can be exactly solved in the analytic bootstrap framework of Belavin, Polyakov and Zamolodchikov. Minimal models are parametrized by two coprime integers 2 ≤ p < q, they are unitary when q = p + 1, and the Ising model is the case (p, q)=(3, 4).

These 2d bootstrap results date back to the 1980s. More recently, the bootstrap method has been successfully used in higher dimensional CFTs, such as the 3d Ising model. While the basic ideas are the same, there are important technical differences between 2d and higher d.

Solving two-dimensional conformal field theories

This is the text of my habilitation defense, which took place on December 21st 2018. The members of the jury were Denis Bernard, Matthias Gaberdiel, Jesper Jacobsen, Vyacheslav Rychkov, Véronique Terras, Gérard Watts and Jean-Bernard Zuber.

In this habilitation defense, I gave a subjective overview of some recent progress in solving two-dimensional conformal field theories. I discussed what solving means and which techniques can be used. I insisted that there is much to discover about Virasoro-based CFTs, i.e. CFTs that have no symmetries beyond conformal symmetry. I claimed that we should start with CFTs that exist for generic central charges, because they are simpler than CFTs at rational central charges, and can nevertheless include them as special cases or limits. Finally, I argued that in addition to writing research articles, we should use various other media, in particular Wikipedia. 

 

Introduction

Two-dimensional CFTs are defined by the presence of a Virasoro symmetry algebra. This symmetry is sometimes enough for solving CFTs, and even classifying the CFTs that obey some extra conditions. For example, we can classify CFTs whose spaces of states decompose into finitely many irreducible representations of the Virasoro algebra: they are called minimal models. In some cases, Virasoro symmetry is not enough, but the CFT can nevertheless be solved thanks to additional symmetries. In particular, we can have symmetry algebras that contain the Virasoro algebra.
Let me discuss a few CFTs that I find particularly interesting. I will classify them according to their symmetry algebras, and characterize these algebras by the spins of the corresponding chiral fields. In this notation, the Virasoro algebra is \((2)\), as its generators are the modes of the energy-momentum tensor, which has spin \(2\). The sum of the spins of the generators gives you a rough idea of the complexity of an algebra.

The Im-flip condition in the two-dimensional Potts model

I have been using this blog for publishing the reviewer reports that I write for journals, since the journals typically do not publish the reports. However, the new journal SciPost Physics does publish the reports for accepted articles. I have recently reviewed an article by Gorbenko, Rychkov and Zan for SciPost Physics, and written about the experience: it would seem that I need not blog about that article, since my report is already online.

However, not everything that I have to say on the article made it into the report. I will now write on two calculations that I did: the first one is a test of one of the article’s main predictions in more general cases, the second one is a direct derivation and generalization of a technical result that they obtain in a roundabout way.

Uniqueness of Liouville theory

The original definition of Liouville theory by Polyakov in the 1980s was written in terms of a Lagrangian, motivated by applications to two-dimensional quantum gravity. In the 1990s however, Liouville theory was reformulated and solved in the conformal bootstrap approach. In this approach, the theory is characterized by a number of assumptions, starting with conformal symmetry. In order to actually define the theory, the assumptions have to be restrictive enough for singling out a unique consistent theory.

After assuming conformal symmetry, it is natural to make assumptions on the theory’s spectrum, i.e. its space of states. For any complex value of the central charge \(c\), the spectrum of Liouville theory is
\[\mathcal{S} = \int_{\frac{c-1}{24}}^{\frac{c-1}{24}+\infty} d\Delta\ \mathcal{V}_\Delta \otimes \bar{\mathcal{V}}_\Delta\ ,\]

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.

Reality of three-point structure constants in CFT, unitary or not

The reality of three-point structure constants in unitary CFT is a crucial ingredient of the numerical bootstrap in more than two dimensions. But what are the precise meaning and the proof of this property? Surely not all operators can have real three-point structure constants, since rescaling an operator by a complex scalar destroys this property. And the proof is not necessarily obvious, because the definition of unitarity as the existence of a positive definite scalar product is not directly related to three-point structure constants.

Fortunately I have received some explanations on these questions from Slava Rychkov. So here is what I understood from his arguments on unitary CFT, plus speculations of my own on non-unitary CFT.