Let $\tilde G$ be a central extension of a group $G$ by $U(1)$.

One common elegant definition is that there should be a short exact sequence of groups: $0 \to U(1) \to \tilde G \to G \to 0$

However, it turns out that one can always find a function $\theta: G\times G \to [0,2\pi)$ such that we can identify $\boxed{\tilde G = G \times U(1)}$ with the group structure $\boxed{(g,e^{i\alpha}) * (h,e^{i\beta}) = (gh,e^{i(\theta(g,h)+\alpha+\beta)})} $.

This choice of $\theta$ is not unique: under the re-identification of $G \to G: (g,e^{i\alpha}) \to (g,e^{i(\rho(g)+\alpha)})$, we get $\boxed{\theta(g,h) \to \theta(g,h)+\rho(g)+\rho(h)-\rho(gh)}$. I will refer to this as a 'gauge transformation' of $\theta$. The information in $\theta$ independent of such choices (i.e. 'gauge invariant') is what is of interest (which is what group cohomology/Schur multiplier encodes).

Now suppose we have a 'loop' in $G$. By this I mean successive applications of group elements such that in the end we are back where we started. The simplest example is a 'loop' associated to two commuting elements: $ghg^{-1}h^{-1} = 1$. Here is why I call it a loop:

loop in the group

Now define the 'integral' of $\theta$ around this loop as follows: $\boxed{\oint \theta := \theta(1,g) + \theta(g,h) + \theta(gh,g^{-1}) + \theta(ghg^{-1},h^{-1})}$

One can check that this is in fact the same for all possible $\theta$. It is a gauge invariant quantity.

More generally, if we have the loop defined by $g_1 g_2 \cdots g_N = 1$, then the integral defined by $\oint \theta := \sum_{n=1}^N \theta \left( \prod_{k=0}^{n-1} g_k, g_n\right) $ is gauge invariant. (Convention: $g_0 = 1$)

My question is the following: can one interpret a central extension of $G$ as being equivalent to some kind of '$U(1)$-bundle' defined on $G$, where $\theta$ plays the role of a sort of connection/gauge field, such that $\oint \theta$ then coincides with its holonomy/Wilson loop?

  • 1
    If $G$ and $\tilde{G}$ are meant to be topological (Hausdorff) groups and the extension is topological, this should be specified. – YCor Jan 15 '16 at 23:42
  • I am indeed not assuming this. For example even $G = \mathbb Z_2 \oplus \mathbb Z_2$ could apply, as this allows for non-trivial loops. That is why some of my words (such as 'bundle') are in scare quotes. The usual place where I see things as bundles/connections/holonomy, is in the context of manifolds, but here it seems that similar concepts apply (e.g. this 'gauge field' $\theta$ whose 'integral' around loops is gauge invariant, etc) whereas the basic context is different. That is exactly why I ask the question: is there a similar treatment as bundles, even though I don't assume a manifold? – Ruben Verresen Jan 16 '16 at 0:05
  • Look at Willerton, arxiv.org/pdf/math/0503266v1.pdf – AHusain Jan 16 '16 at 2:07
  • AHusain, thanks for the link. It does look interesting, but I am not well-acquainted with this type of mathematics. Is there a particular section you refer me to? In section 2.3 (p16-17), Willerton talks about how two-cycles (which my above $\theta$ is) lead to gerbes. Are you implying such a gerbe is the answer to my above question? – Ruben Verresen Jan 16 '16 at 2:30
  • Yes. See page 6 for a summary. You have set n=2. – AHusain Jan 16 '16 at 4:57

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