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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 $\phi(t)$?

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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.

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