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Andoni Royo Abrego

[WSS20] Particles and fields from topological defects in spacetime mesh

Andoni Royo Abrego

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Goals of the Project: Torsion of a connection has been studied as the substrate for matter gauge field since the early days of Einstein-Cartan Theory. In this work we aim to study the minimal form of the ideas shared in such theory, on the setting of the graph rewrite system, where torsion is identified with topological defects. The analysis of different topological defects and their possible dynamics in a two dimensional flat background are aim to be described and hopefully find particle-like behaviour of defects and internal gauge degrees of freedom.

Main result: a plausible interaction between

topological defects is defined in a centered hexagonal grid where these can be regarded as particles and an infinite range interaction field is analyzed. The computational tools for further study have been also created.

Future Work: now that the setup of the interaction is established and the computational tools are created, many interesting experiments can be done to analyze the behaviour of the defects and see 1) how the interaction works in different scenarios and 2) how well does this behaviour match with nature. Before jumping into 3D grids, there are many things left to analyze like many defect problems, rotation of defects, defects with different nature (possibly symmetric), etc.

In most modern theories of physics nature is described by spacetime, a four dimensional smooth manifold, where concepts from differential geometry like curvature and torsion play a key role describing interactions like gravity. Wolfram models, however, use discrete structures (hypergraphs) to describe nature and build physical laws on it, and hence, this differential geometry objects have to be translated to discrete metric spaces. By doing this in a proper way, hopefully, particle-like behaviour, interactions and gauge fields might come out in a natural way.

There are different ways of detecting curvature on a graph, but let me work on this first approach, which I find very natural. There are only three ways of tiling a two-dimensional flat space by regular polygons: four squares, three hexagons and six equilateral triangles in each vertex. Only the triangular grid (or centered hexagonal grid) has all the closest vertices at the same distance though, plus, it can be really easily constructed from the other two just by adding edges and vertices.

In[]:=

GraphicsGrid[{{GridGraph[{4,4}, ImageSizeSmall], ResourceFunction["HexagonalGridGraph"][ {3,3},ImageSize Small],ABCGraph[3]}}]

Out[]=

Pentagons, for example, do not tile the two-dimensional flat space, since the angle of each vertex is of 108º and can not sum 360º. However, if the grid is positively curved the angles of a pentagon get wider and can indeed appear in the grid. Having a centered hexagonal grid, the existence of a pentagon is an indication of positive curvature (similarly, an heptagon is a sign of negative curvature).

In[]:=

sph=DiscretizeGraphics[Graphics3D[Sphere[]]];DiscreteSphere=MeshConnectivityGraph[sph,VertexShapeFunction"Point",EdgeShapeFunction"Line"];HighlightGraph[DiscreteSphere,{NeighborhoodGraph[DiscreteSphere,{0,12},1],{Style[NeighborhoodGraph[DiscreteSphere,{0,133},1],Purple]}}]

Out[]=

D
5
and its Dynamics

At first it might look easy to define an update rule to describe the motion of a

in the hex-grid. However, it is pretty complicated actually. Let me make some definitions first:

NON-PROPAGATING DYNAMIC RULE: it is a rule that defines the motion of a defect such that it does not generate more defects and it can be written by a finite set of rules even if the graph is infinite.

It is quite obvious that a non-propagating dynamic rule has to be able to be executed within a closed subgraph, and hence the search of a non-propagating dynamic rule can be reduced to the search of a rearrangement of the interior of the subgraph that also generates a single defect (in a different position), without changing the nature of the boundary. By “without changing the nature of the boundary”, I mean that if a vertex belonging to the boundary has a certain amount of edges towards the interior of the subgraph, after the arrangement this amount of edges has to be the same. This must be obeyed, otherwise, that vertex would generate a defect after the rearrangement.

In[]:=

DD5=Graph[EdgeList[NeighborhoodGraph[DiscreteSphere,{0,12},6]],VertexLabelsAutomatic,ImageSizeLarge];li=VertexList[DD5][[All,2]];D5=VertexReplace[DD5,Table[{0,li[[i]]} i,{i,1,Length[li]}]];HighlightGraph[D5,{Style[{2527,2551, 5145, 4511,1144,4466,6615,1567,6760, 6039,3959,5956,5632,3230,3089,8927 },Blue],Style[{6,2,3,4,5,62,23,34,45,56},Red]}]

Out[]=

It is not difficult to see that due to the presence of the defect the nature of the boundary is not symmetric (in the direction of the defect), which makes a non-propagating dynamic rule very difficult to find (without loosing the nonplanarity). I think there might not even exist such a rule, no matter how big (non-infinity) the boundary is taken to be. Nevertheless, this suggests that there are non-propagating dynamics rules for pairs of defects (which of course can be drawn inside a symmetric boundary). All the dynamic rules I have attempt generate inevitably more defects. Here there is one example:

Out[]=

Coordinate system for centered hexagonal grids

The coordinate system we are going to use in the centered hexagonal grid is the Cubic Coordinate system, which is obtained from a cubic grid by slicing out the diagonal plane x+y+z=0. This way there are three directions on the grid, and of course one of them is redundant. However, it is convenient to keep them all since we know that the sum of all three coordinates of any vertex in the hex-grid will sum zero. We have created some functions to play with this coordinate system, equipped with a very useful set of options. Here we have some examples of what we can do with them:

Create a hex-grid of desired radius with the Cubic Coordinate system ToolTip:

In[]:=

ABCGraph[8]

Out[]=

Use colors to indicate the values of the coordinates:

