In this project we expanded previous work related to discrete abelian gauge theories through the group SU(2). We examined the SU(2) Hopf bundle on through the Hurwitz group, then conducted several simulations involving Wilson loops in two dimensions for both U(1) and SU(2) and studied their different behaviour on both usual lattices and hypergraphs.
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Overview
Overview
Gauge theories. The SU(2) group
Gauge theories. The SU(2) group
In physics, symmetry plays a fundamental role, providing a mathematical framework for understanding conservation laws and fundamental structures in physical systems. They describe transformations that leave certain properties unchanged, such as rotations, reflections and translations, all of which can be described using the formalism of group theory. One can classify these symmetries in two main categories: discrete and continuous. On one hand, discrete groups consist of isolated symmetry operations, like the cyclic group of rotations of a polygon. On the other hand, continuous groups describe symmetries that depend smoothly on continuous parameters, forming so-called Lie groups. They can be viewed as both algebraic and geometric objects, represented in terms of differentiable manifolds. One particular Lie group is SU(2) (special unitary group), which we will discuss more in detail in what follows. SU(2) consists of all unitary 2x2 matrices with determinant one. Geometrically, it has a deep connection with quaternions, the two being isomorphic. Specifically, each quaternion with +++1maps to the SU(2) matrix
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Fiber bundles
Fiber bundles
A fiber bundle is a structure that enables describing spaces that locally look like the product of two spaces, which, however, may have a more complicated structure globally. A fiber bundle consists of a total space , a base space , a projection map and a fiber , such that, for locally around each the total space looks like a product (U)UF , with an open neighbourhood in containing .
When a fiber bundle can be written globally as a product, the bundle is said to be trivial. The cylinder is a typical example of such a structure, given by the direct product[-1,1]. An example of a nontrivial vector bundle is the tangent bundle of the 2-sphere. Here at each point the tangent plane is , but it's impossible to assign non-vanishing tangent vectors everywhere on the sphere without discontinuities. In less formal language, you can't metaphorically comb the sphere without creating singularities, which is referred to as the hairy ball theorem. Indeed, one cannot write the fiber bundle as a product . Below you can find a visualisation of the above tangent bundle":
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When a fiber bundle can be written globally as a product, the bundle is said to be trivial. The cylinder is a typical example of such a structure, given by the direct product
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Wilson loops
Wilson loops
The idea of Wilson loops is inspired from parallel transport in general relativity, which compares tangent vectors that live in tangent spaces at different points. In the fibre bundle language, this translates to comparing different fiber points through a connection, which is equivalent to introducing a gauge field. A connection is a way to separate a principal bundle in two spaces, called horizontal and vertical subspaces. The later refers to all vectors pointing along a certain fiber, while the first represents all vectors perpendicular to the fiber. This enables one to compare fiber values at different spacetime points by connecting them with curves that are always perpendicular to any given fiber.If we are to give a brief physical interpretation in the context of gauge fields theories, Wilson loops measure how a particle with a given charge behaves when moving along a closed loop under the influence of the given field, so how the particle's internal properties change when travelling along the loop. The behaviour of the Wilson loops as a function of the loop size can give clues about relevant physical phenomena, like quark confinement. In simple terms, if the Wilson loop exponentially decays with the area, it implies confinement, while an exponential decay with the perimeter implies non-confinement properties. However, this also depends on the number of dimensions considered. For our simulation, we restricted ourselves to two dimensions, but the principle can be extended to other dimensions.Explicitly, the formula for Wilson loops is given by:where denotes the closed loop, is the path-ordering operator and Tr is the trace. The loop integral in the exponent denotes the gauge connection integrated along the loop .The reason why area law implies confinement is that it means that the potential has to be some linear function of the distance between two particles, so this means that it becomes increasingly difficult to separate them as the distance increases. We say that measures the holonomy around , that is, how much the internal state of the particle twists as a result of going around the loop. The holonomy describes a mapping of a fiber onto itself upon a horizontal lift along a closed loop.
