# Chern-Simons Current of Left and Right Chiral Superspace in Graphene Wormhole

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## Abstract

**:**

## 1. Introduction

## 2. The Chern-Simons Current for a Superconductor

#### 2.1. The Modified Wilson Loop for Coopers Pairs

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

#### 2.2. Geometry of the Cooper Pairs

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

#### 2.3. The Supersymmetric Support Dirac Network

**Definition**

**8.**

**Definition**

**9.**

## 3. Computational Algorithm for the Chern-Simons Current

- Calculate free energy for ${A}_{1}$, ${A}_{2}$, ${A}_{3}$ and sum them to find the total free energy ${F}^{tot}$. For each modified Wilson loop ${A}_{\mu =k}$, the free energy is calculated by ${F}^{flip}=-2{A}_{k}({A}_{k-1}+{A}_{k+1})$, $k=1,2,3,\dots ,84$.
- If ${F}^{tot}<{F}^{flip}$ then keep ${A}_{\mu}$.
- If ${F}^{tot}>{F}^{flip}$ then flip ${A}_{\mu}$ to the opposite direction.
- Repeat steps until $k=1,2,3,\dots ,84$.

#### Simulation Results of the Chern-Simons Current Along Supersymmetric Support Dirac Network

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The visualization of the modified Wilson loop of connection over graphene lattice of six carbon atoms (

**left**) and the visualization of free electron over supersymmetric structure of graphene hexagonal atoms (

**right**). The electron starts to move freely at ${A}_{3}$ connection with modified Wilson loop localized around the carbon backbone bonding, in the cone, as $d{A}_{1}$ connection over ${A}_{2}$ connection over other carbon ring atoms. The group operator of symmetry breaking between left and right mirror symmetry repeats the pattern of moving free electrons as chiral symmetry breaking gauge group action over the connection of spinor field of Cooper pairs along topological space of invariant property of molecular orbital of $s{p}^{2}$ in graphene carbon atoms.

**Figure 2.**On the (

**left**): the visualization of Wilson loop as spinor field in time series data. On the (

**right**): the visualization of modified Wilson loop over lattice of carbon atoms.

**Figure 3.**On the (

**left**): the tunneling of the Cooper pairs by warping operator through the fifth dimension into a graviphoton. The kernel map of graviphoton projects them to the center of wormhole next lattice vibration of 4 dimensions. On the (

**right**): the modified Wilson loop ${A}_{\mu}={A}_{1}d{A}_{3}+{A}_{2}d{A}_{1}+{A}_{3}d{A}_{2}$ where $AdA:=\int D\left[A\right]A$. The gauge field ${A}_{\mu}$ is a quantum flux attached to the Cooper pairs in spinor field as a connection of spin. It is a holonomy of a supersymmetric support Dirac network (SSDN) for the learning algorithm in a convolutional spinor network.

**Figure 4.**On the (

**left**): the interaction of D-brane with anti-D-brane warping operator to fifth dimension direction $d{t}^{t}$ in the form of graviphoton wave with modified Wilson loop as link between the interaction of curvature in the ribbon graph of the spinor network model. On the (

**right**): the confinement, in quantum foam model, of localized Cooper pairs in tunnel state. This state also can be considered as an entanglement state in time series data when we reverse the time scale of the model by rotating the cone of events.

**Figure 5.**The spinor network of graphene D-branee with 54 carbon atoms is connected to child1 ${X}_{t}$ manifold with 18 carbon atoms and the Chern-Simons bridge with 3 connected carbon atoms. The Chern-Simons manifold is composed by k carbon atoms located as the center of the wormhole. The number $k=d{t}^{*}$ is the amount of extradimensional carbon atoms in this model. We want to find a number k, which can produce a stable wormhole structure with the Chern-Simons current as a supercurrent in the superconductive state.

**Figure 6.**On the (

**left**): the flowchart of the algorithm for random initialized current over 84 carbon atoms. On the (

**right**): the flowchart of the Ising algorithm for the phase transition. We use these algorithms over the fixed structure of carbon lattice atoms as spinor network for Cooper pairs to be localized as the parallel transport of free electrons.

**Figure 7.**The flowchart of all involved modules for the classification of the Chern-Simons current and the prediction of the graphene wormhole size.

**Figure 8.**On the (

**left**): the algorithm of the supersymmetric support Dirac network with the convolutional neural network (CNN) network and the input of 5 layers of tensor correlation matrix from the spinor network of graphene wormhole. The 1st layer is the carbon D-brane with 54 carbon atoms, the 2nd layer is the child1 manifold ${X}_{t}$, the 3rd layer is the Chern-Simons manifold ${X}_{t}/{Y}_{t}$, the 4th layer is child2 manifold ${Y}_{t}$. The last 5th layer is the anti-D-brane layer of graphene with 54 carbon atoms. On the (

**right**): we show the input of adjacent matrix to the CNN for learning and classifying the order parameter.

**Figure 9.**On the (

**left**): the picture shows the Chern-Simons current over carbon lattice of graphene wormhole from two simulations of Metropolis-Hastings algorithm plotted together. We randomly choose the Chern-Simons current at least 84 times over the grid of fixed spinor network of carbon lattice with 84 atoms and we compute the correlation matrix with the size $84\times 84$. On the (

**right**): the plot of FM1 (blue), FM2 (red) and FM3 (yellow) of the Chern-Simons current over carbon lattice of graphene wormhole from two simulations of Metropolis-Hastings algorithm plotted together.

