^{1,2}

^{2,3}

^{1,2}

^{1,2}

^{1,2}

^{1}

^{2}

^{3}

^{3}.

We consider the fermionization of a bosonic-free theory characterized by the

Dualities play an important role in various branches of theoretical physics since long time [

The common feature of dualities is that of relating different physics, which otherwise would not be related one with each other. The example we are dealing with in this paper is the duality which permits to build fermionic degrees of freedom (d.o.f.) out of bosonic ones.

There has been a huge activity in the opposite direction, that is, the formulation in terms of bosons of a (possibly interacting) fermionic theory. The problem has been firstly solved in

Less popular is the reverse process of fermionization where fermionic fields (operators) are obtained from bosonic counterparts. Fermionization has been considered in spin liquid states and heavy fermions [

A stronger motivation for considering the fermionization recently arised again, from condensed matter, and it can be summarized as follows. New states of matter have been predicted (and discovered), called topological insulators (TI) [

For what concerns the

For what concerns the

In conclusion, we know that duality relations in a bosonic theory could give rise to a fermionization procedure. A systematic theoretical framework to get bosonic boundary actions with consistent duality relations relating bosonic fields exists in any space-time

The result is that to any

Here, we pursue the program in

It is worth to mention that our construction gives Weyl spinors in

The paper is organized as follows. In Section

We recall some of the results obtained in [

We conclude that the dynamics on the boundary is completely determined by the

The dimensional reduction on the plane

Summarizing, we are dealing with a nontopological

Since the boundary model (

To do this, we note that, differentiating (

We are dealing with a

Preliminarily, we recall that the fermionization procedure relies on the tomographic transform presented in [

In this section, we review some basic properties of the tomographic transform, which we conveniently use in the fermionization process. For more details on the tomographic representation, we refer to [

The basic ingredient of the tomographic transform in three spatial dimensions is the generalized function

Following [

The duality relation (

In the previous sections, we defined the properties of the tomographic transform and we wrote the duality relation (

Identifying

The crucial observation is that the Lorentz scalar defined in (

At this point, it is straightforward to check that the operator

In [

An additional fermionic field can be introduced:

As we have anticipated, it is possible to reconstruct the spinor

Finally, we are dealing with only one independent tomographic fermionic field, from which we can only construct a Weyl spinor since, as it is well known, for massless fermion in

In this section, as an example, we compute the integrated charge density expressed in terms of the tomographic transformed bosonic fields of our starting model.

The total charge expressed in terms of the tomographic fermionic variables is

The total charge is obtained by subtracting from (

In this paper, we explicitly constructed the fermionic d.o.f. for a

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the support of INFN Scientific Initiative SFT: “Statistical Field Theory, Low-Dimensional Systems, Integrable Models and Applications,” Italian MIUR FIRB2012-Project HybridNanoDev RBFR1236VV and EU FP7 Programme under Grant Agreement no. 234970-NANOCTM. Alessandro Braggio acknowledges the hospitality of the Institute for Nuclear Theory (INT-PUB-13-036) in Seattle, where the work was partially done.