= Bosonic trace vertices from a finite fermionic determinant
{title2=$V_{\rm eff}(x)=-\operatorname{Tr}\log[I-gB^{-1}\sum_kx_kC^{(k)}]$}
Integrating finite <Grassmann variables> out of a quadratic action gives $\det[B-gC(x)]$. If $B$ is invertible, divide by $\det B$ and expand its logarithm to obtain $V_{\rm eff}=\sum_{m\geq1}g^m\operatorname{Tr}[B^{-1}C(x)]^m/m$. Since the bosonic variables commute, an $m$-leg effective <Feynman rule> uses the fully symmetrized trace tensor, with weight $-g^m(m-1)!\operatorname{Sym}\operatorname{Tr}[B^{-1}C^{(k_1)}\cdots B^{-1}C^{(k_m)}]$. The minus sign comes from $e^{-V_{\rm eff}}$. This organizes closed <fermion loops> into bosonic vertices. It is a formal perturbative identity; a singular $B$ prevents the normalized expansion, even though the unfactored determinant polynomial still exists.
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