Quantum systems on a non-simply connected space possess a “large” gauge invariance. Laughlin utilized this to explain quantum Hall effect [1]. Later, it was applied to elucidate a universal relation between filling factor and energy spectrum in quantum many-body systems on periodic lattices (Lieb-Schultz-Mattis-M.O.-Hastings) [2].
Somewhat surprisingly, the large gauge invariance is also deeply related to modern theory of electric polarization developed by Resta et al [3,4]. I will give an overview of applications of the large gauge invariance to condensed matter physics, and also discuss most recent results obtained by combining it with the theory of polarization [5].

References:
[1] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
[2] M. O., Phys. Rev. Lett. 84, 1535 (2000).
[3] R. Resta and S. Sorella, Phys. Rev. Lett. 82,370 (1999).
[4] M. Nakamura and J. Voit, Phys. Rev. B 65, 153110 (2002).
[5] Y.-M. Lu, Y. Ran, and M. O., in preparation