1.-GRACYASK (GRAphs CYcles And  SKyrmions):
Characterisation of topological invariants in discrete lattice geometries via graph re-writing

Sinuhé Perea-Puente (King's College London)

2.-ABSTRACT:We are going to study finite lattice systems in the context of simple re - writing  rules :  Based on discrete manifolds and Cayley graphs, in this project we look for the set and taxonomy of simple re - writing rules that preserve certain properties that can later be understood as conserved measurable magnitudes of a physical system, like the Brouwer degree or winding number . By investigating those toy-models about field configurations of discrete fiber bundles, we will later apply local perturbations to a field configuration, looking for discrete analogs to Noether’ s Theorem . Furthermore, via coloured graphs, with this approach we will aim to characterise (energetically stable) physical - inspired systems in the context of many - particle quantum finite distributions in highly symmetric arrays or low - dimensional spin - like states as domain wall in magnetization theory, being local solutions of minimal information configurations . Even mesoscopic scale is responsible for exotic topological defects, with this discretization we will try to understand the computational first - principle mechanism that allows non - trivial winding numbers and the material and geometries to host pseudo - particle structures like skyrmions, bubbles or merons . We think that this inductive reasoning here would offer a valuable approach for finding new phenomena or even re - interpreting old ones .

3.-Motivation

In several physics systems the whole can be obtained as an exact copy of each of its parts, which facilitates the study of a complex system by looking carefully at its elements, separately. Reductionism offers simplified models which makes the problems easier, but “there’s plenty of room...at the discrete scale”. Here we present a tour for two of its representants: Quivers and skyrmions, studying some of its basic definitions and properties, and an example where both arise together.

For that, we are going to primarily use the quiver module for Mathematica by Taliesin Beynon. Available at  https://github.com/taliesinb/QuiverGeometry, and carefully developed in https://quivergeometry.net.

4.-Introduction: A new (space) hole

4.1.-Start here: To read/run only the first time

tldr: In the following notebook tries to be a illustrative tutorial for showing some results in simple discrete lattice geometries. In principle all the output sections are displayed so unless you want to play around I don’t recommend to run them again, because some of them takes some time to fully  proceed (they are marked with the comment *IDLE*)

First of all, if you get ANY instability during the execution, run the following code:

Quit[]

If you want to interact with this notebook, you will need to download the GitHub repository “quivergeometry”and clone it in your local machine. Then you need to import the module via introducing the directory of the file “init.m”:

Get["C:\\Users\\k20004058\\OneDrive - King's College London\\Documents\\GitHub\\QuiverGeometry\\Kernel\\init.m"] (*Change USERNAME*)

Some initial codes to refresh the kernel:

GenerateQuiverGeometryStylesheet[];ClearBundleData[];GQG;

4.2.-Basic tools: Quiver structures

Our basic construct will be quiver functions that represent base (B) and fibre (F) spaces, being the total space the Cartesian product of the previous E:==B×F.

With the addition of a map π:E->B, we can form a bundle as the triplet (E,π,B) Some examples of basis, trivial (when the map is a projection in the first component) and non trivial bundles are the following:

MatrixForm[{"Increasing size of the quiver geometries->",
"Linear (Z)"->Table[LineQuiver[n,ImageSize->115],{n,4,5}],
"Cyclic/periodic (Zn)"->Table[CycleQuiver[n],{n,3,5}],
"Triangular"->Table[TriangularQuiver[n,ImageSize->70],{n,2,4}],
"Cubic"->Table[CubicQuiver[n,ImageSize->125],{n,2,3}],
"Bundle E=BxF"->TrivialBundleGraph[8,3,ImageSize->230],
"Mobius strip"->MobiusStrip[8]}]

As you may have see here, we are talking about discrete structures, so if we may need to redefine some concepts. When we have defined the spaces, the next step is to find the elements there. The objects for playing will be the graph of functions, represented as a collection of vertices and edges. In this notebook, we are going to consider “smooth”-like discrete functions implying that the connected vertex are only adjacent or in the same fiber section to the previous and following one, or in plain words, you can just go right (through the basis) and one position up, constant o down (through the fiber). We define the set of this elements as (D^1). Below you can interact with a trivial linear bundle, and see the different elements in (D^1).

Manipulate[{TrivialBundleGraph[aa,bb,ImageSize->20*aa],
Part[FindAllBundleSections[TrivialBundleGraph[aa,bb,ImageSize->20*aa,SectionDisplayMethod -> dd]],cc]},
Style["Linear space"Dynamic[aa]Dynamic[bb], 12, Bold],
{{aa,4,"B=Base size"},2,6,1,ImageSize -> Tiny},
{{bb,3,"F=Fibre size"},2,5,1,ImageSize -> Tiny},
{{cc,Random[Integer,{1,9}],"Field config"},1,Length[FindAllBundleSections[TrivialBundleGraph[aa,bb]]],1,ImageSize -> Tiny},
{{dd,"Total","Display"},{"Total","Line","Base","Array"}}]

In the last section of the notebook, we will introduce some physical examples for this abstract mathematical concept. We can see how we can make the system to “evolve” to go from an initial state to a final state via swapping a single vertex up or down, leading us to define a discrete version of homotopy in which two states are related (they are homotopically equivalent) if we can smoothly deform one into the other. The continuous version from [A] “Given two continuous functions f (x), g(x) defined in their respective topological spaces, both functions are homotopic iff exist a continuous function h(x, t) s.t. h(x, 0) = f (x) and h(x, 1) = g(x)”. Here we can just obtain the discrete version if we can obtain go between both in a finite number of (allowed) steps. And how can we characterise these derived paths? That is exactly the idea of rewriting systems [B].

