^{3}.

Supersymmetric quantum mechanical models are computed by the path integral approach. In the

Supersymmetry is a quantum mechanical space-time symmetry which induces transformations between bosons and fermions. The generators of this symmetry are spinors which are anticommuting (fermionic) variables rather than the ordinary commuting (bosonic) variables; hence their algebra involves anticommutators instead of commutators. A unified framework consisting of bosons and fermions thus became possible, both combined in the same supersymmetric multiplet [

Supersymmetric quantum mechanics was originally developed by Witten [

Later, when people started to explore further aspects of SUSY QM, it was realized that this was a field of research worthy of further exploration in its own right. The introduction of the topological index by Witten [

Witten Index was extensively explored and it was shown that the index exhibited anomalies in certain theories with discrete and continuous spectra [

There are also situations where SUSY QM arises naturally, for example, in the semiclassical quantization of instanton solitons in field theory. In the classical limit, the dynamics can often be described in terms of motion on the moduli space of the instanton solitons. Semiclassical effects are then described by quantum mechanics on the moduli space. In a supersymmetric theory, soliton solutions generally preserve half the supersymmetries of the parent theory and these are inherited by the quantum mechanical system. Complying with this, Hollowood and Kingaby in [

The research work in the direction of using supersymmetry to exploit topology occurred in phases: first one started in early 80s with the work of Witten [

This review gives a basic introduction to supersymmetric quantum mechanics and later it establishes SUSY QM’s relevance to the index theorem. We will consider a couple of problems in

The organization of this paper is as follows: Section

The use of Grassmannian (noncommutative) variables [

In compliance with the Bose-Fermi statistics, a collection of Grassmann variables

Let

The above relation implies

Then the set of linear combinations of

Firstly, let us see how a general function of this kind would look like. Let

A function of this kind is even when it only has an even number of

The right derivatives

Applying the above properties, differentiation of a general function will be of the following form:

Left derivative is defined as

It is just a matter of convenience to choose either left or right derivative.

Integration can also be defined over these variables; Grassmannian integration is very useful for supersymmetric localization which will become clear later. This kind of integrals satisfies the following rules:

One interesting feature of Grassmann variables which one can figure out by looking at the properties mentioned above is that the integration for these variables is the same as their differentiation.

Let us apply these rules to the general expression (

The factor

The equivalence of differentiation and integration leads to an odd behavior of integration under the change of variables. Let us consider the case of

This leads to

This can be extended to the case of

Gaussian integrals play an important role in quantum mechanics, quantum field theory, statistical physics, and so forth. These will be quite handy when dealing with sigma model calculations using path integrals later. Therefore, it is worthwhile to recall some properties of Gaussian integrals.

The case of commuting and noncommuting variables will be considered.

The

The above integral (

When the matrix is complex, the meaning of square root and determinant need special care.

We should keep in mind that the variables

The matrix

The disappearance of square root sign from (

Let us now consider a Gaussian integral over

Moreover, each nonzero term in the expansion of the exponential in (

When

Let us consider another integral for

In the above deduction we get a factor

Any sigma model is a geometrical theory of maps from one space to another, for the description of certain physical quantities. Based on different models, such spaces come with extra geometrical structure. In the context of our current study, these spaces will be manifolds. A supersymmetric sigma model contains both bosonic and fermionic fields [

We first consider a simple model where the target space is flat. Doing the path integral will yield a topological invariant of the underlying space. The Lagrangian of the model is follows:

The action is given by

The Witten Index for this simple case can be written as a path integral

In the operator

Using this in (

Now using the integral computations from previous section, we can write

The determinant of an operator by properly regularized infinite product of its eigenvalues

Therefore, we can formally write

We now shift our attention towards supersymmetric sigma models with more interesting target space, that is, a

Studying models on curved manifold by evaluating the functional integral for the Witten Index of an appropriate supersymmetric quantum mechanical system, one can reveal important connection between physics of the sigma model and geometry of the underlying manifold in the form of characteristic classes [

The theory involves bosonic variables

The Lagrangian of this model is

We can further simplify (

The action reads as

The above action is invariant under the following supersymmetry transformations:

Note that the Witten Index is independent of

As mentioned above, the Witten Index is independent of

Now the path integral approach can be employed

The second part corresponding to the fermions after a rescaling

The metric

Now, for the curvature term, which only involves the fermionic zero modes, we have the following integral:

Combining (

The Euler class

Therefore, the integral in (

We can compute the index of Dirac operator [

The path integral expression for the index of Dirac operator is

The components of the metric in (

The symbols

Therefore, at the origin of Riemann normal coordinates, (

The Lagrangian (

The Christoffel connection (

Together with the properties of metric at the origin of the Riemann normal coordinates and the above definition (

Plugging (

Since we are working at the origin of Riemann normal coordinates centered at

The second term in (

The Riemann curvature tensor has the following symmetries:

The above properties imply that the Riemann curvature tensor is symmetric under the exchange of first and second pair of indices and antisymmetric under the exchange of two elements in the first and second pair. The last identity (

Under these properties, (

Using (

The term

The action is

The complete Lagrangian (

A rescaling of time as

We have only expanded the fermions in the curvature term in their corresponding Fourier modes in (

Let us now define a fluctuation in the coordinate system as

Then the second order expansion of the action is

In (

The operator associated with

We can now evaluate the index by doing the following path integral:

We will consider the zero modes

Now we compute the functional determinant in (

Let us focus on the first block; the operator (

If we consider all the

The integration over

We can make the following change of variables to remove

Therefore,

Note that we have dropped infinitely many terms while going through the computations; therefore (

The relation (

In this paper, we started by reviewing some basics about the Grassmann variables. Later in Section

The next two models are supersymmetric sigma model defined on a curved space. First one (

In the case of (

Finally, one could compute these invariants for the restricted Witten Index of the appropriate supersymmetric quantum mechanical systems [

The author declares that there are no competing interests regarding the publication of this paper.

The author would like to thank Maxim Zabzine for introducing the subject and guidance during the stay at Uppsala. The author would also like to thank Yasir Jamil for providing the facilities to carry out this work. In addition, the author would like to thank Ming Chen for providing some useful references and comments. Lastly, the author would like to thank the anonymous referee for his helpful comments and suggestions.

_{4}from supersymmetric localization