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.

I will now discuss a few symmetry algebras, and a few CFTs with these symmetries, while citing a subjectively selected sample of works. Each CFT has a progress bar indicating how far it is from being solved: again, a subjective assessment.

• Beyond the Virasoro algebra, we have the affine Lie algebras \(\widehat{\mathfrak{sl}}_2\) and \(\widehat{\mathfrak{sl}}_3\), and the well-known \(W\)-algebra \(W_3\). The only algebra that is not well-known is the algebra of type \((2, 1, 1)\), which has no name. This is a generalization of both the Virasoro and \(\widehat{\mathfrak{sl}}_2\) algebras, which interpolates between them.
• Liouville theory is a well-known CFT with a continuous spectrum.
• The recently found non-diagonal CFTs involve infinitely many representations of the Virasoro algebra. By non-diagonal I mean that their primary fields can have nonzero spins.
• The \(H_3^+\) model is based on the affine \(\mathfrak{sl}_2\) algebra, it is more complicated than Liouville theory, but we found that you can deduce its solution from Liouville theory. This is the meaning of my upward-pointing arrows: reformulating CFTs in terms of simpler CFTs. Unfortunately, the arrow from the \(W_3\) algebra to the affine \(\mathfrak{sl}_3\) algebra works only in the critical level limit, this is why it is dashed.
• The interpolating CFTs do not have an official name: they interpolate between Liouville theory and the \(H_3^+\) model.

This leads to the three main messages of the rest of this talk:

• The progress bars show an emphasis on solving CFTs: I will discuss what this means.
• My most recent works appear at the bottom of the diagram, in CFTs with Virasoro symmetry only: there is still much to discover about such CFTs.
• All CFTs on the diagram have a parameter: the central charge of the Virasoro algebra. I have not drawn minimal models, which exist for rational central charges only. I will discuss the relations between generic and rational CFTs.

The last message about Wikipedia is unrelated, but I will mention it anyway because it is also an important aspect of my work.

It is good to build models, it is better to solve them

I have been talking of solving CFTs, and I have even drawn progress bars: let me now discuss what this means. Solving a theory means computing observables: we have to say what the observables are, and how we compute them. I will define the observables as

• the spectrum, i.e. the space of states together with the action of the symmetries on that space,
• the correlation functions on the sphere.

These observables are particularly adapted to applications to statistical physics. Other choices of observables are possible, in particular non-local observables such as boundaries and defects, entanglement entropy, or integrated correlation functions.
On the other hand, building a CFT means defining the observables, without necessarily computing them. In this Section I will discuss several methods for building and solving two-dimensional CFTs:

• the Lagrangian method,
• topological recursion,
• the conformal boostrap:
  
  • modular invariance,
  • crossing symmetry:
    
    • analytic,
    • semi-analytic,
    • numerical.

In the bootstrap method, there no analog of the Lagrangian or spectral curve: we use symmetry and consistency axioms for constraining observables.
In quantum field theory, we commonly use Lagrangians for building models, and also for solving them via perturbative computations of functional integrals. In two-dimensional CFT, there is a variant of the Lagrangian method that leads to exact calculations, called the Coulomb gas method. But this method only works for quantities that obey some sort of momentum conservation: it is good for building some CFTs, but not so good for solving them.
In topological recursion, the basic object is not a Lagrangian but a spectral curve. To an \(N\)-point function \(\left<\prod_{i=1}^N V_{\Delta_i}(z_i)\right>\), we can associate the non-commutative spectral curve \[\begin{aligned} y^2 = \sum_{i=1}^N\left(\frac{\Delta_i}{(x-z_i)^2}+\frac{\beta_i}{x-z_i}\right) \quad \text{with} \quad [y, x] =1\ .\end{aligned}\] This spectral curve in principle allows you to perturbatively compute the \(N\)-point function [SR + Chekhov + Eynard 2012]. The computations are not very efficient, but the role of the spectral curve is rather to relate CFT with other integrable models such as matrix models. A particularly powerful version of this idea occurs when the central charge is one, which relates correlation functions to solutions of the Painlevé VI differential equation [Gamayun + Iorgov + Lisovyy 2012], [SR + Eynard 2013].
Let me come to the conformal bootstrap method: the method which I use for actually solving models. The simplest incarnation of the method is called the modular bootstrap. In the modular bootstrap, we use modular invariance of the torus partition function for deriving an equation on the spectrum \(S\) alone, i.e. an equation that does not involve correlation functions: \[\begin{aligned} \operatorname{Tr}_S e^{2\pi i\tau(L_0-\frac{c}{24})} e^{2\pi i\bar\tau(\bar L_0-\frac{c}{24})} = \operatorname{Tr}_S e^{-\frac{2\pi i}{\tau}(L_0-\frac{c}{24})} e^{-\frac{2\pi i}{\bar\tau}(\bar L_0-\frac{c}{24})}\end{aligned}\] where \(L_0\) is an element of the Virasoro algebra, \(c\) the central charge, and \(\tau\) the modulus of the torus. The modular bootstrap has been used successfully for classifying minimal models. However, it is not sufficient for having a consistent CFT, and actually not even necessary, as some CFTs such as generalized minimal models do not exist on the torus. And the torus partition function does not always encode enough information for recovering the space of states. In particular,

• In Liouville theory the partition function does not even depend on the central charge.
• In the Potts model, the spectrum contains indecomposable but reducible representations of the Virasoro algebra, whose structures are not fully determined by the partition function. And their multiplicities in the partition function can be fractional and/or negative.