5.-Bundle taxonomy: the rewriting strikes back

Next question we address then is how can "travel" between different states (this could also potentially lead us to define the neighbourhood of a given element and induce a toy-topology). This will be done introducing the rewriting system graph, that will give us valuable information about the system. We will start by considering a simple case to later start a brief characterisation of some families of bundles, using Cayley graphs.

examplegra=TrivialBundleGraph[3,3]; (*Empty total space*)
examplebun=FindAllBundleSections[examplegra]; (*All the possible sections*)
examplerew=BundleSectionRewritingSystem[examplegra];(*The rewriting system*)
examplecay=RewriteGraph[examplerew,Part[examplebun,1],ImageSize->300]
RewriteQuiver[BundleSectionRewritingSystem @ examplegra,All,VertexSize->30,
VertexShapeFunction->"Vertex"->Function[BundleSectionPlot[#,Method->"Line"]],GraphLegend->Automatic,ImageSize->300]

As you can see above, there is an obvious discrete parity symmetry (we can see left and right parts are quiral, with the mirror inversion), and also we can see that they are common structures between constant bundles (when all the vertices has the same “level”, like the movement of a chess tower). This is related to the concept of connection (in the “Berry phase” sense) under quotient topology and general gauge theory [C], which was more theoretically treated in [D]. An more closer example could be the different branches of the logarithm in complex analysis.

It is also interesting to take the quotient group with this different elements and see that, for example the mirror symmetry behaviour links quiral states, behind the subjacent group  (Subscript[Z, 2]), while the connections will allow us to just consider isomorphic copies of  the simpler cases (and adding manually some extra objects for the cyclic behaviour from the linear, for instance). Furthermore, it is possible to consider discrete analog to Noether’s Theorem, that will be clearer when the physical image is present, to quotient again the possible states and analyse a smaller set (in particular we will see this in the torus structure with translations along base and fiber spaces). In [E] this idea is discussed in the context of the  Ising model. As we can see, this abstract mathematical ideas can have direct corollaries in Physics.

These will be the basic ideas we are going to explore below for a general classification of these kind of quiver graphs.

5.1.-Linear quivers

We are going to begging studying the linear quiver where there is no quotient topology (“gluing” of points in the boundaries). First we are going to introduce a couple of families to later discuss the general structure, characterisation and limiting cases. The first case will be the Rubber band, where we are going to introduce the concept of elasticity.

5.1.1.-Linear rubber  band: (Subscript[L, 2])×(Subscript[L, n])

Manipulate[Module[{rubber},
rubber=TrivialBundleGraph[2,fib];
RewriteQuiver[BundleSectionRewritingSystem @ rubber,All,VertexSize->10*fib,
VertexShapeFunction->"Vertex"->Function[BundleSectionPlot[#,Method->dis]],
ImageSize->100*fib]],
{{fib,3,"Fiber size=2,                             B=Base size"},2,5,1,ControlPlacement->Top},
{{dis,"Array","Display Method"},{"Total","Line","Array"}},ControlPlacement->Bottom]
Dynamic[NumberForm[{
" is the number of connected states in the graph and"VertexCount[RewriteGraph[BundleSectionRewritingSystem[rubber],
Part[FindAllBundleSections[rubber],1]]],Length[FindAllBundleSections[rubber]]" is the total number of states" }]]

As we can see here, this family is quite trivial. It correspond to n copies of (Subscript[Z, 2])×(Subscript[Z, 2]) where n is the size of the fiber (OEIS:A016777). Additionally, we can see how the graph is connected. Finally the motivation of “rubber” bundle can be seen when display method is set to total. There we can see how we can interpret the model as “pushing upwards” a rubber that can only be stretched one place. In the cyclic cases, we will fully “see” this band.

This also motivate us to define some additional spaces that will be useful later. From the set (D^1), we can similarly introduce (D^n), corresponding to rubber bands with elasticity n, as generalisation of discrete continuous functions. In cyclic structures, is easy to see how the most elastic structure would be (D^(floor[F/2])) with F the size of the fiber. In the physical picture of the domain walls, this sets will correspond to Heisenberg exchange terms. Furthermore, we can obtain the precise subset of exact elasticity as the set of rubbers with only elasticity n, obtained as (E^2):=(D^2)\(D^1)and iteratively  (E^n):=(D^n)\(D^(n-1)).

5.1.2.-Tesseract binary machine: (Subscript[L, n])×(Subscript[L, 2])

Manipulate[Module[{binary},
binary=TrivialBundleGraph[bas,2];
RewriteQuiver[BundleSectionRewritingSystem @ binary,All,VertexSize->10*bas,
VertexShapeFunction->"Vertex"->Function[BundleSectionPlot[#,Method->dis]],
ImageSize->100*bas]],
{{bas,3,"Fiber size=2,                             B=Base size"},2,5,1,ControlPlacement->Top},
{{dis,"Array","Display Method"},{"Total","Line","Array"}},ControlPlacement->Bottom]
Dynamic[NumberForm[{
" is the number of connected states in the graph and"VertexCount[RewriteGraph[BundleSectionRewritingSystem[binary],
Part[FindAllBundleSections[binary],1]]],Length[FindAllBundleSections[binary]]" is the total number of states" }]]

This time, it correspond to the hyper-cubical graph where n is the size of the base (OEIS:A000079). Again, the graph is connected. The adjective “binary machine”  can very easily interpret seeing array display method and making the association {red:0,blue:1} each different colour of arrow correspond to the addition of a binary potence.

5.1.3.-The chess king : (Subscript[L, n])×(Subscript[L, n])

Manipulate[Module[{chessking},
chessking=TrivialBundleGraph[king,king];
RewriteQuiver[BundleSectionRewritingSystem @ chessking,All,VertexSize->10*king,
VertexShapeFunction->"Vertex"->Function[BundleSectionPlot[#,Method->disp]],
ImageSize->100*king]],
{{king,3,"F=Fiber size, B=Base size"},2,4,1,ControlPlacement->Top},
{{disp,"Array","Display Method"},{"Total","Line","Array"}},ControlPlacement->Bottom]
Dynamic[NumberForm[{
" is the number of connected states in the graph and"VertexCount[RewriteGraph[BundleSectionRewritingSystem[chessking],
Part[FindAllBundleSections[chessking],1]]],Length[FindAllBundleSections[chessking]]" is the total number of states" }]]

