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Fermionic modes in locally anti-de Sitter spacetime with a compact subspace

We present a complete set of positive and negative energy fermionic modes in a D + 1- dimensional locally anti-de Sitter (AdS) spacetime with a part of spatial dimensions compactified on a torus. Two geometries are considered: the boundary-free geometry and the geometry with an additional 
brane parallel to the AdS horizon. On the brane, the field operator obeys the bag boundary condition. Both regions between the brane and the horizon and between the brane and AdS boundary are considered.