SUSY QM and Atiyah-Singer index theorem 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'.
 A: *

*OP wants to evaluate the index
$$ {\rm Tr}(-)^F e^{-\beta H}=\int_{PBC}\![d\phi]\int_{PBC}[d\psi]~ e^{-S_E[\phi,\psi]},\tag{2.5} $$ 
in Ref. 1 with periodic boundary conditions (PBC) for both the boson $\phi\equiv x$ and the fermion $\psi$. Hence the corresponding Fourier components are labelled by integers$^1$ $n\in\mathbb{Z}$. One may argue that (2.5) does not depend on $\beta$, so that one may consider the $\beta\to 0^+$ limit. 

*Before discussing OP's model with 1 real fermion, we first discuss a model with 2 real fermions $\psi_1^j$ and $\psi_2^j$, or equivalently, 1 complex fermion $\psi^j=(\psi_1^j+i\psi_2^j)/\sqrt{2}$. Following Ref. 2, The Minkowski Lagrangian reads$^2$
$$ L_M~=~ \frac{1}{2}g_{ij}(x)\dot{x}^i\dot{x}^j 
+\frac{i}{2}\sum_{\alpha\in\{1,2\}}\psi^i_{\alpha} g_{ij}(x) \left(\dot{\psi}^j_{\alpha}+\dot{x}^m\Gamma^j_{mk}(x)\psi^k_{\alpha}\right) $$
$$ +\frac{1}{4}R_{ijk\ell}(x)\psi^i_1\psi^j_1\psi^k_2\psi^{\ell}_2\tag{74}$$
with real fermions in Ref. 2, or equivalently
$$ L_M~=~ \frac{1}{2}g_{ij}(\phi)\dot{\phi}^i\dot{\phi}^j 
+i\psi^{\ast i} g_{ij}(\phi) \frac{D\psi^j}{dt}  -\frac{1}{4}R_{ijk\ell}(\phi)\psi^{\ast i}\psi^{\ast j}\psi^k\psi^{\ell}\tag{4.1}$$
with complex fermions in Ref. 1.
We next Wick rotate $\tau=it$. The Euclidean action is 
$$S_E~=~\int_0^{\beta} d\tau ~ L_E.$$ 
Next split an arbitrary virtual path $(x,\psi)$ into zeromodes and non-constant modes:
$$x^i(\tau)~=~x^i_0 +\xi^i(\tau), \qquad 
\psi^j_{\alpha}(\tau)~=~\psi^j_{\alpha 0}+\eta^j_{\alpha}(\tau).  $$
The non-constant modes $(\xi,\eta)$ have Fourier components labelled by non-zero integers $n\in\mathbb{Z}\backslash\{0\}$. Let us rescale from old unprimed variables to new primed variables
$$ \tau~=~ \beta\tau^{\prime},\qquad 
L_E~=~ \frac{L^{\prime}_E}{\beta},\qquad 
S_E~=~ S^{\prime}_E, \qquad g_{ij}~=~ g^{\prime}_{ij}, $$
$$ x^i_0~=~ x^{i\prime}_0,\qquad 
\xi^i~=~ \sqrt{\beta}\xi^{i\prime},\qquad 
\psi^i_{\alpha 0} ~=~\frac{\psi^{i\prime}_{\alpha 0}}{\beta^{1/4}},\qquad 
\eta^i_{\alpha}~=~ \eta^{i\prime}_{\alpha}.$$
One may argue that the total path integral (super)Jacobian of this transformation is precisely $1$, e.g. via zeta-function regularization
$$\prod_{n\in\mathbb{Z}} \sqrt{\beta} ~=~1. $$
We drop the primes from the notation from now on. Only quadratic terms in $(\xi,\eta)$ survive in the Euclidean Lagrangian
$$ L^{(2)}_E~=~ \frac{1}{2}g_{ij}(x_0)\dot{\xi}^i\dot{\xi}^j 
+\eta^{\ast i} g_{ij}(x_0)\dot{\eta}^j   -\frac{1}{4}R_{ijk\ell}(x_0)\psi^{\ast i}_0\psi^{\ast j}_0\psi^k_0\psi^{\ell}_0
+{\cal O}(\beta). $$
The term $\psi^{\ast i}_0 g_{ij}(x_0)\dot{\eta}^j$ vanishes identically because of PBC. 
The above scaling procedure $\beta\to 0^+$ is an example of localization to constant paths 
$$(x^i(\tau),\psi^j_{\alpha}(\tau))~\longrightarrow~ (x^i_0,\psi^j_{\alpha 0})$$ 
in a path integral. The full path integral is given by the Duistermaat-Heckman formula and the method of steepest descent for the Euclidean signature; or equivalently, the WKB stationary phase formula for Minkowski signature, cf. Refs. 4-5. 
The fermionic d.o.f. mimic the de Rham complex. One may show that the Witten index (2.5) becomes the Euler characteristics, cf. the Chern-Gauss-Bonnet theorem, and eq. (4.3) in Ref. 1. $\Box$

