Today at Benasque we had a talk by Arkady Tseytlin based on a paper that appeared a few months ago and written in collaboration with Matteo Beccaria, dealing with partition functions of some free conformal fields. Just before that, Vasiliev presented the basics of higher-spin gauge theories, discussing in particular some aspects of holography. These notes are a good summary of his talk. As we will see, the two subjects are related in many ways, and one can consult for instance the TASI lecture notes by Simone Giombi on this topic. In the following, I will mainly use these three documents.

The basic idea of Higher-spin theory is to generalize the concept of gauge symmetry, which works with massless spin 1 fields, or coordinate invariance, which involves the metric, a massless spin 2 field. Equivalently, one can take as an initial goal the description of a consistent theory with massless fields of arbitrary (integer) spins. The first reaction of a physicist used to supersymmetric QFT is to wonder about how such a theory is allowed, since the Coleman-Mandula theorem is supposed to state that the S-matrix is necessary trivial if massless fields with spins $s>2$ are present, precisely because these fields give birth to very restrictive conservation laws. It is often the ase in physics that the assumptions of a theorem are easily forgotten : in the present case, one crucial assumption is that the QFT is defined in flat Minkowski space-time. It so happens that adding a non-zero cosmological constant, the S-matrix arguments no longer apply, and the Coleman-Mandula theorem is circumvented ! Since this is such an important fact to remember, let us recover it from a different point of view. It is relatively easy to understand higher-spin fields at the linear level, and this has been done in the late seventies by Fronsdal, among others. But extending this to the interacting level, is far from being trivial, and it appeared in the eighties that the terms in the action which are cubic in the fields need higher derivatives. But then by dimensional analysis, these higher derivatives require a dimensionful coupling constant, which was identified with the radius of the background $(A)dS$ in 1987. In fact, the interactions carry inverse powers of the cosmological constant, so they become singular in the flat space limit. Even if we didn't know about the Coleman-Mandula theorem, we would then have reached the same conclusion:

Higher-spin theory requires non-zero cosmological constant.

This is good news, as it solves the apparent paradox, but the full power of the above statement appeared only after the $AdS/CFT$ correspondence was discovered in the late nineties : higher-spin theories are natural candidates for holography !

The natural question is then : what is the dual CFT ? In fact, I have already talked about this previously in this blog, and the answer is in the title : we will obtain a free CFT ! Let us explain very briefly how it works, refering to Giombi's review for more details. Let us consider what is maybe the simplest of all theories : a free massless scalar, with action $$S = \int d^3 x \frac{1}{2} \partial_\mu \phi \partial^\mu \phi \, .$$ This theory is conformal, so it is invariant under the generators of the conformal group (translations, rotations, special conformal transformations and dilatation), but in addition to that, because it is free, it has much more, and the conformal algebra can be extended to an infinite dimensional symmetry algebra. To see that, one can observe that there are HS operators $J_{\mu_1 \dots \mu_s}$, one for each even spin $s$, which are conserved. It is not completely trivial to construct them, but it can be done, solving some differential equation. Then contracting $J_{\mu_1 \dots \mu_s}$ with spin $s-1$ Killing tensors, one obtains ordinary conserved currents, and by the usual construction conserved charges $Q_s$, one for each Killing tensor. One can show that in three space-time dimensions, there are $s(4s^2-1)/3$ charges at spin $s$. Finally, the commutation relations between all these charges define the HS algebra.

In order to have an honest holographic relation, we should be able to take some kind of large $N$ limit in our CFT, so let's slightly modify it, considering instead $$S = \int d^3 x \frac{1}{2} \sum\limits_{i=1}^N \partial_\mu \phi^i \partial^\mu \phi^i \, .$$ This is the so-called $O(N)$ model. All the construction of the previous paragraph can be reproduced, although now the HS operators have $O(N)$ indices,$J_{\mu_1 \dots \mu_s}^{ij}$. But we declare that we are interested in the $O(N)$ singlet sector, meaning that we project all the operators on their singlet part. Then we have a notion of single-trace operators, and we are able to describe the holographic dictionary : $$\begin{array}{c|c} \textrm{CFT} & AdS \\ \hline \textrm{Free } O(N) \textrm{ vector model} & \textrm{Higher-spin gravity in } AdS \\ \textrm{Single trace operator} & \textrm{Single particle state} \\ J_s \textrm{ global symmetry} & \textrm{Massless HS gauge field } \varphi_s \\ J_0 & \varphi_0 \textrm{ with } m^2 = -2/L_{AdS} \\ \textrm{Large } N \textrm{ expansion} & \textrm{Perturbative expansion in } g_{\mathrm{bulk}} \end{array}$$ Note that proving the statements of this paragraph is far from being trivial, required years of work for some of them, and others are still only conjectures ! However, there are strong indications that this is correct, and it "proves" through holography what Vasiliev knew before holography was even discovered : that there exists a consistent theory of higher-spins, with interactions. Moreover, let us stress one point : this is an example of holography without supersymmetry !

We have seen that the $O(N)$ vector model is dual to the simplest HS theory (see also the subtlety mentioned in the previous post, linked above). What about other free CFTs ? Well, we could consider a model where the fields transform in the adjoint representation of $O(N)$. In this case, one big difference with the previous situation is that single-trace operators can have many fields, not just two. This is reflected in $AdS$ by a "string-like" spectrum, with massive fields that can be interpreted as the spectrum of a tensionless string theory. The theory studied by Beccaria and Tseytlin is the next step in this hierarchy : they consider fields transforming in a representation with three indices. The interpretation of the dual spectrum could be that of a tensionless membrane. I may talk more about this holographic correspondence in a subsequent post -- also introducing very useful machinery to deal with symmetric products of representations.