#### Degenerate fields in \(d=2\) CFT

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

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.

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Degenerate fields in \(d=2\) CFT

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

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

Three years and three major revisions after it first appeared on Arxiv and GitHub (why GitHub? see this blog post), my review article on two-dimensional conformal field theory may be mature enough for appearing in book form. But with which publisher?

To answer this question, I should first say why I would want to have a book in the first place, since the text is already on Arxiv.

To answer this question, I should first say why I would want to have a book in the first place, since the text is already on Arxiv.

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?

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?

In the tense negotiations between the German consortium DEAL and Elsevier, there is a new twist: on February 13th, Elsevier announced that it was restoring the access of the affected German institutions to its journals.

Elsevier’s two explanations for this maneuver fall short of being convincing. The first explanation, given to Nature, is that “it is customary [...] to retain access to content after a contracted period is concluded and as long as renewal discussions are ongoing”. Why then cut off access in January, and restore it in February?

Elsevier’s two explanations for this maneuver fall short of being convincing. The first explanation, given to Nature, is that “it is customary [...] to retain access to content after a contracted period is concluded and as long as renewal discussions are ongoing”. Why then cut off access in January, and restore it in February?

The debate about green versus gold open access leaves aside a more fundamental difference: that between legal open access and pirate open access. This difference is essential because, as Bjorn Brembs put it,

In terms of making the knowledge of the world available to the people who are the rightful owners, [pirate] Alexandra Elbakyan has single-handedly been more successful than all [legal] open access advocates and activists over the last 20 years combined.With Sci-Hub, pirate open access is so successful that one might wonder whether legal open access is still needed. The obvious argument that pirate open access is parasitic and therefore unsustainable, because someone has to pay for scientific journals, is easily disposed of: with up-to-date tools, journals could cost orders of magnitude less than they currently do, and be financed by modest institutional subsidies. A better reason why pirate open access is not enough is that it is subject to technical and legal challenges. This makes it potentially precarious, and unsuited to uses such as content mining.

Since several years ago, Wikipedia is being widely used by academics. As a theoretical physicist, I often use it as a quick reference for mathematical terminology and results. Wikipedia is useful in spite of its many gaps and flaws: there was no general article on two-dimensional conformal field theory until I started one recently, the article on minimal models is itself minimal, and googling conformal blocks sends you to a discussion on StackExchange, since there is nothing on Wikipedia.

The paradox is that many academics see these gaps and flaws in the coverage of their own favourite subjects, yet do nothing to correct them. Let me discuss three possible reasons for this passivity: fear of Wikipedia, lack of time, and laziness.

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The jungle outside the ivory tower

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Attracting and retaining academic contributors has long been recognized as a challenge by Wikipedians, to the extent that there are guidelines on how to do it.

The paradox is that many academics see these gaps and flaws in the coverage of their own favourite subjects, yet do nothing to correct them. Let me discuss three possible reasons for this passivity: fear of Wikipedia, lack of time, and laziness.

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