Another example of connected graph, this is a very interesting case. First we will introduce the associated sequence OEIS: A081113 Number of paths of length n-1 a king can take from one side of an nxn chessboard to the opposite side, which is exactly the real image of the bundle, similar to a chess board and moving with the king, funnily enough (E^2) correspond to the movement of the knight. It is possible to set an analogy to the movement inside a cube of dimension n, and we will illustrate in dimension n=3.

bundh3=TrivialBundleGraph[3,3];
bundh4=TrivialBundleGraph[4,4];
r3=RewriteQuiver[BundleSectionRewritingSystem @ bundh3,All,VertexSize->30,
VertexShapeFunction->"Vertex"->Function[BundleSectionPlot[#,Method->"Line"]],GraphLegend->Automatic]
r4=RewriteQuiver[BundleSectionRewritingSystem @ bundh4,All,VertexSize->30,
VertexShapeFunction->"Vertex"->Function[BundleSectionPlot[#,Method->"Line"]],GraphLegend->Automatic];
CubicQuiver[2]

As we can see in the figure, each of the dimension {x,y,z} in the cube can be associated with a colour RGB, and therefore we can simultaneously set a bijection between both structures. Furthermore, the number of vertices of these re-writing graph (4,28,148,690,2976...) is not included in the On-line Encyclopedia of Integer Sequences, so in the near future, it will be about including (if suitable). Till here, we have introduced a subtle classification of the small linear bundles. In the next subsections we are going to study the cyclic bundles and finalise studying the general structure of bigger cases that can not easily visually explained.

5.2.- Cyclic quivers

The intuitive idea behind all this elements is joining the begging and end of the structure (consider the quotient topology). Maybe the most known case is the Möbius strip, projective plane and Klein bottle and torus. Here we are only going to consider the latter one as example of exotic bundles, while we are going to see that the simple combination between a single-cycle and a single-linear quiver is quite trivial. In the case of a cyclic base, while in the case of a cyclic fiber we automatically give our field configurations a loop structure.

5.2.1.- Cyclic rubber bands: (Subscript[C, n])×(Subscript[L, m])

We are firstly going to see the full rubber band case, where the geometric interpretation will be clear.

Manipulate[
Part[FindAllBundleSections[TrivialBundleGraph[CycleQuiver[base],LineQuiver[fibe],LayoutDimension->3,ImageSize->20*base,SectionDisplayMethod->di]],ran],
Style["Cyclic rubber"Dynamic[base]Dynamic[fibe], 12, Bold],
{{base,4,"n=Base size"},3,6,1,ImageSize -> Tiny},
{{fibe,4,"m=Fibre size"},2,5,1,ImageSize -> Tiny},
{{ran,Random[Integer,{1,8}],"Field config"},1,Length[FindAllBundleSections[TrivialBundleGraph[Modulo @ base,fibe]]],1,ImageSize -> Tiny},
{{di,"Total","Display"},{"Total","Line"}}]

Manipulate[
Graphics3D[Table[{ColorData[109][ii], Arrowheads[0.12], 
  Arrow[Tube[{{-Cos[ii*2*Pi/nn],ii/nn/4,-Sin[ii*2*Pi/nn]}, {Cos[ii*2*Pi/nn],ii/nn/4,Sin[ii*2*Pi/nn]}}, 0.02]]},{ii,1,nn}],
  Boxed->False,ViewPoint->{30,100,0},ImageSize->200],
  Style["Spin rotation    C"Dynamic[nn], 12, Bold],
{{nn,7,"B=Base size"},2,10,1,ImageSize -> Tiny,ControlPlacement->Top}]

There is not much new information here. Again we still have a connected structure and the connections or constant levels induce quotient relations where the levels are erased. We can also take the opportunity to set the physical image we are going to discuss later, we can see that the base size is periodic, which can be seen as loops around the unitary circle, so we can set the trivial bijection between the cyclic elements and the set of angles 2π/n, as spin directions in a domain wall, or a polarized beam, for instance.

5.2.2.- Snake bundle: (Subscript[L, n])×(Subscript[C, m])

In this case, we have the cyclic behaviour around the fiber bundle, which allow us to define a loop as a full wrapping of the circumference. Nevertheless, having open boundaries forbids us to obtain a full topological winding number, situation that will happen when we joint the initial and final fibers, leading to a torus.

Manipulate[
Part[FindAllBundleSections[TrivialBundleGraph[LineQuiver[base],CycleQuiver[fibe],LayoutDimension->3,ImageSize->40*base,SectionDisplayMethod->di]],ran],
Style["Snake bundle"Dynamic[base]Dynamic[fibe], 12, Bold],
{{base,6,"n=Base size"},3,6,1,ImageSize -> Tiny},
{{fibe,5,"m=Fibre size"},2,5,1,ImageSize -> Tiny},
{{ran,Random[Integer,{1,8}],"Field config"},1,Length[FindAllBundleSections[TrivialBundleGraph[base,Modulo @ fibe]]],1,ImageSize -> Tiny},
{{di,"Total","Display"},{"Total","Line"}}]

If we impose further conditions on the shape of the bundle we can lead to the hyperbolic PD Sine-Gordon equation whose solutions (solitons, kinks). It is also possible to fully characterise the re-writing graphs in these cases but they are just the same base graph with additional available states due to the connectivity of the space. The general classification of single-cyclic solutions is leave for future work, by defining a quotient with the linear case collapsing all the previous elements into a single one class equivalence, to properly study the structure of the cycle itself.

5.3.- Asymptotic behaviour of single linear and cyclic bundles

One can start by noticing that the growth of the available states in the bundles compared with the available states (from elemental combinatory) is much slower, which allow us to discuss which is the density of elements of  (E^i) in (D^(dim[F])) (the set of all the irrestricted bundles), and compare the different sizes of this sets (with particular interest in (D^(floor[F/2])) , favouring antiparallel configurations, when we introduce the physics image). Another open question is if we can obtain a significance on the dual graphs some elements in the previous rewriting cases.

