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I'm posting this here instead of Physics Stack as my question is on the precise mathematical meaning of a word which is often used in the physics literature.

In theoretical physics (especially string theory), authors often refer to ''lifting'' and I am not certain what the authors mean mathematically, although I am aware it could depend on the context. They often talk of having a special solution (in 7 dimensions, say) and then ''lifting it up'' to a higher number of dimensions. It seems what is meant here is that one embeds a theory in the ''higher dimensional version'' of that theory after truncating certain degrees of freedom as necessary.

Does this word actually have a specific, precise mathematical meaning and does it always need to be used in the context of string theory (ie. string compactifications), or is it just a jargon way of saying that one ''goes from one thing to another in a way which makes sense''?

In one of the appendices of this preprint, for example, the authors talk about ''lifting'' a topological spin QFT to a topological non-spin QFT in a way which implies they just mean ''start with spin and turn it into non-spin'' with no particular formal meaning.

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    $\begingroup$ I might be confused but I think in general, and even in the paper you cite, the word is being used in (at least) two very distinct senses. A common use of lifting is a deformation which eliminates ("lifts") some part of the space of vacua of a theory -- eg you add a potential (maybe we should imagine we literally "lift" some vacua so they are no longer minima). This is related to symmetry breaking. Then there's the sense in which lift is used in the appendix you mention, for modifying a theory to extend its structure (in this case to nonspin manifolds, in other contexts to higher dimensions). $\endgroup$
    – David Ben-Zvi
    Commented Jun 8, 2021 at 1:11
  • $\begingroup$ OK, thanks for explaining. I guess the first one also relates to the way which a potential is often said to have some of its flat directions ''lifted'', which now makes more sense. $\endgroup$
    – Hollis Williams
    Commented Jun 8, 2021 at 14:49
  • $\begingroup$ Now other thing I have seen is that someone will take a particular type of geometric structure and then asks if there are any QFTs which ''lift'' this type of structure. Could you shine some light on how the word is being used here (roughly speaking)? $\endgroup$
    – Hollis Williams
    Commented Jun 8, 2021 at 14:57
  • $\begingroup$ Sorry I don't recognize this usage, could you give an example? $\endgroup$
    – David Ben-Zvi
    Commented Jun 8, 2021 at 15:56
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    $\begingroup$ OK thanks. My impression here is "lift" is used in a typical colloquial fashion, and could be replaced by a word such as "refine" - ie you have some forgetful/lossy operation (passing from a Riemannian manifold to say L^2 functions with the Laplacian, or from a vector space to its dimension) and you want to "lift" to a more refined level (like a 2d CFT of maps into your manifold, or to the level of vector spaces).. $\endgroup$
    – David Ben-Zvi
    Commented Jun 8, 2021 at 17:58

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