U folds from geodesics in moduli space

We exploit the presence of moduli fields in the AdS₃×S³×CY₂, where CY₂=T⁴ or $K3$, solution to type IIB superstring theory, to construct a U-fold solution with geometry AdS₂×S¹×S³×CY₂. This is achieved by giving a nontrivial dependence of the moduli fields in SO(4,n)/SO(4)×SO(n) ($n=4$ for CY₂=T⁴ and $n=20$ for CY₂=K3), on the coordinate η of a compact direction S¹ along the boundary of AdS₃, so that these scalars, as functions of η, describe a geodesic on the corresponding moduli space. The backreaction of these evolving scalars on spacetime amounts to a splitting of AdS₃ into AdS₂×S¹ with a nontrivial monodromy along S¹ defined by the geodesic. Choosing the monodromy matrix in SO(4,n;Z), this supergravity solution is conjectured to be a consistent superstring background. We generalize this construction starting from an ungauged theory in $D=2d$, d odd, describing scalar fields nonminimally coupled to (d-1) forms and featuring solutions with topology AdSd×Sᵈ, and moduli scalar fields. We show, in this general setting, that giving the moduli fields a geodesic dependence on the η coordinate of an S¹ at the boundary of AdSd is sufficient to split this space into AdS_d-1×S¹, with a monodromy along S¹ defined by the starting and ending points of the geodesic. This mechanism seems to be at work in the known J-fold solutions in $D=10$ type IIB theory and hints toward the existence of similar solutions in the type IIB theory compactified on CY₂. We argue that the holographic dual theory on these backgrounds is a $1+0$ CFT on an interface in the $1+1$ theory at the boundary of the original AdS₃.