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In the paper, Cluster Decomposition, T-duality, and Gerby CFT’s , by Hellerman, Henriques, Pantev and Sharpe, in the introduction it says:

"Briefly, the idea is that nearly every stack has a presentation of the form $[X/G]$, where $G$ is a not-necessarily-finite, not-necessarily-effectively acting group acting on a space $X$, and to such presentations, one associates a $G$-gauged sigma model on $X$. The basic problem is that presentations of this form are not unique, and the physics can depend strongly on the proposed dictionary. For example, a given stack can have presentations as global quotients by both finite and nonfinite groups; the former leads immediately to a CFT, whereas the latter will give a massive non-conformal theory."

I'm trying to make sense of the very last claim. I think I have a pretty good understanding how the finite case leads to a CFT, at least in two dimensions. As explained in the same paper on many examples, if $G$ is finite, we can write 1-loop partition function, twisted sectors, etc.

But I'm terribly confused about this "massive non-conformal theory" in connection to infinite groups. My (superficial) understanding of massive theories is limited to the Sine-Gordon model in two dimensions, which in the ultraviolet limit gives a scalar field ($c=1$ CFT). So here's my question:

Is "massive non-conformal theory" they are referring to the same thing as "massive deformation of a CFT", in the sense that the Hamiltonian is deformed with an extra term? If so, what is the relationship between the group $G$ and the deformation? The same paper doesn't discuss a single example for $G$ infinite. Are there any examples in the literature?

I apologize in advance if this is too vague and/or poorly motivated.

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