Proposition. Gauge covariance and transport of selected laws [fdip-006E]
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  • November 1, 2026

For a connection background, the gauge group \(G^V\) changes the frame independently at each vertex. For \(g=(g_v)_{v\in V}\), define \[\begin {aligned} (gU)_e&=g_{h(e)}U_eg_{s(e)}^{-1},\\ (G_Vf)_v&=\rho (g_v)f_v,\\ (G_Ea)_e&=\rho (g_{h(e)})a_e. \end {aligned}\] Then \[\begin {aligned} d_{gU}G_V&=G_Ed_U,\\ Q_{gU}&=G_VQ_UG_V^{-1},\\ \det Q_{gU}&=\det Q_U. \end {aligned}\] In particular, the massive spectra and Gaussian normalizers are gauge invariant.

Proof. 

  • November 1, 2026

On an edge with endpoints \(s,h\), \[\begin {aligned} (d_{gU}G_Vf)(e) &=\rho (g_h)f_h-\rho (g_hU_eg_s^{-1})\rho (g_s)f_s\\ &=\rho (g_h)(f_h-\rho (U_e)f_s). \end {aligned}\] This proves the first identity. Both \(G_V\) and \(G_E\) are isometries, and scalar weights imply \(G_E^*WG_E=W\). Taking adjoints in \(d_{gU}=G_Ed_UG_V^{-1}\) therefore gives \[d_{gU}^*Wd_{gU}=G_Vd_U^*Wd_UG_V^{-1}.\] The mass term commutes with \(G_V\). Multiplicativity of the determinant gives its invariance. The same isometry preserves real Euclidean volume, or the real volume underlying a complex Hermitian space, so changing variables also preserves the corresponding Gaussian integral.

For a fixed incidence presentation, \(U\mapsto gU\) is a group action: \(g_2(g_1U)=(g_2g_1)U\). Its action groupoid has connections as objects and gauge transformations as arrows. If a selector is included, use an orthogonal projector \(P\) on real \(H\), or a complex-linear orthogonal projector on complex \(H\), and transport it by \[P'=G_VPG_V^{-1}.\] The identity, composition and inverse obey this equation by conjugation. Together with relabelings that carry the connection values, these are the connection counterparts of selected-background arrows. The earlier scalar backgrounds and relabeling arrows are recovered with \(F=\mathbb R\), \(G=\{1\}\) and their original selector rank condition.

On the selected space \(X=\operatorname {im}P\), the compressed inverse is \(C_P=(PQ_U^{-1}P)|_X\). The restriction \(G_X:X\to \operatorname {im}P'\) satisfies \[C_{P'}=G_XC_PG_X^{-1}.\] Indeed insert the three conjugation identities for \(P'\), \(Q_{gU}^{-1}\) and \(P'\). Inversion transports the selected precision. For real fields this is exactly the selected Gaussian transport; for complex fields the Gaussian convention is specified in Example [fdip-006I].

Unitarity is essential for this argument. A nonorthogonal matrix representation with an unchanged Euclidean metric need not satisfy \(G_E^*WG_E=W\); changing connection matrices alone would then fail to produce the asserted conjugacy.