Group-Valued Holonomies in Discrete Gauge Theory

• Group-valued holonomies are products of group edge weights around cycles that detect whether local connection data can be globally coherent.
• They provide a discrete analogue of curvature by quantifying deviations from flatness through a scalar contextuality index (κ) computed via invariant group distances.
• They underpin applications in gauge theory and machine learning, acting as regularizers that promote global consistency in complex network models.

Searching arXiv for the cited holonomy/contextuality papers to ground the article. Group-valued holonomies are ordered products of group-valued edge weights around closed paths, used to quantify whether local connection data can be made globally coherent. In the framework of group-valued Boltzmann machines, a bipartite graph $\mathcal G=(V,H,E)$ is equipped with weights $w_{ij}\in G$, where $G$ may be a finite group, $\mathrm{SU}(2)$, $\mathrm{GL}_n(\mathbb R)$, or an infinite-dimensional operator group. The resulting holonomies along cycles define a scalar measure of deviation from flatness, interpreted as a contextuality index $\kappa$: vanishing holonomy corresponds to a flat discrete connection and to the existence of a global section, whereas nontrivial holonomy records an obstruction to gluing local data into a single global description (Magnot, 5 Sep 2025).

1. Formal definition

The basic setting is a graph whose oriented edges carry values in a group $G$. In the group-valued RBM construction, one begins with a bipartite graph $\mathcal G=(V,H,E)$ with visible nodes $V$, hidden nodes $H$, and edges $w_{ij}\in G$0. Each edge $w_{ij}\in G$1 carries a group-valued weight $w_{ij}\in G$2. To define transport along arbitrary oriented paths, one passes to the bidirected graph and sets

$w_{ij}\in G$3

If

$w_{ij}\in G$4

is a cycle, its holonomy is the ordered product

$w_{ij}\in G$5

A cycle is flat if and only if $w_{ij}\in G$6, where $w_{ij}\in G$7 is the identity of $w_{ij}\in G$8 (Magnot, 5 Sep 2025).

To turn holonomy into a scalar quantity, one chooses a distance $w_{ij}\in G$9 on the group. The paper lists geodesic distance on a compact Lie group, the Cayley distance on a finite group, and matrix norms such as $G$0 or $G$1 on matrix groups. The cycle-level deviation from flatness is then

$G$2

Given a prescribed family of cycles $G$3, the global contextuality index is

$G$4

By construction, $G$5. For a sufficiently rich family $G$6, $G$7 if and only if every cycle in $G$8 is flat. In a bipartite RBM, the shortest nontrivial cycles are $G$9-cycles, so these furnish the natural local loci where holonomic inconsistency first appears.

Symbol	Definition	Role
$\mathrm{SU}(2)$0	group-valued edge weight	discrete connection datum
$\mathrm{SU}(2)$1	ordered product around a cycle	parallel transport around $\mathrm{SU}(2)$2
$\mathrm{SU}(2)$3	$\mathrm{SU}(2)$4	cycle-level non-flatness
$\mathrm{SU}(2)$5	average of $\mathrm{SU}(2)$6 over $\mathrm{SU}(2)$7	global contextuality index

2. Flatness, global sections, and discrete bundles

The geometric interpretation is formulated in terms of a discrete principal $\mathrm{SU}(2)$8-bundle. The weights $\mathrm{SU}(2)$9 are treated as edge-labeled connection data, and coherence is defined by the existence of a global section

$\mathrm{GL}_n(\mathbb R)$0

such that

$\mathrm{GL}_n(\mathbb R)$1

If such an $\mathrm{GL}_n(\mathbb R)$2 exists, then for any cycle $\mathrm{GL}_n(\mathbb R)$3,

$\mathrm{GL}_n(\mathbb R)$4

Thus a global section forces every cycle holonomy to be trivial (Magnot, 5 Sep 2025).

The converse is equally important. If $\mathrm{GL}_n(\mathbb R)$5 for all cycles in a connected graph, one may fix a base node $\mathrm{GL}_n(\mathbb R)$6 and define

$\mathrm{GL}_n(\mathbb R)$7

where $\mathrm{GL}_n(\mathbb R)$8. Path-independence follows from flatness, because different paths from $\mathrm{GL}_n(\mathbb R)$9 to $\kappa$0 differ by a cycle whose holonomy is the identity. The paper therefore states the equivalence: flatness of all cycle holonomies is equivalent to existence of a global section, and $\kappa$1 with $\kappa$2 generating the cycle space is equivalent to flatness of the discrete connection (Magnot, 5 Sep 2025).

