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.

République numérique et Open access: les leçons d'une expérience

Après avoir consulté le public au sujet du projet de loi pour une République numérique, le gouvernement français a donné ses réponses aux suggestions qui ont été faites, et le conseil des ministres a adopté un projet de loi modifié en conséquence. Ayant participé à la consultation au sujet de l'article 9 du projet initial, intitulé "Libre accès aux publications scientifiques de la recherche publique", je voudrais ici évaluer dans quelle mesure l'exercice a été utile, et quelles leçons en tirer pour d'éventuelles consultations futures au sujet d'autres projets de loi.

Peut-on former les chercheurs à encadrer des thèses?

Travailler en équipe, communiquer, recruter et encadrer stagiaires, doctorants et postdocs: quelques facettes du métier de chercheur pour lesquelles ils n'ont souvent pas de méthodes de travail bien définies, faute d'y avoir été formés. Mais existe-t-il des méthodes à la fois assez souples pour marcher dans des domaines de recherche variés, et assez simples pour faire l'objet d'une brève formation? C'est ce que je me demandais en m'inscrivant à une formation proposée par le CNRS, intitulée "Accompagner et encadrer un doctorant", que j'ai suivie les 19 et 20 novembre en compagnie de neuf autres chercheurs.

La formation était faite par Simon Thierry, de la société Adoc Mètis, une société formée de trois jeunes anciens chercheurs et d'un doctorant. Ces personnes se sont donné pour mission de développer et de diffuser des méthodes de gestion de resources humaines pour l'enseignement supérieur et la recherche, méthodes inspirées de ce qui se fait dans les entreprises, mais aussi informées par une réflexion et une recherche spécifiques. Le résultat est, à première vue, assez convaincant, et je vais résumer certaines des idées et méthodes proposées.

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:

Research grants that do not finance research

In the quest of research funding bodies to get ever more impact (whatever that means) with ever less money, the French National Research Agency (ANR) has recently made an interesting move: introducing a grant called MRSEI for, as its name indicates, "building European and international scientific networks".

The avowed aim of this grant is to "facilitate the access of French researchers to European and international financing". This grant is therefore not directly for doing research, it is also not a kind of seed funding for starting a project in the hope of later obtaining more funding. It is a grant for obtaining larger grants.

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.

Les critères d'inscription à l'HDR en Physique à l'université Paris-Sud

Bien que le dossier d'inscription à l'HDR soit décrit en détail sur la page de l'université, rien n'indique selon quels critères un candidat sera autorisé à s'inscrire ou non. Le dossier comprend naturellement une liste de publications et des indications sur les étudiants déjà suivis, mais combien faut-il de publications? et quelle expérience d'encadrement est nécéssaire? J'ai reçu récemment quelques explications à ce sujet de Réza Ansari, le correspondant HDR pour la physique théorique entre autres.

Bourses doctorales de l'ED PIF: quelques précisions

L'École Doctorale Physique en Île-de-France (ED PIF) vient de procéder à son premier concours de recrutement des doctorants. Ayant présenté un candidat (malheureux) à ce concours, et ayant eu des échanges avec certains membres du jury, je peux donner quelques éléments sur le fonctionnement du concours, dans le but d'informer les candidats potentiels aux prochaines éditions.