In[]:=

GraphicsGrid[{{ABCGraph[3,ColorFunctionExtract[1]],ABCGraph[3,ColorFunctionExtract[2]],ABCGraph[3,ColorFunctionExtract[3]]}}]

Out[]=

Label the coordinates directly:

In[]:=

ABCGraph[2,LabelingFunctionFunction[#],ImageSizeMedium]

Out[]=

Highlight a specific vertex or list of them:

In[]:=

ABCGraph[2,GraphHighlight{{0,0,0},{-2,1,1},{-1,-1,2}}]

Out[]=

In[]:=

ABCGraph[2,GraphHighlight{Function[First[#]>1]Red,Function[Last[#]>1]Blue}]

Out[]=

D
5
D
7
Defect Pair

defect in a centered hexagonal grid, is a defect where two adjacent vertices have degree five and seven instead. This defect is very well known in Crystallography since it generates a line defect usually referred as disclination.

Out[]=

Since a

defect makes a whole new line appear in the grid, there is not a rule that simply generates a single

defect starting from a centered hexagonal grid without rearranging the whole grid. Nevertheless, instead of letting the “rift” (new line generated by the defect) propagate infinitely or until some boundary, it can terminate in another

defect oriented in the opposite direction, and then, the situation can be generated from a flat hex-grid in a finite and easily computable way. Now, taking a pair of

defects separated by a certain distance, there is a variety of rifts lying inside (or in contact with) the blue diamond in the picture that could connect them and all of them have the same trivially defined distance. Check that by stitching any of the possible rifts connecting the defects (gluing the pair of vertices in the both sides of the rift ), we can close or undo the rift, and the defects would come closer and closer until they annihilate each other and the flat hex-grid is recovered.

Zip dynamics?

The process of stitching a rift is very similar to closing a zipper. The defect acts like the head of a zipper in the sense that when it moves in one direction it opens the jacket (in our case it generates a new rift) and when it moves in the opposite direction it closes it, bringing both edges of the rift together and closing it. It is literally like stitching.

In[]:=

GraphicsGrid[{{Import["/Users/andoniroyoabrego/Documents/WSS2020/PROJECT/Zipper1.png"],Import["/Users/andoniroyoabrego/Documents/WSS2020/PROJECT/Zipper2.png"],Import["/Users/andoniroyoabrego/Documents/WSS2020/PROJECT/Zipper3.png"]}},ImageSizeLarge]

Out[]=

In this illustrations the purple path indicates the chosen rift, the yellow point is the center of the pentagon, the green point is the center of the heptagon. Highlighted black pair of edges are the edges that have been stitched in the previous step and the shadowed parallelogram is the area is going to be stitched in the next step.

It is obvious that by closing/opening the zipper we are transporting the defect along the rift, and hence there is a simple update rule for the dynamics of

defects. I think there is another pretty obvious consequence here: there is an interaction between the defects. Imagine there is a

defect somewhere in an infinitely large hex-grid. Then, any other defect in the hex-grid could find a (finite) set of rifts that would connect with the first defect, and by the stitching the rift they would come closer and closer and eventually annihilate each other. These topological defects generate fields that provide interaction between defects, and these fields are “invisible” to simple observation. Let me explain this better. If a pair of

defects is generated and get separated one from the other by the “opening of the zipper” a long enough distance, one could simply look at the space in between them and see that this space is not different from the flat hex-grid, and he/she would not be able to tell wether there is some “extra space” generated by rifts or not.

Creation/annihilation of
D
5
D
7
defect pairs

Creation of a pair of

defects can be written in a simple rewriting rule by splitting the center of an hexagon, and by applying the same rule to the already existing heptagons, the rift keeps opening.

Out[]=

Strength of the interaction

How could one now characterize the strength of an interaction in this setup? Well, there is something that we know: the longer the distance between two defects, the larger the amount of possible rifts connecting them. Let’s analyze then, how does the number of possible rifts change with the distance. In the end of the day, the number of possible rifts is the number of paths inside the diamond connecting the two pentagons, which is just the combinatorial number in the central line of the Pascal Triangle. This number is called the central binomial coefficient and is given by N(d) =





, where d is the trivial distance between the pentagons.

What can we say about this set up for interactions between topological defects?

Well, this set up has many interesting properties one could analyze and compare with known physics in continuous spacetime, as well as some problems. Let me write down some of them.

- A

defect has an orientation, which is interesting in order to build an analog with spin 1/2 particles. The possible rifts coming out the defect, however, are delimited in an angular direction, which is a big problem, since they are the underlying structure that define the sense of the field. Therefore, the study of rotation of defects is essential in order to go further with this set up, and if analogies with fermions were to be made, spinor-like behaviour should be found.
-The specific interaction proposed here has, in principle, an infinite range. Moreover, since the defect can take many different but equivalent paths it is not deterministic. The multiway graph though, is obviously confluent.

There are many simulations and experiments to be done yet, and now that the complicated tool for that have been created, this ideas can be computed and analyzed much further.

POSTED BY: Andoni Royo Abrego

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[WSS20] Particles and fields from topological defects in spacetime mesh

D5 and its Dynamics

Coordinate system for centered hexagonal grids

Create a hex-grid of desired radius with the Cubic Coordinate system ToolTip:

Use colors to indicate the values of the coordinates:

Label the coordinates directly:

Highlight a specific vertex or list of them:

D5D7 Defect Pair

Zip dynamics?

Creation/annihilation of D5D7defect pairs

Strength of the interaction

What can we say about this set up for interactions between topological defects?

D
5
and its Dynamics

D
5
D
7
Defect Pair

Creation/annihilation of
D
5
D
7
defect pairs