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The headline question: Is it known how to construct an equivalent of the 3-D Ising Fixed point theory on an arbitrary 3-D manifold? Or any non-trivial d > 2 fixed point?

The answer is maybe as simple as a careful conformal mapping of the action . I don't even know whether this is a hard question not.

Let me try to explain what I mean by a fixed point theory on an arbitrary manifold and why I think such a thing should exist. By a fixed point theory I mean a QFT without a lattice scale, or put differently a theory with an actual continuum limit, with correlation function at all distances defined. If we have such a fixed point theory we can say it is the "Ising" if on small scales correlation functions look like the usual Ising correlation functions.

We know how to use RG to construct fixed point theories on the $d$ dimensional plane. But the usual technique relies on the Fourier transform, and is therefore specific to the plane (or maybe other symmetric spaces).

Does a fixed point theory exist for every manifold? It's true in $d = 2$, where I can use analytic maps to construct geometrically non-trivial theories. We can do it with topological QFTs. These are obviously special. It feels like scaling ideas should work - most crudely put a lattice on your manifold and then repeatedly refine the lattice while locally scaling the parameters, so that large scale correlation functions remain constant. But I don't exactly see how geometric information should be encoded in this process and one has a tremendous amount of freedom in local scaling. I haven't figured it out. Maybe the process forces you somehow to a trivial theory or a topological blow up.

Lastly, I haven't specified exactly what geometric information should go on this manifold. The naive assumption is that a conformal geometry is sufficient. But I honestly don't know.

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  • $\begingroup$ Hard or not, that question is very interesting to say the least, +1 :-) $\endgroup$ – Dilaton Jul 30 '13 at 18:40
  • $\begingroup$ Maybe the considerations in Ron Maimon's answer to another question could be interestiting for you too? $\endgroup$ – Dilaton Jul 30 '13 at 18:45
  • $\begingroup$ @Dilaton: Good idea. Its not stuff I've studied, but it feels like it could give a direct construction. $\endgroup$ – BebopButUnsteady Jul 30 '13 at 18:53
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    $\begingroup$ Probably the place to start would be a translationally invariant system with one dimension periodic, and see if this has a non-trivial scaling limit. This is the same as dimensional crossover and finite temperature 2-d, but I can't figure out what the upshot is. $\endgroup$ – BebopButUnsteady Jul 30 '13 at 21:44
  • $\begingroup$ Maybe this would be of interest for you too? $\endgroup$ – Dilaton Aug 19 '13 at 20:36

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