## Abstract

An important measurable quantity in the carbon nanostructures, including the nanotubular part of the graphitic wormhole, is the spin-orbit coupling. We will present in this chapter spin-orbit coupling for the fermions located in exotic graphene structures as is graphene wormhole and also in biological systems. Considering this influence, the two-component Dirac equation is changed into the usual four-component form. As a consequence, the chiral fermions should be detected close to the wormhole bridge. We will show that the smaller is the radius of the wormhole bridge, the stronger this effect should be. Finally, we will describe the role of spinor fields in the time series of genetic code. The reversed transcription process of the gene expression could be defined by a moduli state space model of a coupling spinor field between the gene of a viral particle and the host cell. As a general result, all states of codon can be computed by the Chern-Simons 3-forms.

### Keywords

- spinor network structure
- spin orbit coupling
- Chern-Simons fields
- graphene wormhole
- genetic code

## 1. Graphitic wormhole

The investigation of unique chemical and mechanical properties of nanostructures, e.g., fullerene, graphene, and nanotubes, promises a wide application in many technical areas. The electronic properties of the nanostructures are basically defined by their hexagonal carbon lattice structure and its variations. New promising results are expected with the preparation of more complicated forms as a wormhole. The wormhole is usually composed of two different kinds of nanostructure: two graphene sheets are connected together with the help of a connecting nanotube [1] (Figure 1). This is achieved by a supply of two sets of six heptagonal defects onto both sides of the given nanotube. There exists the restrictions on the form of the nanotube—the chirality must be

The metric tensor of the wormhole is given by

where

### 1.1 Electronic structure

We consider the continuum gauge field theory, i.e., at each point of a molecular surface, we take into account an influence of different gauge fields that enter the Dirac-like equation for an electron

with

and the covariant derivative

In the case of the wormhole with the metric Eq. (1), the effective flux caused by the presence of the defects is included in the gauge field

and after the substitution into Eq. (2), we get the equation

where each sign corresponds to a different Dirac point

for

for

where the energy

The zero modes solve the Dirac equation for zero energy. If one choose the component

for

for

for

for

Recently, in work [2] some peculiarities in the bilayer graphene were analytically predicted. A possible indication of the wormhole could be found in [3, 4], where a new type of zero modes is investigated. These zero modes could be the zero modes studied in this subsection applied to the case of the smallest wormhole.

### 1.2 Case of massive fermions

Up to now we supposed that the fermions appearing in the Dirac equation have the zero mass or that the mass is very small in comparison with their energy, but in [5, 6] it was shown that the Fermi velocity needs to be renormalized due to the elasticity and deformations in a graphene. In our case of the graphitic wormhole, including big deformations, the velocity of fermions close to the wormhole bridge could achieve such values that the relativistic effects can appear or break off the symmetry [7] and the mass of fermions would be non-negligible. The radius of the wormhole and its bridge is very small in comparison with the size of the upper and the lower graphene sheet (Figure 2) and by folding the sheet into a tube they acquire nonzero effective mass as they move along the tube axis. This change of the space topology of graphene from 2D to 1D space compactification is similar to the string theory compactification, and we can imagine a wormhole connecting nanotubes as 1D object.

To include the mass into the Dirac Eq. (2), one can transform the system of equations [8] into the differential equation of the second order

One can suppose cylindrical geometry in order to simplify the equation into

if a radius vector of the point at the surface will have the form

with

A similar form has the dispersion relation associated with the massive 1D Dirac equation

where

where

### 1.3 Spin-orbit coupling in the wormhole connecting nanotube

An important measurable quantity in carbon nanostructures, which includes a nanotubular part of a graphitic wormhole, is a spin-orbit coupling (SOC) [12, 13]. If one considers this influence, two-component Dirac equation could be changed into the usual four-component form, and as a consequence chiral fermions should be detected close to the wormhole bridge.