This equivalence gives group-valued holonomies a precise contextual interpretation. Nontrivial holonomy is not merely a local mismatch; it is an obstruction to global trivialization. In the terminology of the paper, non-zero $\kappa$3 indicates accumulated inconsistency around cycles, exactly the discrete analogue of curvature. A plausible implication is that the holonomy formalism identifies contextuality at the level of discrete transport rather than at the level of only probabilistic compatibility.

3. Non-abelian structure, gauge transformations, and Wilson loops

The formalism is especially sensitive to non-abelian structure. When $\kappa$4 is abelian, cycle holonomies reduce to additive or commutative accumulation of edge labels. When $\kappa$5 is non-abelian, order matters, and commutators contribute nontrivially to $\kappa$6. The paper makes this explicit for $\kappa$7 by considering cycle weights such as

$\kappa$8

for which the product does not collapse to a simple sum of angles. A BCH expansion yields

$\kappa$9

and the commutators $G$0, etc., generate second-order deviations from flatness (Magnot, 5 Sep 2025).

The same section introduces gauge transformations. A node-wise map $G$1 acts by

$G$2

Under this transformation,

$G$3

Therefore, if $G$4 is conjugation-invariant, then both $G$5 and $G$6 are gauge-invariant. The paper identifies this as the natural invariance expected for curvature-like quantities (Magnot, 5 Sep 2025).

This places group-valued holonomies squarely in a gauge-theoretic vocabulary. The holonomies around cycles are described as discrete Wilson loops, and $G$7 is presented as a curvature functional on a lattice gauge theory over the RBM graph. The paper also discusses Berry/Wilson-type scalar variants,

$G$8

together with operator, trace, and Hilbert–Schmidt norm versions of $G$9. The natural significance is that group-valued holonomies admit both intrinsic group-distance formulations and representation-dependent scalar summaries.

4. Computation and representative examples

A practical computation begins by choosing a cycle family $\mathcal G=(V,H,E)$0. The paper recommends either all simple cycles up to a given length, all chordless $\mathcal G=(V,H,E)$1-cycles in a bipartite RBM, or a basis of the cycle space. Using a cycle basis avoids overcounting and isolates independent holonomic obstructions. A step-by-step procedure is given: choose a spanning tree, form the fundamental cycles, compute $\mathcal G=(V,H,E)$2 for each cycle, evaluate $\mathcal G=(V,H,E)$3, and average to obtain $\mathcal G=(V,H,E)$4 (Magnot, 5 Sep 2025).

Two explicit examples illustrate the range from finite groups to Lie groups.

For $\mathcal G=(V,H,E)$5 with addition modulo $\mathcal G=(V,H,E)$6, let

$\mathcal G=(V,H,E)$7

On a $\mathcal G=(V,H,E)$8-visible $\mathcal G=(V,H,E)$9 $V$0-hidden fully connected RBM, the two $V$1-cycle holonomies are

$V$2

so both cycles are flat and $V$3. If $V$4 is changed to $V$5, then

$V$6

hence $V$7, $V$8, and

$V$9

The example shows that a single edge change can induce global inconsistency detectable only after traversing a cycle (Magnot, 5 Sep 2025).

For $H$0, the paper first considers an abelian $H$1 subgroup generated by $H$2: $H$3 Then

$H$4

Using the Frobenius norm,

$H$5

so

$H$6

For

$H$7

one has $H$8, hence

$H$9

If $w_{ij}\in G$00 contains only this cycle, then $w_{ij}\in G$01 (Magnot, 5 Sep 2025).

These examples make clear that the same formal object—ordered transport around a loop—supports discrete parity-type holonomy in finite groups and continuous curvature-like holonomy in Lie groups.

5. Contextuality, gluing obstructions, and related geometric indices

The paper explicitly connects group-valued holonomies to sheaf-theoretic contextuality. In the Abramsky–Brandenburger framework, noncontextuality means that local sections glue to a global section. The holonomy formalism translates this into discrete bundle language: weights are transition functions, and nontrivial holonomies are exactly the obstructions to gluing. On this reading, $w_{ij}\in G$02 quantifies the obstruction to a global section and thus furnishes a geometric measure of contextuality (Magnot, 5 Sep 2025).

