Let $\mathcal{D}(X)$ be the bounded derived category of coherent sheaves on a smooth projective variety $X$. An autoequivalence $\Phi: \mathcal{D}(X) \to \mathcal{D}(X)$ is called phantom if it induces the identity functor on the Grothendieck group $K_0(\mathcal{D}(X))$ and acts trivially on the Hochschild homology of $\mathcal{D}(X)$. In other words, $\Phi$ is invisible to these classical invariants.
What is the geometric meaning of a phantom autoequivalence? Does its existence reflect some hidden geometric structure of $X$?
