Abstract
We derive the Lichnerowicz-Weitzenböck formula for the two-component order parameter superconductor, which provides a twofold view of the kinetic energy of the superconductor. For the one component order parameter superconductor we review the connection between the Lichnerowicz-Weitzenböck formula and the Ginzburg-Landau theory. For the two-component case we claim that this formula opens a venue to describe inhomogeneous superconducting states intertwined by spin correlations and charged dislocation. In this case the Lichnerowicz-Weitzenböck formula displays local rotational and electromagnetic gauge symmetry (SU(2) ⊗ U(1)) and relies on local commuting momentum and spin operators. The order parameter lives in a space with curvature and torsion described by Élie Cartan geometrical formalism. The Lichnerowickz-Weitzenböck formula leads to first order differential equations that are a three-dimensional version of the Seiberg-Witten equations.
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