Conjecture 1.-The density of  (E^i) have the same growth behaviour and it is possible to establish a simple ratio between the amount of elements in each set.

Conjecture 2.-Some dual graph of the rewriting system will relate some elements in the single linear/cyclic quivers.

Manipulate[{Length[FindAllBundleSections[TrivialBundleGraph[bss,fbr]]],
fbr^bss,N[Length[FindAllBundleSections[TrivialBundleGraph[bss,fbr]]]/fbr^bss,4]},
Style["Possible/Total States, Ratio"Dynamic[bss]Dynamic[fbr], 10, Bold],
{{bss,4,"n=Base size"},3,10,1,ImageSize -> Tiny},
{{fbr,3,"m=Fibre size"},2,10,1,ImageSize -> Tiny}] 
MatrixForm[Table[N[Length[FindAllBundleSections[TrivialBundleGraph[i,j]]]/j^i],{i,2,7},{j,2,7}]](*IDLE*)

Similarly, we can see the number of vertices, bundle sections and edges for the linear structures (Subscript[L, n])×(Subscript[L, m]), and if the two first matrices are equal, it means that we have only a single component (having different components that are not related can be understood as states that can not be -smoothly- transformed one into the other, and therefore protected by the topology of the problem)

lim=6; (*Above 8 it takes quite long*)

mV1=MatrixForm[Table[
Length[FindAllBundleSections[TrivialBundleGraph[i,j]]],
{i,2,lim},{j,2,lim}]]

mV2=MatrixForm[Table[{
VertexCount[RewriteGraph[BundleSectionRewritingSystem[TrivialBundleGraph[i,j]],
Part[FindAllBundleSections[TrivialBundleGraph[i,j]],1]]]},
{i,2,lim},{j,2,lim}]]

mE1=MatrixForm[Table[{
EdgeCount[RewriteGraph[BundleSectionRewritingSystem[TrivialBundleGraph[i,j]],
Part[FindAllBundleSections[TrivialBundleGraph[i,j]],1]]]},
{i,2,lim},{j,2,lim}]]

Despite the  first row, column and principal diagonal has been already discussed, the i-row of top matrix, correspond to OEIS:A188861:Number of n X i binary arrays without the pattern 0 1 diagonally, vertically or antidiagonally while the j-column of top matrix correspond to OEIS:A126360: Number of base 6 j-digit numbers with adjacent digits differing by one or less, being particularly interesting the case j=3 Numerators of continued fraction convergents to (Sqrt[2]). It is left to study this connection. For the case of the edges (bottom matrix), none of the non-trivial sequences are included in OEIS. It is also left pending (related to the previous section final statement) the characterisation of the single cyclic structures. Now we are going to see what is the “degree of connection” of the single cyclic/linear bundles, defined as the ratio Vertices/Edges. We trivially know that it belong to [0,1], but what is the behaviour of it for larger sizes?

Code and density matrix of degree of connection:can be ignored

limF=5; (*Above 6 it takes quite long*)
limB=8; (*Above 9 it takes quite long*)
cll = MatrixForm[
  Table[{N[
     VertexCount[
       RewriteGraph[
        BundleSectionRewritingSystem[
         TrivialBundleGraph[i,j]], 
        Part[FindAllBundleSections[
          TrivialBundleGraph[i,j]], 1]]]/
      EdgeCount[
       RewriteGraph[
        BundleSectionRewritingSystem[
         TrivialBundleGraph[i,j]], 
        Part[FindAllBundleSections[
          TrivialBundleGraph[i,j]], 
         1]]],4]}, {i, 2, limF}, {j, 2, limB}]]
         
clc = MatrixForm[
  Table[{N[
     VertexCount[
       RewriteGraph[
        BundleSectionRewritingSystem[
         TrivialBundleGraph[i, Modulo @ j]], 
        Part[FindAllBundleSections[
          TrivialBundleGraph[i, Modulo @ j]], 1]]]/
      EdgeCount[
       RewriteGraph[
        BundleSectionRewritingSystem[
         TrivialBundleGraph[i, Modulo @ j]], 
        Part[FindAllBundleSections[
          TrivialBundleGraph[i, Modulo @ j]], 
         1]]],4]}, {i, 2, limF}, {j, 2, limB}]]
         
ccl = MatrixForm[
  Table[{N[
     VertexCount[
       RewriteGraph[
        BundleSectionRewritingSystem[
         TrivialBundleGraph[Modulo @ i, j]], 
        Part[FindAllBundleSections[
          TrivialBundleGraph[Modulo @ i, j]], 1]]]/
      EdgeCount[
       RewriteGraph[
        BundleSectionRewritingSystem[
         TrivialBundleGraph[Modulo @ i, j]], 
        Part[FindAllBundleSections[
          TrivialBundleGraph[Modulo @ i, j]], 
         1]]],4]}, {i, 2, limF}, {j, 2, limB}]]
         
ccc = MatrixForm[
  Table[{N[
     VertexCount[
       RewriteGraph[
        BundleSectionRewritingSystem[
         TrivialBundleGraph[Modulo @ i, Modulo @ j]], 
        Part[FindAllBundleSections[
          TrivialBundleGraph[Modulo @ i, Modulo @ j]], 1]]]/
      EdgeCount[
       RewriteGraph[
        BundleSectionRewritingSystem[
         TrivialBundleGraph[Modulo @ i, Modulo @ j]], 
        Part[FindAllBundleSections[
          TrivialBundleGraph[Modulo @ i, Modulo @ j]], 
         1]]],4]}, {i, 2, limF}, {j, 2, limB}]]

Plotting the degree of connection of rewriting graphs

The interesting feature is that it seems

ll=ListLinePlot[Table[Flatten[Part[cll,1,i]],{i,1,4}],AxesLabel->{" ","Deg connect"},PlotLabel->"linear/linear",ImageSize->400];
lc=ListLinePlot[Table[Flatten[Part[clc,1,i]],{i,1,4}],PlotLabel->"linear/cycle",ImageSize->400];
cl=ListLinePlot[Table[Flatten[Part[ccl,1,i]],{i,1,4}],PlotLabel->"cycle/linear",ImageSize->400];
cc=ListLinePlot[Table[Flatten[Part[ccc,1,i]],{i,1,4}],AxesLabel->{"Fibre Size-1"},PlotLabel->"cycle/cycle (torus)",ImageSize->400];

GraphicsGrid[{{ll, cl},{lc,cc}}]

Conjecture3.-The sequence is convergence and there exist a(irrational) finite limit of it. To explore via Liouville’s  Approximation Theorem and Farey sequences.Also the limits for a succession itself with another given limit (it can be zero).(The argument is dual for the base/fiber dimensions).

