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Holography: Vasiliev's higher-spin theories and O(N) models

In December 2009, Simone Giombi and Xi Yin of Harvard University wrote a fascinating 90-page paper,

Higher Spin Gauge Theory and Holography: The Three-Point Functions
They have offered a new nontrivial piece of evidence supporting a holographic duality that I will explain momentarily. They calculated three-point functions on both sides, to obtain a function of the spins "s". The result is a ratio of gamma functions involving these spins "s", describing the fields, often shifted by 1/2. And yes, it completely agrees on both sides.

There are some entertaining technical details of their calculation, like the analytic continuation in spins, but you have to read the paper to learn more about them.

What duality am I talking about? Well, on the bulk side, you have Vasiliev's (and Fradkin's - I've met this funny chap) higher-spin theories in AdS4, see the 1999 review
Higher Spin Gauge Theories: Star-Product and AdS Space.
On the boundary side, there is an O(N) model in 3 dimensions. Its matter fields - scalars - transform in the vector representations of O(N). And one should naturally add the "(phi^a.phi^a)^2" quartic self-interaction (although many interesting things already appear in the free theory). See the 2002 paper by Klebanov and Polyakov who conjectured the duality:

AdS Dual of the Critical O(N) Vector Model
The theories on both sides have been known and considered somewhat important before the duality was formulated in the paper above. For obvious reasons, such dualities that relate previously known objects are more interesting than the dualities that have to construct the other side of the equivalence from scratch (especially if the new side is awkward).

Difference from ordinary gravitating vacua

Now, the theories on both sides look somewhat unusual, in comparison e.g. with the "AdS5 x S5" vacuum of type IIB string theory which is equivalent to N=4 SU(N) gauge theory on the boundary. To understand the differences, and why the novelties probably match on both sides, I must say some things about the Vasiliev theories.

The Vasiliev theories in the bulk - which are somewhat manageable and understood in 3+1 dimensions we discuss here as well as 2+1 dimensions (because the higher-spin representation theory of the Lorentz group is more straightforward than it is in higher dimensions) - are conveniently formulated with a large gauge invariance. What is it?

In normal Yang-Mills theories, generalizing electromagnetism, the transformations (or at least their global versions) are generated with a "charge" such as the "electric charge". This quantity is a spacetime scalar i.e. a spin-zero field. The corresponding gauge field must be able to compensate the effects of a spacetime gradient of the transformation parameter - so it must be a spin-one field. Recall that "δ A_m = ∂_m λ".

Similarly, spin-1/2 charges (supercharges) that generate supersymmetry transformations require a spin-3/2 field, the gravitino, to play the role of the "compensator" of the local supersymmetry transformations in the the theory. And if the conserved quantities are spin-one objects - actually only one of them, namely one energy-momentum vector - you need a spin-two field, the metric tensor (or perhaps its unusual generalizations with antisymmetric pairs of indices), to undo the effects of the coordinate transformations - of the local versions of the spacetime translations generated by the energy-momentum vector.

The Coleman-Mandula-like theorems are enough to prove that the conserved quantities with spins exceeding one - which would lead to "gauge fields" whose spin exceeds two (spin-5/2 is already too much!) - can only be included in non-interacting and therefore uninteresting theories. Why? The number of components of the conserved tensors (or spintensors) would constrain the theory too much and the scattering amplitudes would essentially have to vanish.

There exists another loophole and it is given by the Vasiliev theories. There can be higher-spin fields and their corresponding gauge fields. But the higher-spin generators actually have to have nonzero commutators. And their commutators inevitably include generators of an even higher spin. So once you go in this path, you inevitably end with a theory whose fields can have an arbitrarily high spin. The theory's interactions are still limited in some way but the tower of fields make this theory non-trivial, anyway.

Now the real differences

Even the "ordinary" perturbative string theory kind of contains fields of an arbitrarily high spin: the excited string modes. But they're massive, with masses being comparable to the string scale. However, in the Vasiliev theory, all the new high-spin fields are actually massless! Note that the AdS curvature reinterprets what the word "massless" actually does, depending on the fields' spin.

Also, there is a very different dependence of the number of "species" of the fields as a function of the spin. In perturbative string theory, there exists a whole "Hagedorn tower" of states which is getting exponentially dense as you go to higher excited string states. The number of states grows exponentially with the spin - or exponentially with a power of the spin, which is qualitatively similar. The density of black hole microstates which is relevant at strong coupling (or truly high masses) morally continues with this exponential growth that is already seen in perturbative string theory.

But in the Vasiliev theories, the number of distinct fields only grows linearly with the spin. For different dimensions, it could perhaps grow as another power law but there's no exponential growth over there.

