Is there a maximum number of fixed points that a QFT can have?

Question

I was wondering: is there a maximum number of (trivial and non-trivial) fixed points that a QFT can have (as a function of the space-time dimension and field content in the QFT)?

This post imported from StackExchange Physics at 2015-07-08 13:49 (UTC), posted by SE-user Jaswin

Comments

There doesn't seem to be any reason against the existence of infinite number of fixed point.

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See Gukov's recent paper "A New Look at RG Flows"

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@RyanThorngren, there doesn't seem to be any available copy online? 

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Oh sorry, that was the name of his talk. His paper is this one http://arxiv.org/abs/1503.01474

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It depends on the precise definition of "a" QFT. For me, a QFT is a trajectory of the RG flow between two fixed points and so a theory has always two fixed points: one UV fixed point and one IR fixed point (which can be the same if the theory is scale invariant). To be more interesting, the question probably defines "a" QFT has a QFT depending of parameters i.e. a given subspace of the space of all effective theories, stable under the RG flows and the question is about the number of fixed points in this given subspace. Then the answer depends on the precise subspace considered. 

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@40227,  I'm curious why this definition is particular appealing to you. It's mathematically quite possible that a flow can approach a fixed point, then gets repelled from the fixed point( a saddle fix point), then runs to the next fixed point, but gets repelled again when running nearby, and behaves like this over and over again. 

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^Or just think about QED. Because of the Landau pole, there is no UV limit.

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I think Gukov's ideas give a nice approach to this question. One needs to understand what sorts of manifolds can appear as theory space with what sorts of RG flow fields. For example, if the space of parameters is compact, then an infinite collection of fixed points needs to have an accumulation point...

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On the other hand, if the RG flow is the gradient flow of a perfect Morse function (as it is in many of his SUSY examples), then the number of fixed points equals the sum of Betti numbers of the theory space.

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@ Jia Yiyang What you describe is perfectly possible and is not in contradiction with what I have written: passing near a fixed point is different from passing through a fixed point.

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@40227, I understand there's no contradiction, but you said

For me, a QFT is a trajectory of the RG flow between two fixed points and so a theory has always two fixed points

So I'm wondering why this special case of RG flow is particular preferable for a QFT.

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@40227 I would be very interested in a answer along the second point of view you mentioned above, when considering the RG flow in different subspaces of parameter space.

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Note that it is possible to have fractures in RG flows: places where new (continuous or discrete) parameters become tunable and these are not necessarily at fixed points, so the picture of a "flow" as a simple curve can be misleading.