They are used in string theory to construct Conformal Field Theories which describe orbifold limits of Calabi-Yau spaces. See for example Dixon, Harvey, Vafa and Witten, "Strings on Orbifolds I,II" Nucl. Phys. B274 (19860 285 and Nucl. Phys. B261 (1985) 678 for an early application in string theory and Miles Reid in http://arxiv.org/pdf/math/9911165v1.pdf for a more mathematical take on related material.
Jeff Harvey
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