Let me now focus on crossing symmetry of the sphere four-point function, a more complicated equation that involves not only the spectrum \(S\), but also the correlation functions, more precisely the three-point structure constants \(C\):

 The use of crossing symmetry comes in different flavours, depending on how much control we have on the spectrum:

• In the analytic bootstrap, sums over the spectrum have finitely many nonvanishing terms (due to fusion rules), and structure constants can be determined analytically. This is the case in minimal models, and in all 2d CFTs that appeared in my map.
• In what could be called the semi-analytic bootstrap, we know or guess the spectrum, and numerically solve crossing symmetry for the three-point function. We did this in the Potts model in [SR + Picco + Santachiara 2016]. It was later understood that our spectrum was only an approximation of the Potts model’s spectrum, and that it was actually the spectrum of another non-diagonal CFT.
• In the numerial bootstrap, we numerically determine both the spectrum and correlation functions. This flavour of the bootstrap is commonplace in higher than two dimensions. 2d CFTs that were solved analytically provide testing grounds for such techniques.
  

“There’s plenty of room at the bottom”

Let me turn to the simplest possible CFTs: those that only have a Virasoro symmetry algebra. I will will argue that the space of these Virasoro-CFTs is rich and poorly known. First I should define what a Virasoro-CFT is, as by definition all two-dimensional CFTs have Virasoro symmetry. I do not want to ignore or to break a larger symmetry algebra: I want CFTs whose spectrums are small enough that they have a chance of being solvable. For this I require that the multiplicities of indecomposable representation of the Virasoro algebra in the spectrum are bounded – not just finite, bounded. By comparison, minimal models are defined by the condition that the spectrum is made of finitely many irreducible representations: I am now extending this defintion a lot, but hopefully not too much.
Let me discuss some known Virasoro-CFTs. (The bound on multiplicities is \(1\) or \(2\) in all my examples.) I will plot them in the complex plane of central charges, as the central charge of the Virasoro algebra determines some important properties of CFTs, starting with their existence.

• Liouville theory (blue), with its continuous spectrum: the initial Lagrangian definition held for \(c\geq 25\), but the theory can be analytically extended to the whole complex plane minus the half-line \((-\infty, 1)\), and then to that half-line too although not analytically.
• Minimal models (green bars) are parametrized by two coprime integers, \[\begin{aligned} c_{p, q} = 1 - 6\frac{(p-q)^2}{pq} \quad \text{with} \quad 2\leq p < q \end{aligned}\] Unitary minimal models form a small subset (\(q=p+1\)), and unitarity plays no role in classifying minimal models. For each allowed central charge, there are actually \(1,2\) or \(3\) distinct minimal models. Larger bars mean fewer representations in the spectrum.
• Recently found non-diagonal CFTs with infinite spectrums (non-hatched area) [SR + Migliaccio 2017]: they exist for \(\Re c\leq 13\). For each \(c\neq 1\) we actually have two distinct CFTs, which are related by analytic continuation around \(c=1\). We stumbled upon these CFTs when trying to solve the Potts model [SR + Picco + Santachiara 2016], but we now know that they differ from the Potts model.
• This brings us to the \(Q\)-state Potts model (pink): a CFT with a discrete, non-diagonal, really complicated spectrum. When the parameter \(Q\) spans the complex plane, the corresponding central charge spans a nice-looking region: \[\begin{aligned} c= 13 - 6\beta^2 -\frac{6}{\beta^2}\quad \text{with} \quad \frac12 \leq \Re \beta^2\leq 1 \quad \text{such that} \quad Q = 4\cos^2\pi \beta^2 \end{aligned}\] In contrast to all other CFTs that I mentioned, the Potts model has little prospect of an analytic solution, although it may be possible to analytically determine the spectrum.

For a given central charge, we can have up to \(6\) different Virasoro-CFTs. And I did not even include free bosonic CFTs, of which there exist infinitely many at each central charge. Beyond that, there remains much to discover: we know some crossing-symmetric four-point functions that do not belong to any of these CFTs.

