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.

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.

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
\[\mathcal{S} = \int_{\frac{c-1}{24}}^{\frac{c-1}{24}+\infty} d\Delta\ \mathcal{V}_\Delta \otimes \bar{\mathcal{V}}_\Delta\ ,\]