The aim of this project is to attempt for a description of particles as topological objects defined over general hypergraphs. Here, we propose a first description based on the Skyrme model. Our preliminary goal is to obtain a solution that describe a topological soliton (skyrmion) defined over the simplest three-dimensional graph, represented by a tetrahedron. We plan to extend this study to include more complicated structures, suitable for an hypergraphical description of the spacetime.

Introduction & Background

Particles as solitons

Topological solitons are characterized by interesting properties, which have been studied for decades . They arise as solutions to non - linear equations and occurs in several fields of physics, acquiring a great importance in describing non-perturbative properties of many physical systems . More in detail, solitons are localized solutions with a finite energy; in this sense, they behave like particles, whose properties are defined by the topological structures of the underlying manifold . A very interesting property of this description is represented by the fact that fermions arises as solitonic states from bosonic fields . It is the case, for instance, of the baryonic states described by the Skyrme model, in which baryons arise as solitons (called skyrmions) in a mesonic field .

The Skyrme model

The Skyrme model can be understood as a type of Non-Linear Sigma Model (NLσM), with an additional term, called Skyrme term, which is introduced for the stability of solitonic solutions in three infinite spatial dimensions, called skyrmions [1]. More in details, we can define the Lagrangian as ​ ℒ =
K
2

L
μ
μ
L
+
λ
8

G
μν
μν
G

, where​
L
μ
=
-1
U
∂
μ
U
and
G
μν
=[
L
μ
,
L
μ
].
​​The second term of the Lagrangian represents the already mentioned Skyrme term. Here, U (generally named as Skyrme field) is a map from the spacetime to a three dimensional manifold lying in a simple compact Lie group G, representing the underlining flavor group,​ U :
3+1

→ G.​The values of the constants K and λ can be estimated through a comparison with the physical properties of baryons [2]. It is worth noticing here that originally the Skyrme model was defined over an infinite space. Nevertheless, the space in this project is represented by finite graphs or hypergraphs. For this reason, the stability of the skyrmion is assured by the finiteness of the space, allowing us to drop the Skyrme term. For simplicity, in the following sections we will consider the bare NLσM, with λ = 0.Let us observe that in any simple compact Lie group the third homotopy group is non-trivial; in particular,​
π
3
(G)=
. Therefore, we can associate an integer to each solution, represented by the index of the map U. This quantity is a topological invariant, representing the number of skyrmions that compose the system. In a more simple way, this number counts how many times the map U wraps around a three dimensional hole in G and it is defined by the integral​
B=
1
24
2
π
∫L∧L∧L
,​where L is the one-form with components
L
μ
(
L=
L
μ
μ
dx
) and Λ is the wedge product. ​The equations for the Skyrme field U are called Skyrme equations and can be deduced from the Lagrangian, acquiring the form​
∇
μ
μ
L
+
λ
4
[
L
ν
,
μν
G
]=0
,​where
∇
μ
is the covariant derivative, which becomes the usual partial derivative
∂
μ
in a flat spacetime, and
[,]
states for the Lie parenthesis. Thus, the Skyrme equations composes a systems of three coupled partial differential equations. Finding solutions is an hard task and an ansatz is certainly needed. In this project, we make use of a particular ansatz based on the general Euler parameterization for the Lie groups, better defined here below.

Baryons as skyrmions

Skyrmions has been shown to represents a suitable description of baryons, since they are able to reproduce many physical properties. An example is represented by nucleons masses and their interaction radius. Furthermore, this idea is enforced by the works of t’Hooft and Witten, in which they show that the Skyrme model can be linked to QCD, since it can be interpreted as the leading order in the
N
c
expansion, where
N
c
is the number of colors, which is treated as a perturbative parameter.

A first step

As already mentioned, the aim of this project is to check the possibility build these solutions over a spacetime made of simple graphs or hypergraph. This project could represents a first step toward the description of particles in the Wolfram Physics Project [2].​In order to start with a simple setup, we considered the following characteristics.​​Firstly, we adopted a parameterization of the map U already introduced in [4]. The intent of the original work was to study some particular solutions arising from mapping a finite volume, defined as a cube, into G. Here, we analyzed the possibility of defining a very similar parameterization when the space is represented by the most simple graph that can be built in three dimensions. Namely, a tetrahedron. ​​Secondly, we considered G ≡ SU(2), in order to simplify the computations.The map in question is based on the so called general Euler parameterization of a Lie group. In the three dimensional case, when G ≡ SU(2), it becomes​
U(x,y,z)=
xσ
1
e
yσ
3
e
zσ
1
e
,
​​where x, y, and z are general parameters and
σ
i
are matrices are defined in terms of the Pauli matrices that generates the Lie algebra of SU(2), in details,​
σ
i
=

2
τ
i
, where
τ
i
are the Pauli matrices.In the most general case, x, y, and z could be functions of the spacetime. Nevertheless, following [4], we are going to use the ansatz​
U(r,t-y,z)=
(t-y)
σ
1
e
f(r)
σ
3
e
zσ
1
e
, with r, y, z parametrizing the space and t represents the time. f(r) is a general function of r, which is defined by solving the Skyrme equations. As for the case studied in [4], we will see in the next sections that this really simplifies the computations.