**Figure 10.**The result of Metropolis-Hastings and Ising algorithm for supersymmetric support spinor fields ${A}_{1}$ (

**a**), ${A}_{2}$ (

**b**), ${A}_{3}$ (

**c**) of ${X}_{t}$ span by $18\times 18$ carbon lattice. The surface plot shows the result of spinor field.

**Figure 11.**On the (

**left**): the Ising algorithm result below ${T}_{c}$ for the spinor network in the iteration $k=3$ of child1 manifold. Above ${T}_{c}$, the spinor network will break down and separate. On the (

**right**): the average tensor correlation between $A1$, $A2$, $A3$ connections.

**Figure 12.**On the (

**left**): the plot of maximum likelihood function of Laplace probabilistic principal component analysis (PCA) of FM1. The maximum probability is at $k=50$ of carbon lattice atoms. We use this result to calculate the height of the wormhole structure in the nanotube. On the (

**right**): the plot of Laplace PCA of spinor fields ${A}_{1}$ (red), ${A}_{2}$ (blue), ${A}_{3}$ (yellow), all have only one component of carbon atom.

**Figure 13.**On the (

**left**): the plot of layer1 of frequency mode modulation of Holo-Hilbert transform (FM1) of the first 75 of 84 carbon lattice atoms for the input to tensor correlation algorithm to find a spinor network. In the (

**middle**): the plot of layer2 of frequency mode modulation of Holo-Hilbert transform (FM2). On the (

**right**): the plot of PPCA of FM2 of the Chern-Simons current.

**Figure 14.**On the (

**left**): The plot of average Chern-Simons current density over $54\times 54$ lattice of carbon atoms. The numerical simulation is performed with the Ising algorithm using the spinor network of ${A}_{2}$ in the temperature range $T=50\phantom{\rule{0.166667em}{0ex}}\mathrm{K},\dots ,512\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. It shows the fluctuation of current density at high temperature. In low temperature, there is no fluctuation in the simulations. In the (

**middle**) panel, it is shown the plot of the average Chern-Simons current density for the temperature range $T=100\phantom{\rule{0.166667em}{0ex}}\mathrm{K},\dots ,295\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. We cannot notice the fluctuation in this plot. We need more zoom to data in smaller range to see the nature of current fluctuation at high temperature. On the (

**right**): the plot of average Chern-Simons current density over $54\times 54$ lattice of carbon atoms. The numerical simulation is performed with the Ising algorithm using the spinor network of ${A}_{1}$ (red), ${A}_{2}$ (blue), ${A}_{3}$ (yellow) in the temperature range from $T=1\phantom{\rule{0.166667em}{0ex}}\mathrm{K},\dots ,295\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$.

**Figure 15.**The picture at top panel is generated from the Metropolis-Hastings algorithm draw in an adjacent matrix of spinor ${A}_{1}$ (

**a**), ${A}_{2}$ (

**b**), ${A}_{3}$ (

**c**), at temperature $T=137\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. We glue the pictures into three channels of RGB image (

**d**). On the bottom panel there is the picture output in the hidden layer after applying convolutional operator. The class of these input data for the prediction of superconductivity state are labeled with the real value 0 for the CNN to learn.

**Figure 16.**The picture at top panel is generated from the Metropolis-Hastings algorithm draw in an adjacent matrix of spinor ${A}_{1}$ (

**a**), ${A}_{2}$ (

**b**), ${A}_{3}$ (

**c**), at temperature $T=512\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. We glue three pictures into three channel RGB image (

**d**). The class of these input data for the prediction of normal state are labeled with value 1.

**Figure 17.**The tensor network of FM3 for the carbons with numbers $k=82$, $k=83$ and $k=84$ (from the (

**left**) to the (

**right**)).

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## Share and Cite

**MDPI and ACS Style**

Capozziello, S.; Pinčák, R.; Bartoš, E. Chern-Simons Current of Left and Right Chiral Superspace in Graphene Wormhole. *Symmetry* **2020**, *12*, 774.
https://doi.org/10.3390/sym12050774

**AMA Style**

Capozziello S, Pinčák R, Bartoš E. Chern-Simons Current of Left and Right Chiral Superspace in Graphene Wormhole. *Symmetry*. 2020; 12(5):774.
https://doi.org/10.3390/sym12050774

**Chicago/Turabian Style**

Capozziello, Salvatore, Richard Pinčák, and Erik Bartoš. 2020. "Chern-Simons Current of Left and Right Chiral Superspace in Graphene Wormhole" *Symmetry* 12, no. 5: 774.
https://doi.org/10.3390/sym12050774