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.
These fields were originally characterized in terms of the corresponding representations of the Virasoro algebra: a primary field is degenerate if the corresponding representation has a singular vector. A better characterization, that works not only for the Virasoro algebra but also for larger symmetry algebras such as W-algebras, is that a primary field is degenerate if its OPE with any other field yields only finitely many primary fields. In other words, it is better to characterize degenerate representations not in terms of their structure, but in terms of their fusion products. From this characterization, it follows that OPEs of degenerate fields yield degenerate fields. So, while there exist infinitely many degenerate fields, it is enough to study a few basic fields that generate all the others via their OPEs. In Virasoro-symmetric CFT, there are \(2\) basic degenerate fields called \(V_{(2,1)}\) and \(V_{(1,2)}\), and infinitely many degenerate fields called \(V_{(r,s)}\) with \(r,s\in\mathbb{N}^*\).

There are two types of results that can be derived using degenerate fields: universal results on conformal blocks, and results on correlation functions of specific models. These results are obtained by inserting a degenerate field in a block or correlation function of interest, see this article for examples of results on conformal blocks, and this review for computing correlation functions in Liouville theory and minimal models. Degenerate fields can be used irrespective of their appearance in the spectrum of a given model: in particular, they can be used for solving Liouville theory, whose spectrum does not include any degenerate field.

OPEs of our basic degenerate fields with a primary field of momentum \(P\) look like \[V_{(2,1)} V_P \sim V_{P-\frac{b}{2}} + V_{P+\frac{b}{2}} \quad , \quad V_{(1,2)} V_P \sim V_{P-\frac{1}{2b}} + V_{P+\frac{1}{2b}}\] where the momentum \(P\) and parameter \(b\) are functions of the conformal dimension and central charge respectively. Analytic bootstrap equations that are deduced from degenerate fields relate momentums by shifting them as in the above OPEs. In particular, for \(b^2\notin \mathbb{R}-\mathbb{Q}\), the shifts involve two incommensurable quantities \(\frac{b}{2}\) and \(\frac{1}{2b}\), and this is why Liouville theory’s correlation functions can be completely determined.

Weight-shifting operators in \(d\neq 2\) CFT

In a recent article, Karateev, Kravchuk and Simmons-Duffin have introduced weight-shifting operators in \(d\neq 2\) CFT, whose similarity with \(d=2\) degenerate fields is striking. These operators correspond to finite-dimensional representations of the conformal algebra, so an OPE of such an operator with any other field only involves finitely many primary fields. In any given dimension, there exists a finite set of basic weight-shifting operators that generate all the others: for example, in \(d=3\), this set is made of just one operator, which is associated to the spinor representation. And weight-shifting operators can be used in any CFT, whether or not they belong to the spectrum.

The main motivation for introducing weight-shifting operators is to simplify calculations of conformal blocks, more specifically to reduce conformal blocks that involve spinning fields, to simpler conformal blocks. Then a natural question is whether weight-shifting operators could also contribute to solving specific models. To answer this question, let us look at the OPE of a weight-shifting operator \(W_j\) with a primary field \(V_\Delta\) of conformal dimension \(V_\Delta\). According to eq. (2.12), the OPE shifts dimensions by half-integers, \[W_j V_\Delta \sim \sum_{i=-j}^j V_{\Delta+i}\] where \(j\in\frac12 \mathbb{N}\) describes the properties of \(W_j\) under conformal transformations, and \(i\) runs by increments of one. It follows that analytic bootstrap equations deduced from weight-shifting operators would shift dimensions by integers. But in reasonably non-trivial CFTs, differences of field dimensions are not always integer. So there is little hope of fully solving CFTs by adapting the \(d=2\) analytic bootstrap method. Adapting this method would only be useful in CFTs with series of fields whose dimensions differ by integers. Examples of such CFTs include \(d=2\) Virasoro-symmetric CFTs, if we forget the Virasoro symmetry and only use the global conformal symmetry.

Conclusion

Weight-shifting operators are promising tools for computing conformal blocks, and less promising for solving specific CFTs. If they indeed lead to much simpler computations than other methods, one may wonder why they were intoduced only recently, given that their \(d=2\) analogs have been widely used since the 1980s. It seems that the flow of ideas between the subfields of \(d=2\) CFT and \(d\neq 2\) CFT has not been optimal. It is probably the practitioners of \(d=2\) CFT who are to be blamed for not explaining their techniques well enough, and not working out the generalization to \(d\neq 2\) themselves.