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, $$

where $\psi^k$ are real Grassmann variables. This is supersymmetric under

$$ \delta x^i =\epsilon \psi^i, \qquad \delta\psi^i=\epsilon \partial_t x^i.$$

We want to compute

$$ \operatorname{Tr}(-)^Fe^{-\beta H}=\int_\text{periodic}[dx][d\psi]

\exp \left(-\int_0^\beta dt \mathcal L\right),$$

in the limit $\beta \to 0$.

My question is: to see that the lagrangian for quadratic fluctuations around constant configurations $\xi^i=x^i -x^i_0$, $\eta^i=\psi^i-\psi^i_0$ (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,$$

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: http://inspirehep.net/record/195891 or http://inspirehep.net/record/190192