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.

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

On single-valued solutions of differential equations

This post is about the issue of solving a nonlinear matrix equation that I raised on MathOverflow. This matrix equation determines the existence of single-valued solutions of certain meromorphic differential equations. The motivating examples are the BPZ differential equations that appear in two-dimensional CFT. For more details on these examples, see my recent article with Santiago Migliaccio on the analytic bootstrap equations of non-diagonal two-dimensional CFT.

With weight-shifting operators, \(d\neq 2\) looks increasingly like \(d=2\) in CFT

When working on conformal field theory, your life is very different depending on whether the dimension is two or not. In \(d=2\) you have that infinite-dimensional symmetry algebra called the Virasoro algebra, and in some important cases such as minimal models you can classify your CFTs, and solve them analytically. In \(d\neq 2\), your symmetry algebra is finite-dimensional, and you mostly have to do with numerical results. This not only makes you code a lot, but also incites you to make technical assumptions that are physically restrictve, such as unitarity.

Degenerate fields in \(d=2\) CFT

What makes \(d=2\) CFT solvable in many cases is the existence of degenerate primary fields.

Differential equations from fusion rules in 2d CFT

In two-dimensional conformal field theory, correlation functions are partly (and sometimes completely) determined by the properties of the fields under symmetry transformations. In particular, correlation functions of primary fields are relatively simple, because by definition primary fields are killed by the annihilation modes of the symmetry algebra. On top of that, there exist degenerate primary fields that are killed not only by the annihilation modes, but also by some combinations of creation modes. As a result, correlation functions that involve degenerate primary fields sometimes obey nontrivial differential equations, for example BPZ equations. Usually, these equations are deduced from the relevant combinations of creation modes, called null vectors.
Determining null vectors in representations of a symmetry algebra is often complicated, as the algebraic structures of the relevant algebras and of their representations can themselves be complicated. Even in the case of the Virasoro algebra, it is not easy to explicitly determine null vectors. It is however much easier to determine which representations do have null vectors, using the fusion product. For example, if we know degenerate representations \(R_{(1,1)}\) and \(R_{(2,1)}\) with null vectors at levels \(1\) and \(2\) respectively, we can deduce that the fusion product \(R_{(2,1)}\times R_{(2,1)}\) is degenerate and contains \(R_{(1,1)}\). The remainder of \(R_{(2,1)}\times R_{(2,1)}\) must therefore be a degenerate representation, which can be identified as \(R_{(3,1)}\), and has a null vector at level \(3\). (See Section 2.3.1 of my review article for more details.)
An important idea is therefore that it is not the structures of the algebras and representations that matter, but rather the structure of the category of representations, in other words their fusion products. This idea has in particular been developed in the works of Fuchs, Runkel and Schweigert. But how does this help us compute correlation functions, and determine the differential equations that they obey? In other words, can we determine differential equations from fusion products, without computing null vectors?

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.

Free bosons and Virasoro null vectors

In a recent article, Manabe and Sulkowski have proposed a method for deriving Virasoro null vectors, starting with certain deformed matrix integrals. In this blog post I will look for a conformal field theory interpretation of this method.

 

Quick reminders on Virasoro null vectors.

 

A null vector of the Virasoro algebra is labelled by two integers \(r,s\geq 1\), whose product is the level of the null vector. This null vector occurs in the Verma module with a specific conformal dimension \(\Delta_{r,s}\), and it can be written as
\[|\chi_{r,s}\rangle = L_{r,s} |\Delta_{r,s}\rangle\] where \(|\Delta_{r,s}\rangle\) is the primary state of our Verma module, and \(L_{r,s}\) is a level \(rs\) creation operator.

Relations between conformal field theories with affine and $W$-algebra symmetries

This is a commentary of the recent article by Creutzig, Hikida and Ronne, which I was asked to review for the Journal of High-Energy Physics. I am grateful to the authors for helpful correspondence.

It has been known for a long time that $W$-algebras can be obtained from affine Lie algebras by Drinfeld-Sokolov reduction. The reduction eliminates a number of generators of the affine Lie algebra, leaving a $W$-algebra with fewer generators (but more complicated relations). The reductions of algebras are most useful when they can be promoted into relations between correlation functions of conformal field theories. For example, the reduction from the $\widehat{\mathfrak{sl}}_2$ affine Lie algebra, to the Virasoro algebra, can be promoted into a relation between correlation functions of the $H_3^+$ model and Liouville theory, two CFTs whose symmetry algebras are $\widehat{\mathfrak{sl}}_2$ and Virasoro respectively.

Generalizing the $H_3^+$-Liouville relation to models with larger symmetry algebras could be helpful for understanding, or even solving, such models. In order to find such generalizations, there are two approaches:

Toric Virasoro conformal blocks

This is a commentary of a recent article by Nikita Nemkov, based on the report I wrote for the journal JHEP. I am making this text public because it might be useful to the community, but is kept confidential by the journal (as is unfortunately common practice). This blog post omits the parts of the report that deal with technical details and suggested improvements. Only the general commentary is reproduced, in a slightly modified form. Making it public implies renouncing anonymity. But I had already renounced anonymity by engaging in private correspondence with the author while studying his article. This made the process easier and more efficient, and I am grateful to Nikita Nemkov for his prompt and detailed answers. As a result, the article underwent very important improvements between Arxiv's second version and the JHEP version. I had never seen an author make such extensive improvements following a referee's suggestions.

Virasoro conformal blocks in closed form

In a recent article, Perlmutter investigated closed-form expressions for Virasoro conformal blocks. As a complement to that article, let me discuss what is known on such expressions, and what they are good for.

The definition of conformal blocks is the subject of an interesting discussion at Physics.StackExchange. Basically, conformal blocks are the universal building blocks of correlation functions, and are determined by conformal symmetry.

Conformal blocks at Physics.StackExchange

I realized that the first Google hit for "conformal blocks" was a discussion at Physics.StackExchange about "A pedestrian explanation of conformal blocks".
This discussion is quite interesting and there are a number of good quality answers. But these answers were written in the span a few days, and they do not amount to a complete or satisfactory explanation of conformal blocks.

Such an explanation should probably be written as a Wikipedia article. But before writing it, one should probably rewrite the article on conformal field theory, and more generally build a decent set of articles on that subject. So, as a quick fix, I just added my own explanation of conformal blocks to the discussion in question.

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.

Modular invariance in non-rational CFT

Modular invariance of the torus partition function is often the first -- and sometimes the only -- thing people check about a proposed CFT. There is a good reason for this: computing the torus partition function only requires knowing the characters of the representations which appear in the spectrum, whereas other consistency checks, such as crossing symmetry of the sphere four-point function, involve much more complicated conformal blocks.

However, modular invariance is neither sufficient, nor necessary for a CFT to be consistent.