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?
