# Is an SCFT moduli space always an algebric cone?

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Given a generic superconformal field theory, it is always true that the moduli space of vacua is an algebraic cone? Intuitively this should be the case since dilatations are symmetries of the theory.
Is this correct?

Is there a proof of this fact?

What is an algebraic cone? Can you give some reference? The moduli space of vacua of a SCFT usually is described by the Coulomb and Higgs branches and these do not have to be any kind of cones.

There is a definition of algebraic cone here
https://en.wikipedia.org/wiki/Cone_%28algebraic_geometry%29
or otherwise the standard reference is the book by Hartshorne "Algebraic Geometry".

In other terms it is an algebraic variety with a conical singularity.

An easy example can be C^2/Z_2.

I know that in a moduli space, in general, there could be present Coulomb branches, Higgs branches, Tensor branches and Mixed branches...
My question came from the fact that in all the examples I know the different branches are cones.

For example in page 8 of this paper
https://inspirehep.net/record/1353155
the authors clearly say that the Higgs Branch of 3d N=4 SQED is a Hyperkahler cone.

I wanted to know if this "being a cone" is a generic feature of SCFTs.

Ok, you see the term "algebraic" in cone is what confused me. Now the answer is no in general, at least I do not see why the moduli stack of vacua of SCFT's would have a conical structure always. In the paper you mention they talk about the Higgs branch of a SCFT which is usually some hyperkahler manifold in 3d and 4d (I am not sure about other dimensions) and in most cases I have seen. But I don't think that the Coulomb branch has to even have any hyperkahler structure or a conical structure. The example you say is usually called orbifold in physics by the way.

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