Generic cases are simpler than special cases

Historically, the solving of two-dimensional CFTs has started with rational theories: theories whose spectrums are made of finitely many representations of the Virasoro algebra. But are these really the simplest theories?
To answer this question, let me consider fusion rules of degenerate representations. Let me define a degenerate representation as a representation whose fusion product with any irreducible representations is a sum of finitely many irreducible representations. Let me accept that for generic central charges, there exist two degenerate representations \(R_{\langle 2,1\rangle}\) and \(R_{\langle 1,2\rangle}\) whose fusion products with any Verma module are sums of two Verma modules: \[\begin{aligned} R_{\langle 2,1\rangle} \times V_P = \sum_\pm V_{P\pm \frac{\beta}{2}} \quad , \quad R_{\langle 1,2\rangle} \times V_P = \sum_\pm V_{P\pm \frac{1}{2\beta}}\ .\end{aligned}\] Here the momentum \(P\) and parameter \(\beta\) are functions of the conformal dimension and central charge respectively, chosen for making fusion products simple. If I assume that the fusion product is associative, then I can deduce that fusion products of degenerate representations are degenerate, and that there exist degenerate representations \(R_{\langle r, s\rangle}\) for \(r,s\in\mathbb{N}^*\) with the momentums \(P_{\langle r,s\rangle} =\frac12(\beta r -\beta^{-1}s)\). I can also find the fusion products of all these degenerate representations with one another and with Verma modules, in particular \[\begin{aligned} R_{\langle r_1,s_1 \rangle} \times R_{\langle r_2,s_2 \rangle} = \sum_{r_3\overset{2}{=}|r_1-r_2|+1}^{r_1+r_2-1}\ \sum_{s_3\overset{2}{=}|s_1-s_2|+1}^{s_1+s_2-1} R_{\langle r_3,s_3 \rangle}\ , \qquad r_i,s_i\in\mathbb{N}^*\ . \label{rrsr}\end{aligned}\] These are the fusion rules of generalized minimal models, valid at generic central charges.
Now let me consider cases when two degenerate representations coincide: say \(R_{\langle r,s\rangle} = R_{\langle p-r, q-s\rangle}\). This constrains the central charge to be \(c_{p,q}\), and we find a finite set of representations that is closed under fusion: \[\begin{aligned} R_{\langle r_1,s_1 \rangle} \times R_{\langle r_2,s_2 \rangle} = \sum_{r_3\overset{2}{=}|r_1-r_2|+1}^{\min(r_1+r_2,2p-r_1-r_2)-1}\ \sum_{s_3\overset{2}{=}|s_1-s_2|+1}^{\min(s_1+s_2,2q-s_1-s_2)-1} R_{\langle r_3,s_3 \rangle}\ , \quad \left\{\begin{array}{l} 1\leq r_i\leq p-1 \\ 1\leq s_i\leq q-1 \end{array}\right.\end{aligned}\] These are the fusion rules of minimal models. They can be deduced from the fusion rules of generalized minimal models, but they are more complicated, as they depend on the central charge via \(p,q\). The structures of degenerate representations are also more complicated in minimal models. However, their structures are not really needed for computing correlation functions: what matters is fusion rules. The situation is similar with larger symmetry algebras such as W-algebras.
So it is tempting to study generic central charges first, and to recover the rational cases as limit. We saw that this could be done at the level of fusion rules, where taking the limit is straighforward although not particularly easy. However, at the level of correlation functions, this can be more subtle for several reasons:

• The finiteness of the limit can require cancellations of divergences: this happens with conformal blocks in Zamolodchikov’s recursive representation.
• Representations can become logarithmic or otherwise more complicated.
• There can be several different ways of taking the limit \(c\to c_{p,q}\), due to identities such as \(R_{\langle r,s\rangle} = R_{\langle p-r, q-s\rangle}\).

The subtleties are not bugs, but features: they allow rich structures to hide in relatively simple theories that exist at generic central charges [SR 2018].

To reach many readers, write in Wikipedia

In a few words, I would like to say why writing in Wikipedia is part of my work. Results are useless if they are not communicated effectively, the question is how best to do it. Let me consider a few written media that we can use:

By open media I mean texts that are not only publicly available, but also written in a widely readable style. In principle all these media can be useful, depending on what we want to say and to whom. We are certainly biased towards writing research articles, because careers are built on them. This does not prevent us from using other media though.
Among these other media, Wikipedia stands out because everybody reads it, but very few physicists write in it. Or, to reformulate this more positively, whatever you write there will find many readers. This is supported by data: see the pageview statistics for four Wikipedia articles that I have created or rewritten. The 10-20 daily views of each article mean thousands of yearly readers. And the number of readers is not very sensitive to an article’s quality: I did not see much change after greatly improving the article on Liouville theory. So Wikipedia shapes how two-dimensional CFTs are viewed by people like students or researchers from other fields, whether the articles are good or bad. Making them good is therefore important.
But too few physicists write in Wikipedia. I tried to write good articles, but I would admit that they are probably biased towards my own points of view, you are all welcome to help rectify that.