There exist differences on the boundary side, too. In the N=4 d=4 gauge theory, the fields transform in the adjoint representation of SU(N). However, in the O(N) model dual to the Vasiliev theory, they transform in the fundamental representation of O(N).

In the context of noncommutative geometry or Matrix theory, the difference between adjoint and fundamental representations is simple to describe (although not necessarily relevant for our Vasiliev AdS/CFT pair). If the matrix indices are chosen to encode the dependence of a function on two emergent noncommutative dimensions - think of a fuzzy sphere or a fuzzy torus - then the adjoint representation of U(N) produces fields that live in the bulk of a "membrane" while the fundamental representation of U(N) may give birth to fields that live on one-dimensional curves inside the membrane. For example, the O(N) heterotic E8 Matrix theory for the 11D space with one Hořava-Witten boundary has 16 fundamental fermionic fields which produce the extra E8 degrees of freedom on the boundaries of cylindrical boundaries.

However, the difference between the fundamental and adjoint representations seems to be somewhat more dramatic in this Vasiliev case. The absence of the adjoint representation eliminates the whole Hagedorn tower of states and replaces it with a simpler "linear" (or "power-law") tower of higher-spin states.

It's very questionable whether such a theory or vacuum without gravity and without the corresponding growth of the density of states should be considered a part of string theory - whether it is another solution of some unified equations of motion. We don't have the most universal definition of "string theory" yet so we only "add" things that are manifestly connected with the old ones but my answer would probably be "no" at this point: it doesn't seem to be "the same theory". Nevertheless, the holographic duality still seems to be working here. In some sense, the theory is less complex than the vacua of string theory so one should try to understand where the holography comes from as comprehensibly as possible.

Automatic holography of AdS spaces

There is one feature of the holography that may be puzzling: the bulk theory is not a typical theory of gravity, with the black holes etc. So why is it holographic in the first place? How can something that looks more like a "local field theory in the bulk without gravity" be equivalent to a field theory in a lower dimension?

Well, the Vasiliev theory clearly can't be just another generic local field theory of the usual type. The infinite collection of the fields must actually yet paradoxically "reduce" its number of degrees of freedom in a similar way as in the gravitating vacua of string theory.

But maybe, this is not really necessary. It's because holography is kind of automatic in any theory defined on anti de Sitter space. It's because the volume of a large (much larger than the AdS radius) "spherical" region of "AdS_{d}" is actually proportional to the surface "area" (multiplied by the AdS curvature radius). So most of the volume "sits" near the boundary, anyway. That's where the warp factor squeezes most of the volume.

In this sense, one could expect any theory, and not just a gravitating theory, on the AdS space to be equivalent to a lower-dimensional theory. However, it's still true that it won't be a simple theory with a finite number of fields - like the O(N) model - if you start with a generic local field theory in the bulk. The Vasiliev theory is special in this regard. It's special because of some features that make it analogous to string theory even though the detailed realization of the "X-factor" needed to get a nice local boundary dual are different in both qualitative cases.

During the last 20 years, theoretical physics has made a stunning progress in the understanding of the equivalence of - and the transmutations between - seemingly very different physical objects and environments. Various dualities and transitions have taught us that many things that we used to consider very different are in fact completely equivalent descriptions of the same physics, at least in certain situations. Charges are the same things as momenta and windings, electrically charged particles and magnetic monopoles are two descriptions of the same things, confinement may look like the Higgs mechanism, and so on.

However, this unified perspective on all conceivable physical phenomena is not quite complete yet. That's why it's important to look at different types of physical models that are seemingly as constrained as the vacua of string theory (or almost) but the precise character of the constraints differs in some technical aspects. The Vasiliev theories may be an example of this cutting edge. Because they're similar to the normal stringy vacua but they may be simpler from some viewpoint, we should be able to fully understand these "toy models", shouldn't we?

And that's the memo.

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reader Brian G Valentine said...

I measured the distance between hyperbola verticies on (Escher's representation of) Klein's model of the Lobachewskian plane to find a representation of the metric Escher used to prepare his drawing

reader Eugene said...

Hi, Lubos! I would like to comment a little bit since it seems to me that you have wrote that there is no gravity in Vasiliev theory ("local field theory in the bulk without gravity"). In fact, there is a spin-two field in Vasiliev theory, so the gravity is included. There are also black-hole type solutions in this theory. One more comment is that the number of states with some fixed spin does not grows with the spin, it is a constant, e.g. the minimal theory containts the files of each spin from 0 to infinity in one copy. Thanks for this topic!