supersymmetry and the de Rham complex

Question

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.

Answer 1

I think there is a typo in the references to "Supersymmetry and Morse theory", [21] should be replaced by [22]="Constraints on supersymmetry breaking". The quantization of non-linear sigma models and its relation with the de Rham complex is discussed in Section 10 of this paper.

In addition to the big "Mirror symmetry" book already mentionned in the comments, a useful reference is in the book "Quantum Fields and Strings: A Course for Mathematicians" (http://bookstore.ams.org/qft-1-2-s/ ), Part 1, Supersolutions, Chapter 3 (p261-276), where one can find a detailed derivation of the supersymmetry of the Lagrangian, and Chapter 4 (p279-289), where a more conceptual point of view on this Lagrangian is given via some superspace formalism (in particular, in superspace formalism, the expression of the Lagrangian looks as simple as a single quadratic term, and one can recovers all the complicated terms such as the Riemann curvature-4-fermions terms by a simple systematic expansion in components).

Here is a brief sketch of the relation with the de Rham complex. Fields of the theory are $\phi^i$, $\psi^i$, $\overline{\psi}^i$. Upon canonical quantization, the Hilbert space of the quantum theory is the $L^2$-space of differential forms on $M$, quantum supercharges are $d$ and $d^{*}$ operators and so the quantum Hamiltonian is the Laplacian $\Delta=\{d,d^{*}\})$.

The relation between the explicit form of the Lagrangian and the Laplacian=quantum Hamiltonian is quite subtle because of the general phenomenon of ordering ambiguities in quantization. It is a good exercise to write explicitely the classical Hamiltonian from $L$ but it will not tell you immediately what the quantum Hamiltonian is the Laplacian because there are ordering ambiguities. The point of the supersymmetric story is that in this case the Hamiltonian is constructed from the simplest objects which are the supercharges, and that there is no ordering ambiguity to construct the quantum supercharges.