Topological Casimir effect for fermionic condensate in AdS spacetime with compact dimensions

We investigate combined effects of gravitational field and spatial topology on the fermionic condensate (FC) for a massive Dirac field in locally anti-de Sitter (AdS) spacetime with a part of spatial dimensions compactified to a torus. For general phases in the periodicity conditions along compact dimensions the topological Casimir contribution is explicitly extracted and the renormalization is reduced to the one for purely AdS spacetime. The FC is an even periodic function of the magnetic flux enclosed by compact dimension with the period of flux quantum. The topological contribution vanishes on the AdS boundary and dominates in the total FC in the region near the AdS horizon. For proper lengths of compact dimensions smaller than the AdS curvature radius the influence of the gravitational field is weak and the leading term in the corresponding expansion coincides with the FC for a locally Minkowski spacetime with compact dimensions. For large proper lengths the decay of the topological FC follows a power law for both massless and massive field, in contrast to an exponential decay in Minkowski bulk for massive fields. Applications are discussed for 2D Dirac materials.