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

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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@ 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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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.

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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.