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SU(2) Hopf fibration
SU(2) Hopf fibration
The Hopf fibration shows how one can build higher-dimensional spheres from lower-dimensional ones. To illustrate the concept, we will first briefly discuss the simplest case, the decomposition of the 3-sphere into a collection of circles parametrised by points on the 2-sphere, denoted by ↪ where π is the projection map. To unpack this, note that the 3-sphere can be realised as a unit sphere defined by
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An -action on defined by simultaneous phase rotation Note that the result of this action stays in . The orbits of this action form circles and the space of orbits is diffeomorphic to the 2-sphere .A projection map realizing this quotient is given explicitly by , which defines a map from to .Analogously, the 7-sphere can be defined as the unit sphere in . There exists an -action on where the 3-sphere can be identified with the unit quaternions. This action can be written in terms of quaternionic multiplication. Explicitly, by identifying ≅ we write points on the 7-sphere as pairs of quaternions satisfying Then, the action of the 3-sphere on the 7-sphere is given by:The orbits of this action are 3-spheres and the space of orbits is diffeomorphic to the 4-sphere.Thus, we obtain the fibration ↪, known as the quaternionic Hopf fibration. The projection map can be written explicitly in terms of quaternionic coordinates as which defines a map from the 7-sphere to the 4-sphere. This includes the point at infinity when 0
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SU(2) Hopf fibration
SU(2) Hopf fibration
We started with generating a discretisation of the 7-sphere as a semi-regular polytope with 240 points, obtained through embedding it in the space and parametrising it using quaternions. Then, the discrete gauge group is a nonabelian 24-elements subgroup of SU(2), called the Hurwitz group:The action of maps points from the 7-sphere to other points on the 7-sphere. Next, we can construct the orbits under the action of , which are 10, each containing 24 points, as acts without fixed points. The fiber invariant defined above as takes one of the values , which can be thought of as points on the 4-sphere. These form the base space of the bundle. Each point is adjacent to 8 others, so excluding the opposite point. One can show that there is no global section by showing that a connection cannot be extended to all fibers, and so the bundle remains nontrivial. The code below first reconstructs the quaternions and orbits:
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(*OrbitDecomposition*)ClearAll["Global`*"];(*1.Buildthequaternions*)e0={1,0,0,0};e1={0,1,0,0};e2={0,0,1,0};e3={0,0,0,1};Q8={e0,-e0,e1,-e1,e2,-e2,e3,-e3};signTuples=Tuples[{1,-1},4];halfSum=Table[(s[[1]]*e0+s[[2]]*e1+s[[3]]*e2+s[[4]]*e3)/2,{s,signTuples}];q24=DeleteDuplicates[Join[Q8,halfSum]];Print"✔ # of quaternions: ",Length[q24];(*2.Buildthe24diagonalleft-multiplicationactionson*)quatLeft[{a_,b_,c_,d_}]:={{a,-b,-c,-d},{b,a,-d,c},{c,d,a,-b},{d,-c,b,a}};actions=Table[ArrayFlatten[{{quatLeft[q],ConstantArray[0,{4,4}]},{ConstantArray[0,{4,4}],quatLeft[q]}}],{q,q24}];Print"✔ # of actions: ",Length[actions];(*Expect24*)(*3.Buildthe240E_8roots*)e[i_]:=UnitVector[8,i];rootsA=Flatten[Table[s1*e[i]+s2*e[j],{i,1,8},{j,i+1,8},{s1,{1,-1}},{s2,{1,-1}}],3];rootsB=Select[Tuples[{1/2,-1/2},8],EvenQ[Count[#,-1/2]]&];rootsE8=Join[rootsA,rootsB];Print"✔ # of rootsE8: ",Length[rootsE8];(*4.Partitionintoorbits*)orbitMap=<||>;orbits={};Do[root=rootsE8[[i]];keyRoot=ToString[root,InputForm];If[KeyExistsQ[orbitMap,keyRoot],Continue[]];(*Generatetheorbitunderthe24actions*)orbitVecs=DeleteDuplicates[Table[actions[[j]].root,{j,Length[actions]}]];orbIdx=Length[orbits]+1;AppendTo[orbits,orbitVecs];(*Recordeachvector’skey*)Do[orbitMap[ToString[orbitVecs[[k]],InputForm]]=orbIdx,{k,Length[orbitVecs]}];,{i,Length[rootsE8]}];Print"✔ # of orbits found: ",Length[orbits];
They can also be visualised individually:
Wilson loops simulations
Wilson loops simulations
On lattices
On lattices
The simulation below initializes the link variables on the lattice to be close to the identity, representing a nearly trivial vacuum configuration, but with perturbations of randomness to simulate small field fluctuations. This is done by adding small random noise to the identity quaternion and then normalizing to ensure it stays on the SU(2) group manifold. Each link on the lattice is assigned such a random SU(2) matrix, both in the horizontal and vertical directions. The lattice has periodic boundary conditions, effectively wrapping the edges around to form a torus. It turns out that if we eliminate this condition, the behaviour of the simulation remains almost the same.