It is worth noticing the local minima reached for dimension F=3 that we will see that in those cases all the states are available (cycle/cycle) and almost all for the linear/cycle case. This will correspond to the next section. Till now, all the fundamental group were  trivial, because we can “pull” the defect as we wanted, but what happen in the case of a torus? Here are going to see how topologically protected structures arise naturally.

6.-The return of the (topo) logy: Winding numbers in hopfions

Under the previous sections we have seen that all the graph were connected, which means all the states were reachable from any other (trivial fundamental group). But now we are going to study the torus, where intuitively we can see that loops around the two dimensions will not be possible to constraint to a single point (the domain is not simply connected). This will be exactly the host for loops, that will be characterised by its winding number (number of loops around the dimension), which has sign. Note also that in our case, due to the discrete-continuity conditions, the trivial states are already loops in one dimension, so we lose one external degree of freedom. Let’s have an overall view of the connectivity of the smooth discrete bundles by running a manipulate counting and showing the connected components of the rewriting graph.

WARNING: Be careful! Very heavy manipulate, unstable size and  the section bundle selection is global, so the important situations may happen at the edges:

Manipulate[Module[{toro43,rtoro43,ctoro43,ee},{toro43=TrivialBundleGraph[Modulo @ aa, Modulo @ bb,ImageSize->25*aa],

rtoro43=RewriteQuiver[BundleSectionRewritingSystem @ toro43,All,GraphLegend->Automatic,LayoutDimension->3,ImageSize->300];
$BundleSectionDisplayMethod = dd;
ctoro43= ConnectedComponents[rtoro43];
ee=Length[ctoro43];

Flatten[Table[{ctoro43[[i]][[Min[Length[ctoro43[[i]]],cc]]],Length[ctoro43[[i]]]},{i,1,Length[ctoro43]}]]}],

Style["Torus "Dynamic[aa]Dynamic[bb]->Dynamic[ee]"Components",  {Length[ctoro43]}, 12, Bold],

{{aa,4,"B=Base size"},2,12,1,ImageSize -> Tiny},
{{bb,3,"F=Fibre size"},2,12,1,ImageSize -> Tiny},
{{cc,Random[Integer,{1,Length[ctoro43[[1]]]}],"Section bundle"},1,Length[ctoro43[[1]]],1,ImageSize -> Tiny},
{{dd,"Total","Display Method"},{"Total","Array"}}]

If you have played around enough, you will have discovered that here we can have disjoint components, and all these can be uniquely characterised by a single (integer) Chern number, a topological invariant that fully (topologically) characterised the section, in this case the winding number, that represents the amount of loops around the free axis. Note, for example that for F=3, the cardinal is (3^B) because all the states are available! Let’s take a look to some particular cases seen in the previous figure

Remarkable (small) individual cases and interpretation

There exists a minimum threshold below the base is not enough long to wrap around the fibre (we will require dimB>ndimF for hosting 2n {+,- depending on the direction} loops types or topological defects)

t34=TrivialBundleGraph[Modulo @ 3,Modulo @ 4];
FindAllBundleSections[t34]; (*To see them*)
r34=RewriteQuiver[BundleSectionRewritingSystem @ t34,All,LayoutDimension->3,ImageSize->300,ArrowheadShape->None]
c34= ConnectedComponents[r34];
{"Connected components of (Subscript[T, 34])=",Length[c34]}

So here we still don’t have enough basting for wrapping around the fibre dimension. But, when we increase the dimension of the base:

t44=TrivialBundleGraph[Modulo @ 4,Modulo @ 4];
FindAllBundleSections[t44]; (*To see them*)
r44=RewriteQuiver[BundleSectionRewritingSystem @ t44,All,LayoutDimension->3,ImageSize->300,ArrowheadShape->None]
c44= ConnectedComponents[r44];
{"Connected components of (Subscript[T, 44])=",Length[c44]}

We can see here that we have disconnected (and isolated!) components. Let’s try to look explicitly at them:

Rest[ConnectedComponents[r44]]

We can see that they correspond to a single wrapping around the fibre dimension (again, with the chess analogy, like the movement of a bishop). More interestingly it is easy to see that the belong to the group of permutation matrix of dimension 4, isomorphism with the symmetric, Sn → GL(n, Z2). Furthermore, doing some cycle and permutation basic Algebra we can see that the elements correspond to <e,(14)(23),(13)(24),(12)(34),(13),(24),(1432),(1234)> that it is isomorphic to the dihedral Group D4. We observe that each element is “frozen”, is locked on its position because all the section is used to wrap, and the internal degrees of freedom are exhausted. It is worth studying whether the Galois Theory modifies somehow this result for n>5.

