In Alvarez-Gaume's paper "Supersymmetry and the index theorem" there is given a certain supersymmetric Lagrangian whose quantization, apparently, leads to the de Rham Laplacian on the exterior algebra of a manifold. For what it's worth, the Lagrangian in question is

$$L = \frac{1}{2} g_{ij}(\phi) \dot{\phi}_i \dot{\phi}_j + \frac{\sqrt{-1}}{2} g_{ij}(\phi)\overline{\psi}^i \gamma^0 \frac{D}{dt} \psi^j + \frac{1}{12} R_{ijkl} \overline{\psi}^i \psi^j \overline{\psi}^k \psi^l$$ where $\phi$ is a map from $\mathbb{R}$ into a target manifold $M$, $g_{ij}$ is a metric on $M$, and $\psi^i$ is a spinor field on $M$ (I guess).

The same Lagrangian appears in several places, e.g. in Witten's paper on "Supersymmetry and Morse theory" (although seemingly with $1/12$ replaced by $1/8$). Witten says ``How canonical quantization of [this Lagrangian] leads to the exterior algebra was discussed in [21].'' But I cannot find anything of the sort in [21]="Dynamical breaking of supersymmetry", nor in any other source that I know about.

Can anyone give an explanation, or a reference, for $L$ above is related to the de Rham Laplacian? I am happy with a physics level of rigor but I am hoping to see some details spelled out.

Many thanks for any help.