An observation on the range of f(r), y and z.

The parameters y and z should be defined over a certain range. In the same way, the function f(r) should have some specific boundary conditions. The reason of this constraint lie in the fact that the map U needs to cover an entire, closed cycle in order to have a defined topology. This problem has already been addressed in [5]. We mention here that this condition is satisfied when​
0≤y≤4π,0≤z≤4πandf(0)=0,f(
L
r
)=π
. ​
L
r
is the range of r, that can be taken arbitrary. For simplicity, we are going to slightly modify the ansatz above as follows​
U(r,t-y,z)=
4(t-y)
σ
1
e
f(r)
σ
3
e
4
zσ
1
e
, where now, the first and third exponent have been multiplied by a factor 4. This way, the ranges of y and z restrict to​
0≤y≤π,0≤z≤π
. This will be useful for the future parameterization. Furthermore, also the parameter r will be considered in the following range​
0≤r≤π
, giving
f(0)=0,f(π)=π
.

First results

The code relative to this parameterization defined above is reported here below.
σ represents the Pauli matrices.
In[]:=
σ=

2
Table[PauliMatrix[k],{k,3}];​​MatrixForm/@σ
Out[]=

0

2

2
0
,
0
1
2
-
1
2
0
,

2
0
0
-

2

Φ and Θ will be used in order to define the first and third exponential of the parameterization of the map U.
In[]:=
Φ[t_,z_]:=4(t-z)​​Θ[y_]:=4y
Then, U can be defined as here below.
In[]:=
U[f_,y_,z_]:=Simplify[MatrixExp[z σ[[1]]].MatrixExp[f σ[[3]]].MatrixExp[y σ[[1]]]]

The parameterization

In[]:=
xr=π;​​=Polyhedron[{{0.,0.,0.},{xr,0.,0.},{0.,xr,0.},{0.,0.,xr}},{{2,3,4},{3,2,1},{4,1,2},{1,4,3}}];​​Graphics3D[]​​
Out[]=

The coordinates

The choice of the parameters characterizing the map U has been chosen lying along the perpendicular edges of the tetrahedron, named as X, Y ad Z.

The metric

The metric in the new coordinates r, y and z and the time t is reported here below.
The following part of the code is devoted to the definition of the metric corresponding to the parametrization of the spacetime in the parameters (t,r,y,z). gS corresponds to the spatial part of the metric and g is the full metric tensor.

The Vielbein

Equations of motion

Using the parameterization introduced above, the Skyrme equations really simplifies, leading to an ordinary differential equations for f(r).
This first part of the code is only devoted to the choice of the metric tensor we want to use and the computation of its inverse.
Here below, the various parts that compose of the Skyrme equation are defines. Γ represents the Levi-Civita tensor, which is non zero for a general metric, but is zero in our case.
The LHS of the Skyrme equation Eq = 0 becomes

Baryonic charge

Integrating over the volume of the tetrahedron, one arrives to the following result
As expected, the baryonic charge is an integer number, stating that the model presents solutions that are topologically non-trivial, which could accounts for a first description of graphical particles .

A comment on the results and future development

In this work we defined a parameterization for a very simple graph that enable us to find topologically non-trivial solutions to the Skyrme model, which can potentially lead to a description and classification of topological particles over hypergraphs. In this scope, we plan to extend this description for more complicated structures, starting by studying the properties of the map U arising from gluing simple hypergraphs.

Acknowledgments

Firstly, I would like to thanks Xerxes D. Arsiwalla and Stephan Wolfram for this very interesting opportunity and for having introduced me to this world. Furthermore, I would really like to thanks my mentor Adiba Shaikh for her help, support and patience. Another great thanks goes to the all the organizers of the school to all the speaker and participants, for having increased my knowledge with their talks and comments.

References

[1] N.S. Manton, P. Sutcliffe, Topological solitons, Cambridge Monographs on Mathematical Physics (2004).
[2] https://www.wolframphysics.org/technical-introduction/
[3] G. S. Adkins, C. R. Nappi, E. Witten, Static Properties of Nucleons in the Skyrme Model, Nucl.Phys.B 228 (1983) 552.
[4] S.L. Cacciatori, F. Canfora, M. Lagos, F. Muscolino, A. Vera, Cooking pasta with Lie groups, Nucl.Phys.B 976 (2022), 115693.
[5] S.L. Cacciatori, F. Dalla Piazza, A. Scotti, Compact Lie groups: Euler constructions and generalized Dyson conjecture, Trans.Am.Math.Soc. 369 (2017) 7, 4709-4724.

CITE THIS NOTEBOOK

Graphical Exploration of Topological Particles in the Skyrme Model​
by Federica Muscolino​
Wolfram Community, STAFF PICKS, January 12, 2024
​https://community.wolfram.com/groups/-/m/t/3101578