We can see that this is a general feature of the torus with equal size of base and fibre, because all the elements are aligned to wrap (there is not degeneracy).

t55=TrivialBundleGraph[Modulo @ 5, Modulo @ 5,ImageSize->80];
FindAllBundleSections[t55]; (*To see them*)
r55=RewriteQuiver[BundleSectionRewritingSystem @ t55,All,LayoutDimension->3,ImageSize->300,ArrowheadShape->None]
c55= ConnectedComponents[r55];
{"Connected components of (Subscript[T, 55])=",Length[c55]}

And what happens when we start to have internal degree of freedom due to a larger base function? We can have “constant” sections, that can be associated with rewriting graphs [”C”->”CC”], with C each of the colours available on the system.

t54=TrivialBundleGraph[Modulo @ 5, Modulo @ 4,ImageSize->80];
FindAllBundleSections[t54]; (*To see them*)
r54=RewriteQuiver[BundleSectionRewritingSystem @ t54,All,LayoutDimension->3,ImageSize->300,ArrowheadShape->None]
c54= ConnectedComponents[r54];
{"Connected components of (Subscript[T, 54])=",Length[c54]}

Here we obtain our first clear example of structures with winding number +1 and -1. Let’s plot some of the elements of them:

$BundleSectionDisplayMethod = "Total";
Table[{RandomChoice[c54[[i]],3],Length[c54[[i]]]},{i,1,Length[c54]}]

Manipulate[Module[{toro,rtoro},
toro=TrivialBundleGraph[Modulo @ tora, Modulo @ torb];
rtoro=RewriteQuiver[BundleSectionRewritingSystem @ toro,All,GraphLegend->Automatic,MaxVertices->5000,MaxEdges->90000,LayoutDimension->3,ImageSize->400,ArrowheadShape->None]],
Style["Torus "Dynamic[tora]Dynamic[torb], 12, Bold],
{{tora,7,"B=Base size"},2,9,1,ImageSize -> Tiny},
{{torb,4,"F=Fibre size"},2,9,1,ImageSize -> Tiny}] (*IDLE*)


comp=MatrixForm[Table[{
Length[ConnectedComponents[RewriteQuiver[BundleSectionRewritingSystem @ TrivialBundleGraph[Modulo @ i, Modulo @ j],All]]]},
{i,1,6},{j,1,6}]]

It is worth playing around with the cases (Subscript[T, 45]),(Subscript[T, 64]),(Subscript[T, 65]),(Subscript[T, 66])  and, if you have enough patience, (Subscript[T, 94]) where the second winding number emerges.  Finally we can see how the connectivity drops in this case, defining again another “degree of completion” based on the ratio greater connected component divided by the total number of elements.

MatrixForm[Table[{N[VertexCount[RewriteGraph[
BundleSectionRewritingSystem[TrivialBundleGraph[Modulo @ i, Modulo @ j]],
Part[FindAllBundleSections[TrivialBundleGraph[Modulo @ i, Modulo @ j]],1]]]/
Length[FindAllBundleSections[TrivialBundleGraph[Modulo @ i, Modulo @ j]]]]},{i,2,7},{j,2,7}]] (*IDLE*)

We can see clearly the requirement about the size of the fibre and the base, the completion of the  (Subscript[C, 3])×F graph. It is clear that the columns will go towards 0, as the different winding numbers follow the integer number, but some extra research is still needed.

7.-The Physics Awakens: From domain walls to skyrmions energetic stabilization

Here we are going to try to use all the new ideas obtained in the previous sections in a particular case relevant to the Physics. We are going to primarily rely on the torus structure (unique, for the moment capable to host the topological solitons). Nowadays, topological structures in Physics has arisen a very profound research field, in  the context of Condensed Matter, Optics or Acoustic. Example of this are baby skyrmions and hopfions [F], obviously in the continuous (real) space. So here we are going to introduce the analogy via the micro magnetic domain walls that will lead us to the skyrmions, that recently has been extended in other areas.

Conjecture 4. The Möbius strip will not introduce new winding structures, while the projective plane and the Klein bottle structure yes.

7.1.-Basic ingredients

If we consider the Picture 9 analogy and recall the concept of Neel and Bloch domain walls, in particular in the context of Micromagnetism when they arise as the interface transition that separates two magnetic domain (or how the system evolves to wrap the broken symmetry insider the magnetic domains). The idea of the angular displacement, it also inspired this concept in other context, as topological solutions that arise when a discrete symmetry is broken (sudden change in a physical parameter). Be that as it may, the fact is that we can prepare a complete analogy between domain walls and some types of bundle sector via the following map:

Manipulate[Module[{h,skyrmion},{TrivialBundleGraph[6,6];
Part[FindAllBundleSections[TrivialBundleGraph[6,6]],rot],
h=π/2*If[rot==1,1,0];
skyrmion=Flatten[Table[{{s Cos[θ]-Cos[θ+h]Sin[-π s]/15,s Sin[θ]-Sin[θ+h]Sin[-π s]/15,-Cos[-π s]/15},
{s Cos[θ]+Cos[θ+h]Sin[-π s]/15,s Sin[θ]+Sin[θ+h]Sin[-π s]/10,+Cos[-π s]/15}},
{s,0,1,0.1},{θ,0,0.0000000001*2 π,π/20/(s+0.00001)}],1]; 
Show[{Graphics3D[{ColorData["Rainbow"][(Flatten[#][[6]]-Flatten[#][[3]])],Arrowheads[0.04],Arrow[Tube[#,0.005]]}&/@skyrmion]},
ImageSize->200,Boxed->False,ViewPoint->{10,100,0}] 
}],
Style["Domain wall analogy with bundle sections", 12, Bold],
{{rot,1,"Neel or Bloch Type, considering the projection into the y axis"},{1-> "Bloch", 96 -> "Neel"},ImageSize -> Tiny,ControlPlacement->Top}]

Manipulate[Module[{h,skyrmion},{TrivialBundleGraph[6,6];
Part[FindAllBundleSections[TrivialBundleGraph[6,6]],rot],
h=π/2*If[rot==1,1,0];
skyrmion=Flatten[Table[{{s Cos[θ]-Cos[θ+h]Sin[-π s]/15,s Sin[θ]-Sin[θ+h]Sin[-π s]/15,-Cos[-π s]/15},
{s Cos[θ]+Cos[θ+h]Sin[-π s]/15,s Sin[θ]+Sin[θ+h]Sin[-π s]/10,+Cos[-π s]/15}},
{s,0,1,0.1},{θ,0,0.0000000001*2 π,π/20/(s+0.00001)}],1]; 
Show[{Graphics3D[{ColorData["Rainbow"][(Flatten[#][[6]]-Flatten[#][[3]])],Arrowheads[0.04],Arrow[Tube[#,0.005]]}&/@skyrmion]},
ImageSize->200,Boxed->False,ViewPoint->{10,100,0}] 
}],
Style["Domain wall analogy with bundle sections", 12, Bold],
{{rot,96,"Neel or Bloch Type, considering the projection into the y axis"},{1-> "Bloch", 96 -> "Neel"},ImageSize -> Tiny,ControlPlacement->Top}]

While all the intermediate states will be hybrid states of the previous. Even more interesting is discovering how now we can see translation along the fiber dimension as global rotation, and this discrete symmetry will allow us to assure that the energy (in a level system, basically it correspond to the integral area) is conserved (in the Heisenberg model) while the translations along the base dimension (global shift) correspond to displacement of the domain wall, another common topic on Micromagnetism . So we could take the quotient to eliminate the cardinal by 36. In the next lines we are going to naturally relate these domain wall analogy with the skyrmions.

