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

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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
retagged Jul 8, 2015

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

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.

@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.

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...

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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