To make it simple, take the spin foam formalism of ($SU(2)$) 3D gravity. My question is about the choice of the data that will replace the (smoothly defined) fields $e$ (the triad) and $\omega$ (the connection) on the disretized version of space-time $\mathcal{M}$: the 2-complex $\Delta$: why choose to replace $e$ by the assignment of elemnts $e\in su(2)$ to each 1-cell of $\Delta$, and elements $g_{e}\in SU(2)$ to each edge in the dual 2-complex $\mathcal{J}_{\Delta}$? I mean, these are both $su(2)$-valued 1-forms, thus, roughly speaking, assigning elements of $su(2)$ to vectors of the tangent bundle $\mathcal{TM}$, in other terms assigning elements of $su(2)$ to infinitesimal displacements represented by the 1-cells of $\Delta$. I can understand the choise for $e$, but not for $\omega$, why this is so? Why it is not the inverse choice?
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$\begingroup$ What is $\mathcal{J}_\Delta$? That may help people answer this. But note that this question may not attract much attention here. Wait and see. $\endgroup$– David Roberts ♦Commented Dec 2, 2011 at 2:34
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$\begingroup$ I edited the question to specify all the symbols contained therein $\endgroup$– PedroCommented Dec 2, 2011 at 10:39
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$\begingroup$ Are you referring to a particular paper or set of papers here as a reference? You might also have more luck on the theoreticalphysics.stackexchange.com theoretical physics stackexchange. $\endgroup$– j.c.Commented Dec 2, 2011 at 17:12
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$\begingroup$ Yes, for example arxiv.org/abs/gr-qc/0301113, I also posted the question in theoreticalphysics.stackexchange.com thanks. $\endgroup$– PedroCommented Dec 2, 2011 at 23:42
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