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In e.g., http://arxiv.org/abs/arXiv:0712.3526 the author claims:

Since the massless higher-spin field theories involve infinite-dimensional gauge symmetries, one expects that such theories may be ultraviolet finite.

This statment is connected to the statement that one believes that Vasiliev theory is UV-complete.

How exactly is the connection between infinite gauge symmetries and UV finiteness and why do we believe that this is true?

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First of all there is no proof of this statement. It is just a general expectation that the more symmetries you have the more reason to expect better quantum properties. This works with SUSY, the more SUSY you have the better the theory is at the quantum level, say $N=4$ SYM, or $N=8$ SUGRA that some people still have hope to be well-defined.

If you involve AdS/CFT consideration than 4d Vasiliev theory is conjectured by Klebanov and Polyakov to be dual to free/critical vector model, which are well defined quantum field theories (especially the free one :) ), so one should expect 4d Vasiliev not to have any problems. For example, for boundary conditions corresponding to the free model one should have all loop corrections to vanish. This has not yet been checked.

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  • $\begingroup$ Thank you for the answer. So there seems to be a connection between more gauge symmetries and better quantum properties. Could you maybe explain (or be more explicit) about the systematics of that claim. There might be some (general) arguments for that claim? $\endgroup$ – ungerade Apr 3 '14 at 19:04
  • $\begingroup$ Apart from usual SUSY arguments that fermions cancel bosons in loops you can involve the argument that the more symmetries you have the harder it is to write down a possible counterterm. $\endgroup$ – John Apr 3 '14 at 19:23
  • $\begingroup$ 1.) One of the problems with the SUSY argument is, that Vasiliev theory is not supersymmetric and so it does not apply. 2.) I'm not sure if I understand you but let me rephrase your second statement to see if i maybe do: It is usually hard to find the right counterterm (of the many possible ones) and symmetries help (since they restict the possiblities of possible counterterms) to find the right one which finall helps to renormalize the theory. $\endgroup$ – ungerade Apr 4 '14 at 9:07
  • $\begingroup$ you can supersymmetrize Vasiliev theory and it can have as many SUSY as you like, but having SUSY is not a point about higher-spins, they should be well-defined on their own. SUSY was just an example of how having more symmetries can improve quantum properties of a theory. I agree with your rephrasing of 2). There is also an argument that the symmetries of higher-spin theory are the maximal ones, so it cannot be a broken phase of even more fundamental. But again there is no proof, these are just expectations. $\endgroup$ – John Apr 4 '14 at 17:19

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