The notion of skyrmions, an object in between Physics and Mathematics in the context of solitons, discovered in the context of Nuclear Physics but later applied in many disciples will be presented with special interest in two of its properties: topological protection and locally energetically favourable, whose relation with the Wolfram Project is direct[G].

We will start by commenting the figure of Tony H. Skyrme from which the skyrmions take its name. English physicist from the second part of the last century, and member of the Manhattan project, he focused on nuclear physics in which he initially work on short-range forces in the nuclei, an later develop a theory of non-linear sigma model nuclear forces, lately known as the Skyrme model, where the nucleus was filled with a “medium” caused by three pion fields, remarkably predicting the incipient QCD theory.

As it is well-known, classically in mechanics, particles correspond to an ever-lasting building blocks with non-zero mass, but in QFT approximation, in a closer look, they consist on wave-like excitations of a fundamental field. The idea now is how we made this functions not to disappear, as they would naturally decay.

Here is where the concept of soliton will be related with the skyrmions and again by another British author, the Scottish  engineering J.S Russell. In the beginnings of S.XIX, building water channels in Great Britain, he discovers that although in general dumped boats needed to be pulled because its water waves below rapidly decay, in certain narrow parts, the boat just continues autonomously its travel for really long distances (kilometres!). So something was happening with the water “wave of translation”.

In a more general way, we can characterise this solitons  by the following properties: they are localised (i.e. finite energy and therefore finite size), stable (they have a constant shape in time as a solution of the PDE, and via quantum counterpart it seems to be also true in the discrete case) and algebraic (meaning that the can interact with other solitons and recombine, collide and emerge with a phase shift or collapse). The quantity that fully characterizes the skyrmions corresponds to the number of times the field configuration wraps the unitary sphere  and it’s given by the skyrmion number (analogous to winding number):

n==(1/(4π))∫M·(((∂M)/(∂x))×((∂M)/(∂y)))dxdy===(1/(4π))∫∫((dθ)/(dr))((dϕ)/(dα))sinθdαdr==(1/(4π))(Subscript[[cosθ], θ(r==∞)]^(θ(r==0)))(Subscript[[ϕ], (Subscript[ϕ, f])]^(Subscript[ϕ, i]))==p·m

where M stands for any smooth (continuous) field configuration (it is known that, at least in ferromagnetism, this is no longer a Chern number), and p and m for polarity and vorticity respectively. The sign of it can be positive or negative, depending on the orientation of the vector fields, which leads to talk about (positive) skyrmions, with a positive winding number, or anti-skyrmions, with a negative one, in analogy with the “topological charge” image. It is invariant under smooth transformation and therefore skyrmions are topologically protected when the space is smooth enough, and therefore we prove it would not decay in time.

The last question is how can we derive the skyrmions from the domain walls. And the answer is based on the Toulous-Kleman relations. In essence, domain walls separate one-dimensional topological defect, as being condensed in one dimension, it’s space parameter is trivial, but if the loop the domain wall around the center, we can entangle a  zero-dimensional defect, which is exactly the definition of skyrmion (the unique undetermined quantity is the value in the center, corresponding to the aisle point in the Alexandroff compactification of the stereographic projection, that is the vorticity). Let’s see graphically

Manipulate[Module[{hh,sky},
hh=π/2*hel;
sky=Flatten[Table[{{st Cos[θ]-Cos[θ+hh]Sin[-π st]/20,st Sin[θ]-Sin[θ+hh]Sin[-π st]/20,-Cos[-π st]/20},
{st Cos[θ]+Cos[θ+hh]Sin[-π st]/20,st Sin[θ]+Sin[θ+hh]Sin[-π st]/20,+Cos[-π st]/20}},
{st,0,1,0.1},{θ,0,dim*2 π,π/20/(st+0.0001)}],1]; 
Show[{Graphics3D[{ColorData["Rainbow"][(Flatten[#][[6]]-Flatten[#][[3]])],Arrowheads[0.016],Arrow[Tube[#,0.002]]}&/@sky]},
ImageSize->500,Boxed->False]],
Style["Domain wall transformation to skyrmion", 12, Bold],
{{dim,0.01,"Extra dimension"},0,1,0.01},
{{hel,0.01,"Character, 0=Neel, 1=Bloch"},0,1,0.01,ControlPlacement->Top}]

Each of the intermediate states between the pure Neel and Bloch states can be identify with our bundle sections, but the general classification is presented as future lines of development.

8.-Concluding remarks, future work and discussion

But how can actually be created the skyrmions? The answer is given by Thomson: If you cannot measure it, you cannot improve it. When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind. So the topological protection must be substituted with the notion of a “big-enough” potential barrier, and therefore can be created (or destroyed) applying “enough large” perturbations. Also, topologically stable don’t immediately means energetically stable, and a formal study of the Hamiltonian must be considered.

And the Hamiltonian is solved finding local variation minimization (non-trivial, metaestable) and global (trivial, ground-state) solutions of the variational principle. Therefore, we could associate to each bundle section an energy value corresponding to the difference of energy with the trivial state, corresponding to a ferromagnetic state with all the spin in the same direction, that is, a constant bundle! Then we could measure the “curvature” of each of the states. This parallel favouring is thanks to the direct exchange o Heissenberg term, but it is easy to see how this will NOT give rise to local minima, just global. Therefore, there should be an opposite term that favours antiparallel structures (diamagnetic), and the fine tuning of each of the weight could give rise to the skyrmions, as local solutions of the Hamiltonian.

