Minimal string theories and their limits

This text is an introduction to recent work by Collier, Eberhardt, Mühlmann and Rodriguez:

• arXiv:2309.10846 – The Virasoro Minimal String

• arXiv:2409.18759 – The complex Liouville string: the worldsheet

• arXiv:2410.07345 – The complex Liouville string: the matrix integral

I am grateful to the authors for helpful discussions and correspondence, and to SciPost for invitations to review two of these articles. (As always with SciPost, the reviews are online.) I am also grateful to the string theory group at IPhT Saclay for inviting me to discuss this subject in their journal club.

 

Minimal string theories

In the worldsheet approach, a string theory may be constructed from a two-dimensional conformal field theory with the central charge c = 26, where we consider primary fields of conformal dimensions $\Delta=\bar{\Delta}=1$. These conditions on the central charge and conformal dimensions, called respectively criticality and marginality, are necessary for the string theory to be independent from the parametrization of the worldsheet. Alternatively, if we take the physical spacetime to be the worldsheet itself rather than the target space, we obtain a model of two-dimensional quantum gravity. In this interpretation, criticality allows gravity to remain topological at the quantum level.

Which conformal field theories give rise to string theories that are simple enough to be tractable, but complicated enough to be interesting? A simple recipe is to take a product of two theories: a theory called the matter CFT, which can in principle be arbitrary, and Liouville theory, which allows arbitrary complex values of c and Δ, allowing us to fulfill criticality and to build a marginal field from any diagonal field of the matter CFT. The resulting string theory is tractable provided the matter CFT is, given that Liouville theory is exactly solved.

Does this covariant function belong to some 2d CFT?

In conformal field theory, correlation functions of primary fields are covariant functions of the fields’ positions. For example, in two dimensions, a correlation function of N diagonal primary fields must be such that
$$\begin{aligned} F(z_1,z_2,\cdots , z_N) = \prod_{j=1}^N {|cz_j+d|^{-4\Delta_j}}  F\left(\tfrac{az_1+b}{cz_1+d},\tfrac{az_2+b}{cz_2+d},\cdots , \tfrac{az_N+b}{cz_N+d}\right) \ , \end{aligned}$$
where z_j ∈ ℂ are the fields’ positions, Δ_j ∈ ℂ their conformal dimensions, and $\left(\begin{smallmatrix} a& b \\ c& d \end{smallmatrix}\right)\in SL_2(\mathbb{C})$ is a global conformal transformation. In addition, there are nontrivial relations between different correlation functions, such as crossing symmetry. But given just one covariant function, do we know whether it belongs to a CFT, and what can we say about that CFT?

In particular, in two dimensions, do we know whether the putative CFT has local conformal symmetry, and if so what is the Virasoro algebra’s central charge?

Since covariance completely fixes three-point functions up to an overall constant, we will focus on four-point functions i.e. N = 4. The stimulus for addressing these questions came from the correlation functions in the Brownian loop soup, recently computed by Camia, Foit, Gandolfi and Kleban. (Let me thank the authors for interesting correspondence, and Raoul Santachiara for bringing their article to my attention.)

Doesn’t any covariant function belong to multiple 2d CFTs?

In conformal field theory, any correlation function can be written as a linear combination of s-channel conformal blocks. These conformal blocks are a particular basis of smooth covariant functions, labelled by a conformal dimension and a conformal spin. (I will not try to say preciely what smooth means.) In two dimensions, we actually have a family of bases, parametrized by the central charge c, with the limit c = ∞ corresponding to global conformal symmetry rather than local conformal symmetry.

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.

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.