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Geometric phases in adiabatic Floquet theory, abelian gerbes and Cheon's anholonomy

We study the geometric phase phenomenon in the context of the adiabatic Floquet theory (the so-called the $(t,t')$ Floquet theory). A double integration appears in the geometric phase formula because of the presence of two time variables within the theory. We show that the geometric phases are then identified with horizontal lifts of surfaces in an abelian gerbe with connection, rather than with horizontal lifts of curves in an abelian principal bundle. This higher degree in the geometric phase gauge theory is related to the appearance of changes in the Floquet blocks at the transitions between two local charts of the parameter manifold. We present the physical example of a kicked two-level system where these changes are involved via a Cheon's anholonomy. In this context, the analogy between the usual geometric phase theory and the classical field theory also provides an analogy with the classical string theory.