One can reflect on two sources of SOC: (1) the interatomic one that preserves the

Considering the SOC we can write the Dirac equation for the nanotube in the form

where

The expression

where

Next one can take

where

For the interatomic source of the SOC, one has

with the difference of energies of the relevant

the transformed Hamiltonian

The operators

In our model, the SOC is induced by the curvature, and it is described with the help of two strength parameters, namely,

for the case of single-wall carbon nanotube with different magnitude. Here,

### 1.4 Graphene black hole

The effects connected with the deformation of a graphene and a consequent change of the distance of the carbon atoms in the layer are described in [14]. It causes the rotation of the

## 2. Spinor fields in biological systems

One of the present problems in genetic engineering is the prediction of biological gene variation and the representation of corresponding genetic code. This issue emerges in the plotting graphs related to the connection curvature of a docking processes. The docking process is important in the genes of the protein structure and could be adopted instead of using a very long alphabet notation as the string sequence and the comparison of the sequences of docking. From this point of view, methods of quantum field theory, general relativity, and related tools can be of high interest. The equilibrium between the supersymmetry and the mirror symmetry of the left-handed and right-handed DNA, RNA, nucleic and amino acid molecules can be explained by anti-de Sitter (AdS) correspondence in the Yang-Mills theory and the Chern-Simon currents in biology as the curvature of the spectrum in genetic code of the protein curvature.

Today, a genetical structure is studied by standard alphabet codes

There are still attempts to perform empirical data analysis of the genetic variation [23] and to detect the pattern matching over the gene sequence by using algorithm over a standard alphabet code as their time series representation. It seems that one main problem in this field is how can we predict the genetic variation and the gene structure in the viral particle and other organisms, or in the context of new representation, the question is how we can explain the intuition behind a definition of new time series data of gene, e.g., involved in the Batalin-Vilkovisky cohomology of DNA and the viral gene structure. The Chern-Simons current and the anomaly over a superspace of cell membrane can be applied to diagnose new gene diseases, the cloning technology or the gene therapy in medicine. Moreover, a presented method can be improved also in view of describing a useless trash area of DNA, which is considered as unknown part of human genome.

On the other side, another approach based on the usage of a spinor field in the Kolmogorov space of the time series data [24] over the genetic code can represent the gene structure as the ghost and anti-ghost fields of the codon and anticodon. This can be achieved in the frameworks of supersymmetry [25] and G-theory [22]. Results of the works show that all calculations over the codon can be assumed as a new superspace of the time series representation of the gene structure [24].

In [26, 27, 28], we introduced a new representation of the genetic code in the time series using a modeling by strings and D-branes. By applying a spinor field to a superspace in time series data [29], the method allows us to develop supersymmetry for living organisms. In particular, it is possible to control the anomalies in the codon and anticodon ghost fields and construct an algebraic approach for the trash DNA.

The “gravitational” analogy of the Chern-Simons currents in a gravitational physics, emanating from a system of DNA-RNA transcriptions, could have interesting counterparts also in biology. A representation of codons in human genome, derived from the Chern-Simons currents, can be useful in biology to explain the source of connections over protein-docking states. In this perspective, adopting cohomology in biology can be useful as a new modeling tool for plotting genes with spinor field in time series data. Especially, the junk area of DNA, with repeated inactive genes, can be represented by the Chern-Simons currents with extended structures of knot states in a Laurent polynomial of knots.

Further we discuss the role of spinors in the time series of genetic code. We can denote

In the retroviral RNA of the observed state space

The reversed transcription process of the gene expression is defined by a moduli state space model of a coupling spinor field between the gene of a viral particle and the host cell (Figure 3)

One can define

If

As known from biochemistry, there exist only

If

where

For an equilibrium state of evolution of an organism, we have no change of covariant derivative for the tensor field

We can define a *J* of the connection between the fields is defined by the Chern-Pontryagin density for the interaction of behavioral fields. The current varies from the curvature of docking between the behavior of the curvature over amino acid *k* in

The connection between genes is

where

One can translate a genetic code in a codon in three steps. The translation operator of group is given by a behavior matrix in Lie group, a group of supermanifold of living organism with action in three times. It generates a codon representation as an adjoint representation over gene expression, and it is a precise definition of genetic code with parity two of ghost field and anti-ghost field in the Chern-Simons current for the representation of a gene

Let a knot serve as a representation of anticodon in t-RNA topological structure for amino acid

The above term gives the asymmetric property of chiral molecule of DNA and RNA, twisted from the left hand to the right hand in a supersymmetry breaking as knot polynomial related to the connection

