Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and Lagrangian
$$ \mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k \partial_t + \Gamma^i_{mk} \partial_t x^m\right)\psi^k, \tag{1}$$
where $\psi^k$ are real Grassmann variables. This is supersymmetric under
$$ \delta x^i =\epsilon \psi^i, \qquad \delta\psi^i=\epsilon \partial_t x^i.\tag{2}$$
We want to compute
$$ \operatorname{Tr}(-)^Fe^{-\beta H}=\int_\text{periodic}[dx][d\psi] \exp \left(-\int_0^\beta dt \mathcal L\right)\tag{3}$$
in the limit $\beta \to 0$.
My question is: to see that the lagrangian for quadratic fluctuations around constant configurations
$$\xi^i=x^i -x^i_0, \qquad\eta^i=\psi^i-\psi^i_0,\tag{4}$$
(namely the one surviving in $\beta \to 0$ limit) is
$$ \mathcal L^{(2)}=\frac12 g_{ij}(x_0) \partial_t \xi^i \partial_t \xi^j - \frac14 R_{ijkl}\xi^i\partial_t\xi^j \psi_0^k\psi_0^l +\frac{i}2\eta^a\partial_t\eta^a.\tag{5}$$
What are the right substitutions to make, besides using Riemann normal coordinates and vielbein $e_i^a e_j^b \eta_{ab}=g_{ij}$?
References: Friedan and Windey or Alvarez-Gaume'.
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