At each plaquette (1x1 loop), the code computes the holonomy around the loop, from which the Wilson loop is calculated. Though this particular simulation operates near the continuum limit with weak fields, the same framework can be extended to explore more complex, strongly fluctuating regimes that are central to understanding phenomena like confinement and phase transitions.
If we repeat the procedure for U(1), the graph below is more shifted towards 1, as would be expected from an abelian group.
At each plaquette (1x1 loop), the code computes the holonomy around the loop, from which the Wilson loop is calculated. Though this particular simulation operates near the continuum limit with weak fields, the same framework can be extended to explore more complex, strongly fluctuating regimes that are central to understanding phenomena like confinement and phase transitions.
If we repeat the procedure for U(1), the graph below is more shifted towards 1, as would be expected from an abelian group.
The simulation below tests a toy model of the area law in 2D - it randomly updates the gauge field matrices on the edges to minimize the action, which is a quantity related to the energy stored in the field configurations. Once the simulation reaches equilibrium, the expectation value of Wilson loops of various sizes is measured. Incidentally, the results look like what we would expect from actual confinement (the area law is respected for SU(2) but not for U(1)), but that is not the case and has to do with the constraint that bidimensionality poses. However, the simulation builds the intuition and provides the pipeline to extend the study to higher dimensions.
On hypergraphs
On hypergraphs
The above construction can be easily transferred to hypergraphs. For this subsection we use a hypergraph with a given rewriting rule indicated below instead of our lattice and perform the simulation again. First, we extract the number of vertices, hyperedges and specify a specific range for cycle range (here between 3 and 9):
Most notably, there are more values around 0 as the cycle length goes up and the distribution looks fairly smooth.
If we repeat the procedure for U(1), the values tend more towards the extremities of the domain:
If we repeat the procedure for U(1), the values tend more towards the extremities of the domain:
The results of both simulations are consistent with the (non)abelian structure of the two groups. In two dimensions, the results are mainly caused by the two groups' structure and geometric constraints. Increasing the dimensionality to 4D would enable studying more nontrivial effects caused by actual confinement.
To generate an ensemble of energy‑weighted field configurations, we can use the Metropolis Monte Carlo method, like in the first part of this section. It starts with some initial assignment of gauge variables on every link of the 4D lattice. Then, one link at a time, applies a small random change to its value and calculate how much the change would raise or lower the system’s total energy. If the proposal lowers the energy, it is always accepted - if it raises the energy it is only accepted with a certain probability, reflecting thermal fluctuations. By repeating this cycle, the overall collection of configurations visited ends up reflecting the true balance between energy minimization and thermal fluctuations. Measuring Wilson loops in that ensemble then reveals a genuine area‑law decay tied to the flux tubes of an SU(2) gauge field in four dimensions.
To generate an ensemble of energy‑weighted field configurations, we can use the Metropolis Monte Carlo method, like in the first part of this section. It starts with some initial assignment of gauge variables on every link of the 4D lattice. Then, one link at a time, applies a small random change to its value and calculate how much the change would raise or lower the system’s total energy. If the proposal lowers the energy, it is always accepted - if it raises the energy it is only accepted with a certain probability, reflecting thermal fluctuations. By repeating this cycle, the overall collection of configurations visited ends up reflecting the true balance between energy minimization and thermal fluctuations. Measuring Wilson loops in that ensemble then reveals a genuine area‑law decay tied to the flux tubes of an SU(2) gauge field in four dimensions.
Concluding Remarks
Concluding Remarks
Acknowledgments
Acknowledgments
I am grateful to my mentor, Dugan Hammock, for all the support provided throughout this project, as well as to the whole WSS team for this experience.
References
References
1
.Manton N. S., Connections on discrete fibre bundles. Comm. Math. Phys. 113 (1987), no. 2, 341-351. https://projecteuclid.org/euclid.cmp/1104160217
2
.Wolfram Physics Project: Technical Documentation. https://www.wolframphysics.org/technical-introduction/
3
.Makeenko Y., A Brief Introduction to Wilson Loops and Large N, Phys.Atom.Nucl. 73 (2010) 878-894 https://inspirehep.net/literature/823994
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
An Investigation of Discrete SU(2) Gauge Theory through the Hopf Fibration and Wilson Loops
by Ioana Milea
Wolfram Community, STAFF PICKS, July 9, 2025
https://community.wolfram.com/groups/-/m/t/3497643
by Ioana Milea
Wolfram Community, STAFF PICKS, July 9, 2025
https://community.wolfram.com/groups/-/m/t/3497643