This geometric interpretation differs in construction from the probabilistic contextuality measures developed in Contextuality-by-Default. In CbD, one begins from context-labeled random variables that are stochastically unrelated across contexts, constructs maximal couplings for same-content variables, and defines contextuality as the gap between the sum of maximal pairwise similarities and the best total similarity realizable in a single overall coupling. For the Alice–Bob system, the contextuality index is

$w_{ij}\in G$03

with $w_{ij}\in G$04 exactly in the noncontextual case and $w_{ij}\in G$05 exactly in the contextual case (Dzhafarov, 2021). A plausible implication is that $w_{ij}\in G$06 and $w_{ij}\in G$07 address the same global-versus-local tension from different mathematical directions: one through discrete transport and flatness, the other through couplings of random variables.

A second geometric comparison is furnished by the coset-based contextuality measure

$w_{ij}\in G$08

where $w_{ij}\in G$09 is the number of lines in an incidence geometry and $w_{ij}\in G$10 is the number whose cosets are mutually commuting. In this approach, contextuality is the fraction of lines whose commutator structure fails to match the quantum commutation structure. For Mermin’s square, $w_{ij}\in G$11; for the split Cayley hexagon, $w_{ij}\in G$12 (Planat, 2014). Both $w_{ij}\in G$13 and $w_{ij}\in G$14 are geometric and cycle- or line-sensitive, but $w_{ij}\in G$15 is formulated directly in terms of group-valued transport and conjugation-invariant distances rather than defective incidence lines.

More recently, projector geometry has produced another state-independent contextuality index,

$w_{ij}\in G$16

where

$w_{ij}\in G$17

That framework also interprets contextuality as a configuration-level obstruction to joint classicality and proves that $w_{ij}\in G$18 is a necessary configuration-level condition for observable contextuality (Günhan et al., 26 Apr 2026). This suggests a broader pattern: contemporary contextuality theory increasingly expresses nonclassicality through geometric obstructions—failure of global gluing, nontrivial holonomy, noncommuting coset coordinatizations, or overlap-matrix curvature.

6. Applications and scope

Within machine learning, the most immediate application is to group-valued Boltzmann machines. Allowing weights to take values in groups such as $w_{ij}\in G$19, $w_{ij}\in G$20, or operator groups is proposed as a way to model projective transformations, spinor dynamics, and functional symmetries, with applications to vision, language, and quantum learning. In that setting, minimizing $w_{ij}\in G$21 during training encourages a flat, globally consistent connection. The paper describes this as a regularizer promoting geometric consistency, a robustness prior in adversarial or noisy environments, a structural bias in quantum or spin-based architectures, and a diagnostic of topological memory through persistent non-zero holonomies across homologically distinct cycles (Magnot, 5 Sep 2025).

The dependence of $w_{ij}\in G$22 on the cycle family $w_{ij}\in G$23 is a substantive modeling choice. Using a cycle basis emphasizes independent obstructions. Using all cycles up to a fixed length emphasizes local inconsistencies. Using longer cycles captures larger-scale curvature. The paper further states that adding edges adds potential cycles and can reveal more inconsistencies, while removing edges reduces the cycle space and can lower $w_{ij}\in G$24 by ignoring previously measured obstructions (Magnot, 5 Sep 2025).

Stability is another practical feature. If $w_{ij}\in G$25 is Lipschitz with respect to the group coordinates, then $w_{ij}\in G$26 varies continuously with the weights; small perturbations of the weights induce small changes in $w_{ij}\in G$27. This is particularly relevant for matrix-group implementations, where one may take $w_{ij}\in G$28 near the identity or $w_{ij}\in G$29 more generally.

In the narrower conceptual sense, group-valued holonomies provide a unification of discrete bundle theory, gauge invariance, sheaf-theoretic gluing, and contextuality diagnostics. Their central content is simple but structurally rich: local group data may appear compatible edge by edge, yet fail globally when transported around a loop. That failure is recorded by holonomy; its scalar aggregate is $w_{ij}\in G$30; and flatness corresponds precisely to the existence of a globally coherent trivialization.