Quantum gauge theories involve (functional) integration over a Lie group. Is there any meaningful generalisation to (non-manifold) topological groups?

Consider for example the Whitehead tower $$ \cdots \to \mathrm{Fivebrane}(N) \to \mathrm{String}(N) \to \mathrm{Spin}(N) \to \mathrm{SO}(N) \to \mathrm{O}(N). $$

The last three groups are Lie, and they lead to well-defined QFT's. By untwisting $\mathrm{Spin}(N)$ one gets $\mathrm{String}(N)$, which is defined as the cover with trivial $\pi_3$ (and so it is not a Lie group). Can one do functional integration over this group? Does it lead to a well-defined and physically reasonable theory?

I guess that, at the perturbative level, all groups in the tower are essentially equivalent. But higher homotopy groups do play a role in gauge theory, and so the different groups are not equivalent at the non-perturbative level. For example, by going from $\mathrm{Spin}(N)$ to $\mathrm{String}(N)$, one loses instantons. Is this correct?

  • $\begingroup$ This might be a very silly question and caused by some very basic misunderstanding, but let me ask it anyway: if I understand correctly, in gauge theories the gauge potential is a Lie-algebra valued differential form. In fact it is the pullback of a connection form on a principal $G$-bundle by a section (gauge choice) and such a connection form is $\mathfrak{g}$-valued. If the group is not a Lie group, where the gauge potential would take values on? $\endgroup$ – user1620696 Apr 10 at 17:32

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