^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

^{3}.

In this work, we analyze an extended

Back in

Essentially, the investigations associated with central charges in field theories can be divided into three parts. In the first one, the main subject was the classification of the multiplets related to massless and massive representations. In this part, the papers by Salam and Strathdee [

The second part corresponds to investigations at the classical level. One of the remarkable contributions was the work by Witten and Olive [

Another interesting classical point of view showed up after the dimensional reduction of supersymmetric models. Some works have discussed that central charges can be seen as an inheritance of dimensional reduction [

The third part concerns the quantum aspects. Specifically, by means of quantum effects, the central charge has appeared as a quantum anomaly in superalgebra. From this perspective, many situations have been analyzed, such as nonlinear sigma models, kink, monopoles, domain walls, and vortices [

In the context of supersymmetric mechanics with central charge, some investigations of field-theoretical models may correspond to (quantum) mechanical systems, namely, in the study of superconducting cosmic strings, localized fermions on domain walls, and gapped and superconducting graphene [

At this point, it is worthy to emphasize that we are using the term “central charge” based on the generators with all vanishing commutation relations. However, there are situations in which new bosonic generators have been added to superalgebra with some nonvanishing commutators. For example, we address the cases of weak supersymmetry [

In this work, some investigations of

The paper is organized as follows: in Section

In

With the aforementioned relations, one can check that

In order to implement this extended supersymmetry through a superfield approach, we first introduce two (complex) Grassmann parameters,

The superalgebra above can be realized in a differential representation. To achieve this, we define

Moreover, we point out an important comment: it is worthy to introduce a deformation

By using these (deformed) covariant derivatives instead of the usual

At this moment, it is advisable to point out that, in this section, the introduction of central charge is not associated with an extra coordinate in superspace. In other words, the superspace is parametrized by

The introduction of supercharges and deformed derivatives in connection with central charge is not exclusive to this work. For instance, in [

As already mentioned, one of the goals in this work is to study in what multiplet we can introduce a nontrivial central charge transformation related to the superalgebra (

Let us first establish the supersymmetric transformation of these components. By using

Now we can proceed to fix the central charge transformation. In the case of real superfield, we note that

Finally, we remember that, in

The nonlinear sigma models in connection with extended supersymmetric mechanics have been a subject of intense investigation. In [

Having established the superspace formulation, let us investigate a possible application. Here, we propose what we refer to as our deformed nonlinear sigma model in terms of the deformed derivatives, a multiplet of real superfields

By taking

In this subsection, let us revise a particular one-dimensional nonlinear sigma model described in [

In the supersymmetric description of this model, we have two real superfields

According to [

Finally, we notice that the Lagrangian (

In this section, we present another point of view to investigate this extended supersymmetry. Our proposal is to introduce a new coordinate

Here, we shall generalize the procedure described in [

By using the superalgebra, (

Starting with the group element

Now we consider an infinitesimal transformation and obtain the differential representation to supercharges and central charge. By using infinitesimal parameters

A general operator

At this point, we realize that the central charge behaves like a momentum operator of the “extra dimension”

In order to obtain the covariant derivatives, we consider an alternative multiplication to the right of the group elements,

These covariant derivatives exhibit the same representation of the deformed derivatives in the mechanical case (now with

Having established the supercharges and covariant (deformed) derivatives, we turn our attention to the discussion of superfield and supersymmetric transformation. We define a real (bosonic) superfield as

It is important to mention that, in this formulation, we do not have a classical mechanics description, because now we deal with the component fields

Let us obtain the component transformations with this kind of supersymmetry. By taking the Taylor expansion of the right-hand side of (

Once

Finally, we emphasize that in our formalism the chiral and antichiral superfields do not depend on

In this subsection, we discuss an application of the previous formalism. We investigate a particular model in two dimensions. Using the covariant derivatives (

This action leads to the following Lagrangian density

Hence, we obtain a Klein-Gordon profile (first two terms), so we again note that

Since the auxiliary field

The equations of motion are given by

Here, we notice that these equations of motion can accommodate topological configurations. Let us initially consider a trivial fermionic sector,

It is interesting to highlight that we can take advantage of the supersymmetric structure to get a set of solutions with nontrivial fermionic sector. We shall adopt a similar procedure discussed in [

First, one may verify that, by applying supersymmetric transformations, (

Finally, let us present a particular case. If we fix the arbitrary function

By using the supersymmetric pertubation, (

Finally, it is worthy to comment that, by construction, this fermionic solution has a trivial condensate,

Initially, we have discussed the

In the second part, we have considered an implementation of superalgebra (

Finally, we point out some possible investigations related to this supersymmetry in two dimensions. First, in order to accommodate charged matter and gauge fields, one could analyze the introduction of other multiplets, such as complex bosonic (fermionic) scalars and vectors superfields. Moreover, the connection between this superalgebra and other two-dimensional (Poincaré) supersymmetries remains a subject of further investigation. In particular, one could study the usual supersymmetry in two dimensions and redefine or drop out some Lorentz (boost) generators. Remember that, in two dimensions, it is possible to have a Majorana-Weyl fermion and implement the heterotic

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

We would like to thank Prof. J.A. Helayël-Neto for useful comments and reading the manuscript. LPRO is supported by the National Council for Scientific and Technological Development (CNPq/MCTIC) through the PCI-DB funds, grant no.

^{N−1}

^{n−1}model