Enlarged geometries of gauge bundles (Q5941839)

The anholonomity object of the basis \((R,\theta)\) on a differentiable variety [see \textit{M. Rahula}, New problems in differential geometry (1993; Zbl 0795.53002), Ch. II] submits two structure equations and Jacobi identities: \[ [R_K R_L]= R_J c^J_{KL},\;d\theta^J= -1/2 c^J_{KL} \theta^K\wedge \theta^L, \] \[ R_{[_{I}c^J_{KL}]}+ c^J_{[_{I/S/}c^{S}_{KL}]}=0. \] In the structure \(\Delta_\nu\oplus \Delta_h\) of connection this object splits into 6 subobjects: \[ c^a_{bc}(= f^a_{bc}),\;c^\lambda_{\mu\nu}(= f^\lambda_{\mu\nu}),\;c^a_{\mu b}(= C^a_{\mu b}),\;c^a_{\nu\mu}(= \beta^a_{\mu\nu}),\;c^\lambda_{bc}= c^\lambda_{b\mu}= 0. \] The fields \(R_a(= X_a)\) lie on an integrable vertical distribution \(\Delta_\nu: c^\lambda_{bc}= 0\). The fields \(R_\mu(= Y_\mu)\) on horizontal \(\Delta_h\) are admitted by \(\Delta_\nu: c^\lambda_{b\mu}= 0\). The subobject \(C^a_{\mu b}\) defines an infinitesimal transformation of the vertical basis by a shift along \(Y_\mu\), and \(\beta^a_{\mu\nu}\) corresponds to the curvature. The subojects \(f^a_{bc}\) and \(f^\lambda_{\mu\nu}\) correspond to anholonomy of two bases. In particular \(X_a\) can be generators of a Lie group \(G\) with structure coefficients \(f^a_{bc}\). Three kinds of commutation relations are deduced: those of a gauge theory, those of an extended gauge theory, and those of a gravitational model. When \(\dim\Delta_\nu= \dim\Delta_h\), as in the case involving spacetime and the electroweak theory, a tetrad-like field is introduced to represent the isomorphism \(\Delta_\nu\approx \Delta_h\). The same objects of usual geometry are found and suggest a relation to gravitation and lead to reasonable dynamical equations. See also [\textit{R. Aldrovandi}, J. Math. Phys. 32, 2503-2512 (1991; Zbl 0744.53033)].