14 Matrix holonomy and finite gluing
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Let \(\Gamma\) be a connected finite graph. At every vertex \(v\), place finite dimensional complex vector spaces \(V_v\) and \(W_v\) of the same dimension. For every oriented edge \(e:u\to v\), let
be invertible. The reversed edge carries the inverse map.
Definition 14.1: Path transport
For an oriented path
define
For a loop based at \(r\), these are the based holonomies.
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Theorem 14.2: Matrix holonomy criterion
Fix a root \(r\) and an isomorphism
The following conditions are equivalent.
There are isomorphisms \(U_v:V_v\to W_v\), with the declared \(U_r\), such that
\[ U_vA_e=B_eU_u \]for every edge \(e:u\to v\).
For one fundamental cycle basis based at \(r\),
\[ U_rA_\gamma=B_\gamma U_r. \]The same identity holds for every based loop \(\gamma\).
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Proof of Theorem 14.2
If the vertex maps exist, multiply the edge equations around a loop. Every intermediate vertex map cancels, giving condition three. Condition three implies condition two.
Conversely, choose a spanning tree. Starting from \(U_r\), define \(U_v\) by transport along the unique tree path from \(r\) to \(v\). The edge equations hold on every tree edge by construction. Every non tree edge closes one fundamental cycle. The edge equation on that edge is equivalent to the root holonomy identity for its fundamental cycle. Condition two therefore gives all remaining edge equations.
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Corollary 14.3: Unbased gauge classification
There exists some family of vertex isomorphisms if and only if the two holonomy representations
are conjugate.
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Proof of Corollary 14.3
Apply Theorem 14.2 and allow \(U_r\) to vary.
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Corollary 14.4: Scalar and projective gluing
If every based loop acts by a scalar, finite gluing is equivalent to equality of the scalar holonomies on a fundamental cycle basis. The number of independent tests is
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Proof of Corollary 14.4
A connected graph has a spanning tree with \({\left\lvert V\right\rvert}-1\) edges. Every remaining edge gives one fundamental cycle. In the scalar case, conjugacy has no effect.
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Proposition 14.5: Projective cocycle form
Suppose the edge maps are defined only up to scalar, in the setting where all based holonomies of both systems are scalar. Choose representatives. The ratio of the two scalar holonomies on each loop is unchanged by vertex rescaling. It defines a character of the first homology group when the scalar transport is abelian.
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Proof of Proposition 14.5
A vertex rescaling appears once with positive exponent and once with negative exponent around a closed path, so it cancels. Multiplication of loops multiplies the scalar ratios. In the abelian case, the ratio factors through the abelianization of the fundamental group, which is the first homology group.
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Proposition 14.6: Exact edge residual from cycle defect
Fix a spanning tree and construct the vertex maps by tree transport from \(U_r\). Let \(e:u\to v\) be a non tree edge. Let \(p_u\) and \(p_v\) be the tree paths from the root to \(u\) and \(v\). Define the fundamental cycle defect
Then the edge residual is exactly
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Proof of Proposition 14.6
Substitute
into the left side and factor \(B_{p_v}\) on the left and \(A_{p_u}^{-1}\) on the right.
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Corollary 14.7: Quantitative finite gluing
With the notation of Proposition 14.6,
Thus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.
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Proof of Corollary 14.7
Take operator norms in the exact identity.
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Remark 14.8
The theorem is an exact finite gluing result. It assumes that the packet spaces and edge transports have already been constructed on both sides.