*We now return to OP's question. OP's Euclidean Lagrangian [which corresponds to eq. (73) in Ref. 2] is obtained by imposing the additional condition $$\psi_1^i~=~\psi_2^i~=~\psi^i/\sqrt{2}.$$ The quartic curvature term disappears by symmetry. The rescaling from old unprimed variables to new primed variables now reads
$$ \tau~=~ \beta\tau^{\prime},\qquad 
L_E~=~ \frac{L^{\prime}_E}{\beta},\qquad 
S_E~=~ S^{\prime}_E, \qquad g_{ij}~=~ g^{\prime}_{ij}, $$
$$ x^i_0~=~ x^{i\prime}_0,\qquad 
\xi^i~=~ \sqrt{\beta}\xi^{i\prime},\qquad 
\psi^i_0 ~=~\frac{\psi^{i\prime}_0}{\sqrt{\beta}},\qquad 
\eta^i~=~ \eta^{i\prime}.$$
One may argue that the total path integral (super)Jacobian of this transformation is still precisely $1$. Only quadratic terms in $(\xi,\eta)$ survive in the Euclidean Lagrangian
$$ L^{(2)}_E~=~ \frac{1}{2}g_{ij}(x_0)\dot{\xi}^i\dot{\xi}^j 
-\frac{1}{4}R_{ijab}(x_0)\psi^a_0\psi^b_0\xi^i\dot{\xi}^j+\frac{1}{2}\eta^a  \dot{\eta}^a+{\cal O}(\beta).\tag{78}$$
cf. Ref. 2. Here $a,b$ are flat indices (aka. as vielbein indices). The calculation (78) is simplified by using Riemann normal coordinates, cf.
$$ \Gamma^{k}_{ij}(x)~\sim~ \frac{1}{2} R^k{}_{i\ell j}(x_0) \xi^{\ell} \tag{5.10} $$ 
in Ref. 3. 
The fermions now form a Clifford algebra, so that the index (2.5) becomes the Dirac/A-roof genus, cf. eqs. (79)-(81) in Ref. 2.
$\Box$
References:


*

*L. Alvarez-Gaume, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. 90 (1983) 161.

*L. Alvarez-Gaume and E. Witten, Gravitational Anomalies, Nucl.Phys. B234 (1984) 269.

*D. Friedan and P. Windey, Supersymmetric Derivation of the Atiyah-Singer Index and the Chiral Anomaly, Nucl.Phys. B235 (1984) 395.

*R.J. Szabo, Equivariant Localization of Path Integrals, hep-th/9608068.

*S. Cordes, G. Moore and S. Ramgoolam, Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories, hep-th/9411210; Chapter 12.

*H. Oogori, Lecture 8: Supersymmetry and index theorems.
--
$^1$ Anti-periodic boundary conditions (ABC) corresponds to half-integer modes.
$^2$ We have corrected some index typos in eq. (74) of Ref. 2. Another typo in Ref. 2 is that $\tau$ in eq. (74) should be $t$.  Later Ref. 2 forgets to remove an $i$ in front of the Wick-rotated kinetic term for the fermions in eq. (78). 
