Instantons and no wormholes in $AdS_3\times S^3 \times CY_2$

We study supergravity instantons sourced by axion (and saxion) fields in the Euclidean $AdS_3\times S^3 \times CY_2$ vacua of IIB supergravity. Such instantons are described by geodesic curves on the moduli space; the timelike geodesics can describe Euclidean wormholes, the lightlike geodesics describe (generalisations of) D-instantons and spacelike geodesics are sub-extremal versions thereof. We perform a concrete classification of such geodesics and find that, despite earlier claims, the wormholes fail to be regular. A subclass of the lightlike geodesics is supersymmetric and, up to dualities, lift to Euclidean strings wrapping 2-cycles in the CY$_2$. The dual of these instantons are expected to be worldsheet instantons of the D1-D5 CFT.