Hi, I'm actually physics student and I've not been able to understand how the following integration has been performed:
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The fields e and A have support on these discrete structures. The
$su(2)$-valued 1-form field$e$is represented by the assignment of an$e \in su(2)$to each 1-cell in$\Delta$. The connection field A is represented by the assignment of group elements$g_e \in SU(2)$to each edge in$\mathcal{J}_\Delta$The partition function is defined by
${\cal Z}(\Delta)=\int \prod_{f \in {\cal J}_{\Delta}} de_f\prod_{e \in {\cal J}_{\Delta}} dg_e e^{i {\rm Tr}\left[e_f U_f\right]}$where $de_f$ is the regular Lebesgue measure on $R^3$, $dg_e$ is the Haar measure on $SU(2)$, and $U_f$ denotes the holonomy around faces, i.e., $U_f=g^1_e\dots g^{N}_e$ for $N$ being the number of edges bounding the corresponding face. Since $U_f \in SU(2)$ we can write it as $U_f=u^0_f\ {{1}} + F_f$ where $u^0_f\in C$ and $F_f \in su(2)$. $F_f$ is interpreted as the discrete curvature around the face $f$. Clearly ${\rm Tr}[e_f U_f]={\rm Tr}[e_f F_f]$. An arbitrary orientation is assigned to faces when computing $U_f$. We use the fact that faces in ${\cal J}_{\Delta}$ are in one-to-one correspondence with $1$-cells in $\Delta$ and label $e_f$ with a face subindex.
Integrating over $e_f$, we obtain
${\cal Z}(\Delta)=\int \ \prod_{e \in {\cal J}_{\Delta}} dg_e \prod_{f \in {\cal J}_{\Delta}}{\delta}(g^1_e\dots g^{N}_e),$
where $\delta$ corresponds to the delta distribution defined on ${\cal L}^2(SU(2))$.
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The details are not important, what I need to understand is how the integration over the $e_f$'s has been performed
thanks
