Supersymmetric generalisation of the bosonic $\sigma$ model in QM I am reading some lecture notes which demonstrate how various models in SUSY QM can be used to obtain topological invariants such as the Euler characteristic from the Witten Index. 
The following lagrangian has been used directly, said to be the supersymmetric generalization of the bosonic $\sigma$ model. What is the motivation to consider this Lagrangian? How do I obtain the lagrangian? What is the sigma model being considered and how does it generalise?
Please provide a direct answer or references as deemed necessary. 
(I googled to learn more, but most of the reviews start from lagrangians in TQFT which I have no knowledge about. I would like a more elementary explanation for the lagrangian. )
$\phi^i(t)$ are maps from $R$ or $S^1$ to a Riemannian manifold $M$ with metric $g_{ij}$.
$$L =
\frac{1}{2}g_{ij}(\phi)\dot{\phi}^i \dot{\phi}^j+\frac{i}{2}g_{ij}\bar{\Psi}^i \gamma^0 \frac{D\Psi^j}{dt} + \frac{1}{12}R_{ijkl}\bar{\Psi}^1 \Psi^j \bar{\Psi}^k\bar{\Psi^l}
$$
$\frac{D}{dt}$ is the covariant derivative with $\dot{\phi}$ as a connection, and $\bar{\Psi}^i_{\alpha}=\bar{\Psi}^i_{\beta}\gamma^0_{\beta \alpha}$
How can I understand this from the physics of quantum mechanics? and what are the maps $\phi(t)$?
 A: Before going into the details, let me tell you that this type of
actions describe the deepest connection between geometry and physics
and generalizations of these types of theories are still under active
research even today.
The Lagrangian describes $N=1$ supersymmetric quantum mechanics on a
Riemannian manifold
The bosonic part of this Lagrangian is the kinetic term of a particle
moving on a Riemannian manifold $\mathcal{M}$ having a metric $g$. As
very well known the trajectories of the particles are the geodesics of
the manifold. The functions $\phi^i$ are just the coordinates on the
manifold.
The fermionic parts of the Lagrangian make the Lagrangian invariant under
the (N=1 supersymmetry) transformations:
$$\delta \phi^i =\epsilon \bar{\psi}^i$$
$$\delta \psi^i = -i \gamma^0 \dot{\phi}^i \epsilon - \Gamma^l_{jk} \bar{\epsilon} \psi^j \psi^k$$
Please see the following article  by Luis Alvarez-Gaume',
The supersymmetry operator can be written in terms of the canonical
momenta:
$$ \pi_i = g_{ij} \phi^j$$
as:
$$Q= i \pi_i \bar{\psi}^i -  \gamma^0 \Gamma_{ijk}  \bar{\psi}^i  \psi^j \psi^k $$
It can be easily checked that this operator generates the correct
supersymmetry transformation given the canonical Poisson brackets:
$$\{\phi^i, \pi_j\} = \delta^i_j$$
$$\{\psi^i, \psi^*_j\} = g_{ij}(\phi)$$
The inclusion of the fermionic coordinates in the Lagrangian gives spin
to the particle moving on the Riemannian manifold . This fact was
discovered by Berezin and Marinov in 1975, please see their original
article (they consider  the case of flat space time).
Most importantly, when the theory is quantized, then if we add to the
canonical quantization rules
$$\pi_i \rightarrow i \frac{\partial}{\partial \phi^i}$$
canonical quantization rules for the fermionic coordinates
$$\psi^i \rightarrow \gamma^i$$
i.e., quantize the Grassmann algebra to Dirac matrices or Clifford
algebra (please do not get confused with the gamma matrices in the
classical action which must be treated as numerical coefficients), then
the supersymmetry operator becomes the Dirac operator(in curved space). This is the reason
why this action describes a spinning particle.
Also, the square of the supersymmetry operator is the Dirac Hamiltonian:
$$ QQ^{\dagger}+Q^{\dagger}Q = H = \pi_i \dot{\phi}^i + i g_{ij} \bar{\psi}^i \dot{\psi}^j - L$$
The four fermion term expresses the fact that in curved space, the Dirac Hamiltonian differs from the scalar Hamiltonian. In differential geometry, this Dirac Hamiltonian is the Laplacian on forms.
One of the important applications of these types of actions is that they are used to provide quantum mechanical proofs of the various index
theorems.
Now, it is not difficult to think the following generalizations. If supersymmetric
quantum mechanics in $0+1$ dimensions describes a spinning particle, then
a supersymmetric sigma model in $1+1$ dimension will describe a spinning
string. In fact, Witten used this observation to compute the index of the
Dirac operator on a loop space.
One can think of the particles as probes for studying the geometry and
topology of the spaces they are confined to move on. A classical particle
can be used to study the geodesics. Upon quantization, more information
can be obtained for example the energies which constitute the spectrum of the Laplacian can give topological and geometrical information. When the
particle is given a spin, then even more topological information can be
deduced due to these index theorems, for example spinning particles can see holes and handles in the manifold.
Specifically, the model under consideration can be used to prove the
Atiyah-Singer theorem for the Dirac operator index  (The difference
between the number of zero modes of $Q$ and $Q^\dagger$)  on a Riemannian manifold and evaluate the result by means of the partition function path
integral. Please see the following article by Friedan and Windey
$$ ind(Q) = \int \mathcal{D} \phi \mathcal{D} \psi e^{i \int_{PBC} L dt}$$
(PBC denotes periodic boundary conditions) The zero modes of the Dirac
Hamiltonian are just the harmonic forms which generate the de-Rham
complex of the manifold.
Finally, if we replace the particle by a string still we can probe much more
topological and geometric information about the manifold.
