Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a pair: bicategory [InlineEquation not available: see fulltext.] and endobifunctor [InlineEquation not available: see fulltext.]. For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor Σ q such that Σ q α: = q - deg α Σ α for any 2-morphism α and coincides with Σ otherwise. Applying the quantized trace to the bicategory of Chen–Khovanov bimodules we get a new triply graded link homology theory called quantum annular link homology. If q= 1 we reproduce Asaeda–Przytycki–Sikora homology for links in a thickened annulus. We prove that our homology carries an action of [InlineEquation not available: see fulltext.], which intertwines the action of cobordisms. In particular, the quantum annular homology of an n-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups depend on the quantum parameter q.
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