By using the new parameter of knot

one can induce a spinor field for representation of genetic code, where

A Chern-Simons current

and

The explicit definition of curvature over the connection of genetic code has also new meaning of the genetic spectrum current

An example of our approach can serve as a case of phenylalanine (

where we explicitly define the differential form of genetic code for

For a translation in reversed direction of antigen shift and drift in gene evolution theory, we can use the definition of the group action of reversed direction of time by the

Then a numerical representation for spinor field of curvature in the gene expression by the Chern-Simons action is defined as follows:

where

The derivation of the Chern-Simons current can be done by a simple algorithm [30], i.e., the Chern-Simons current maps the string of genetic code into numerical values by explicit formulas. It can be used to plot the time series data directly into the superspace of gene expression. We transform the alphabet string values, which cannot be computed in the classical standard definition of genetic code, into the Chern-Simons current of time series data of genetic code with

## 3. Circular Artin braid group representation for spinor field in genetic code

In each cell division, the telomeres are shortened [31], and total length of DNA is changing. As the result of shorter biological clock from cell division, the living things die. In order to understand cell cocycle and division mechanism of telomerase aging, one can explain the source of cancer as a source of age acceleration and its relationship to telomere shortening mechanism. It is a source of braid group operation [32] so-called self-diffeomorphism in the genetic code. The age acceleration is a relative measurement between the chronical clock and the biological clock in telomere. Up to now, scientists understand that a telomere and telomerase are the locations of ancient viruses that rely on DNA in the chromosomes of living organisms. Telomere is composed of the repeated sequence of

Here we assume that all genetic code cannot be completely separated and biological clock in telomere length is parametrized by a hidden state of the number of

and

therefore one can write eight bases for spinor field in the genetic code in braid form as follows

We may also use

In a

### 3.1 Classification of loop braid group in genetic code

We classify three types of loop braid group operations; it is a representation of an anyon for protein folding. For two-dimensional representation of D-brane in loop braid group for the genetic code, we define abelian anyon for biology in (2 + 1) dimensions, the extra dimensions used to represent the homotopy path of protein folding.

The loop braid group,

For a group operation of the genotype

It is isomorphic to the mapping class group of the infinitely punctured disk, a discrete set of punctures limiting to the boundary of the disk.

By an analogy with the action of the symmetric group by permutations in various mathematical settings, there exists a natural action of the braid group on

This sequence splits, and therefore pure braid groups are realized as iterated semi-direct products of free groups.

The braid group

The braid group operation gives

implying that

If

If

Thus the elements

## 4. Conclusions

The spin-orbit coupling is an important quantity, which is measurable in the carbon nanostructures, including the graphitic wormhole (or its nanotubular part); it can also help to identify the wormhole structure in details. SOC in a graphene could be induced by the nonzero curvature; in the particular case of the wormhole with negative curvature, the chiral fermions penetrating through the connecting nanotube in the wormhole structure could be created. The two-component Dirac equation is changed into the usual four-component form. As a consequence, the chiral fermions should be detected close to the wormhole bridge; the effect is stronger if the radius of the wormhole bridge is smaller. Moreover, one can detect permanently oriented flow when the chiral fermions prefer only one direction of the massive or massless fermionic current from the upper graphene sheet to the lower one, depending on the wormhole curvature.

We also describe the role of spinor fields in the time series of genetic code. The reversed transcription process of the gene expression could be defined by a moduli state space model of a coupling spinor field between the gene of a viral particle and the host cell. As a general result, all states of codon can be computed by the Chern-Simons 3-forms. The Chern-Simons current, coming from ghost and anti-ghost fields of supersymmetry theory, can be used to define a spectrum of gene expression in new time series data where a spinor field, as alternative representation of a gene, is adopted instead of using the alphabet sequence of standard bases

The reported results of the work have promissory perspective for their extension to interdisciplinary areas as machine learning, econophysics, or biological sciences.

## Acknowledgments

The work is partly supported by Scientific Grant Agency VEGA Grant No. 2/0009/19 and No. 2/0153/17. R. Pinčák would like to thank the TH division in CERN for hospitality.