Berry’s geometric phase is a correction to the wave function arising in the study of adiabatic quantum systems; it has been discovered by M. V. Berry. There are analogous effects for other wave phenomena; there has been also much earlier work of Pancharatnam on the related phenomenon in optics. The origin of the Berry’s phase is in nonflatness of a parallel transport which appears in the corresponding phase factors.
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A relation to Chern-Bott connection is explained in 4.1 of lecture notes
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wikipedia: geometric phase
<http://www.mi.infm.it/manini/berryphase.html>
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Maxim Braverman, The Berry phase and the phase of the determinant, arxiv/1310.6332
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