So we could assignee a weight to each vertex, corresponding to the energy required to evolve from each neighbourhood and develop Brownian motion (or any Montecarlo method, is similarity with the neural networks). That was exactly how skyrmions were experimentally discovered, via Dzaloshinki-Moriya interaction a term that opposed to the Heisenberg term.

In fact there are more terms in a arbitrary material, each with a different relative weight:

H==-J∑(Subscript[S, i])(Subscript[S, j])-K∑(Subscript[S, i]^z)(Subscript[S, j]^z)-B∑(Subscript[S, i]^z)-(∑_i)D·((Subscript[S, i])(Subscript[S, j]))+(Subscript[H, RKKY])

where the summation is extended to neighbours. S corresponds to the spin directions, and the other terms correspond to the following: The first term corresponds to direct exchange or short-range exchange with J representing the integral of exchange and under symmetry condition can be associated with the divergence of the magnetization. The next term is associated with the anisotropy, and it is related with the concept of  “easy axis” having characteristic coefficients K depending on the different symmetries in the material. In third position, we found the magnetic field or Zeeman energy.

Later on, the Dzaloshinki-Moriya interation, which is a asymmetric exchange energy that arise due to the spin-orbit interaction in absence of spatial inversion between a ferromagnetic material and a heavy metal and favors non-uniform structures, and where D represents the coupling coefficient and its key for skyrmion stabilization. Finally, theRuderman-Kasuya-Kittel-Yosida interaction or indirect exchange that happens between heavy materials with d,f shells with delocalized s ones, which polarized and causes a new coupling. Its mathematical expression is omitted for clarity.

Can we apply this approach to our DISCRETE system? Can this mesoscopic objects, quantic? The idea is now trying to see the “boundaries” of this skyrmion theory. We could go to bigger scales, but that would abandon the scope of the modulus so we are going to put our attention in diminishing the size, to reach quantum-like scale. As in this regime the properties of the skyrmion will collide with quantum phenomena like tunnelling, annihilation process that destroys the topological protection in low-dimensional regimens or Heisenberg's uncertainty principle and maybe affect the winding number measurement and precision, and therefore it loses its integer-like behaviour. But there exists theoretical models [H] and we are currently developing a numerical simulator that try to obtain this topological structures in the quantum regime, with the previous theory we can introduce different rewriting rules and relative weigths (for example rewriting rule exclusive for functions in (E^i) ), to try to see if we can deduce the skyrmions naturally from the model.

Furthermore, we have started with Micromagnetism, but currently skyrmions are being discovered in many complex physical systems, from light [I], acoustic, and could be potentially apply to SPP excitation via dipole unidirectional coupling [J], among others.

Despite the conjectures here introduced, we have seen the a link to topological Physics of the quiver theory. Furthermore, critical concepts like topological defects have not been properly addressed [K], while it is quite interesting to see what happen when some vertex are more difficult to reach that others and therefore the rewriting graphs takes alternative paths, leading to energy storared around them.  Also the mirror symmetry has not been fully exploted towards deeper knowledge about discrete Noether theorem, where the different cardinality of the initial  and final spaces (the one of the discrete symmetry and the real one where we measure the physical magnitudes with permission of the falsation of the continuum hypothesis), will naturally lead to topological protection, as there are no such more available states, like in the quantum model for the atom.

Finally I want to mention that I am fully convinced that the limit of Physics when the technology tends to infinity is Mathematics, and we can always find a “physical sense” of discarded Mathematics solutions, like fully-complex solutions of real-number problems [J], in other words

(lim_(Technology->∞))Physics==Mathematics

9.-Keywords

Quiver

Discrete Geometry

Fibre bundles

Graph rewriting

Skyrmions

Quantum Skyrmions

Winding Number

Topological defects

Cayley graphs

Domain Walls

10.-Acknowledges

11.-Bibliography

[A] Munkres J. R., Topology: A First Course, 6th ed., Prentice-Hall, 1974.

[B] Wolfram S., “A Class of Models with the Potential to Represent Fundamental Physics” Complex Systems, 29 (2), 2020 pp. 107-536.

[C] Graham Van Goffrier “[WSS20] Full Discretization of Local Gauge Invariance.” (Last date visited July 2022) https://community.wolfram.com/groups/-/m/t/2030337.

[D] Matthew Maddock “[WSS21] Gauge field theories on discrete principal fibre bundles.” (Last date visited July 2022) https://community.wolfram.com/groups/-/m/t/2312018.

[E] Capobianco S., “Can anything from Noether’s theorem be salvaged for discrete dynamical systems?” International Conference on Unconventional Computation, 6 (6), 2011 pp. 77-88.

[F] Kobayashi M., Nitta M., “Torus knots as Hopfions” Physics Letters B, 728, 2014 pp. 314-318.

[G] Vladyslav Kuchkin “[WSS21] Non-trivial homotopies of arbitrary hypergraphs and their solitons.” (Last date visited July 2022) https://community.wolfram.com/groups/-/m/t/2311778.

[H] Lohani V., Hickey C., Masell J., Rosch A., “Quantum Skyrmions in Frustrated Ferromagnets” Phys. Rev. X   9 (041063), 2019.

[I] Shen Y., Zhang Q., Shi P., Du L., Zayats A. V., Yuan X., “Topological quasiparticles of light: Optical skyrmions and beyond.?” Unplublished  arXiv:2205.10329, May 2022.

[J] Perea-Puente, S., Rodriguez-Fortuño “Dependence of evanescent wave polarization on the losses of guided optical modes” Phys. Rev. B, 104 (085417), 2021.

[K] Keith Patarroyo “[WWS21] Defect movement in lattice rewriting systems.” (Last date visited July 2022) https://community.wolfram.com/groups/-/m/t/2164049.