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We study T^{11-D-q}xT^q/Z_n orbifold compactifications of 11D supergravityand M-theory by a purely algebraic method. Using the mapping between scalarfields of toroidally compactified maximal supergravity and generators of theU-duality symmetry, we express the orbifold action as a finite order innerautomorphism and compute the residual real U-duality algebra surviving theorbifold projection for all dimensions D=1,...,10-q. In D=1, these invariantsubalgebras are shown to be described by Borcherds and Kac-Moody algebras witha degenerate Cartan matrix, modded out by their centres and derivations. Wefurther construct an alternative description of the orbifold action in terms ofequivalence classes of shift vectors, finding that a root of e_{10} can alwaysbe chosen as the class representative in D=1. In the case of Z_2 orbifolds ofM-theory descending to type 0' orientifolds, we argue that these roots can beinterpreted as pairs of magnetized D9- and D9'-branes ensuring tadpolecancellation. More generally, we provide a classification of all such rootsgenerating Z_n product orbifolds for n<7.Comment: 108 pages, v2:Section 9.4 returned to intended position 11, v3:final  corrected version published in JHE

Hidden Borcherds symmetries in Bbb Znorbifolds of M-theory and magnetized D-branes in type 0' orientifolds

Hidden Borcherds symmetries in Bbb Z_norbifolds of M-theory and magnetized D-branes in type 0' orientifolds