$U$ folds from geodesics in moduli space

We exploit the presence of moduli fields in the ${\mathrm{AdS}}_3\times S^3\times C{Y}_2$, where $C{Y}_2=T^4$ or $K3$, solution to type IIB superstring theory, to construct a $U$-fold solution with geometry ${\mathrm{AdS}}_2\times S^1\times {\mathrm{S}}^3\times C{Y}_2$. This is achieved by giving a nontrivial dependence of the moduli fields in $\mathrm{SO}(4,n)/\mathrm{SO}(4)\times \mathrm{SO}(n)$ ($n=4$ for $C{Y}_2=T^4$ and $n=20$ for $C{Y}_2=K3$), on the coordinate $\eta$ of a compact direction $S^1$ along the boundary of ${\mathrm{AdS}}_3$, so that these scalars, as functions of $\eta$, describe a geodesic on the corresponding moduli space. The backreaction of these evolving scalars on spacetime amounts to a splitting of ${\mathrm{AdS}}_3$ into ${\mathrm{AdS}}_2\times S^1$ with a nontrivial monodromy along $S^1$ defined by the geodesic. Choosing the monodromy matrix in $\mathrm{SO}(4,n;\mathbb{Z})$, 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 ${\mathrm{AdS}}_d\times S^d$, and moduli scalar fields. We show, in this general setting, that giving the moduli fields a geodesic dependence on the $\eta$ coordinate of an $S^1$ at the boundary of ${\mathrm{AdS}}_d$ is sufficient to split this space into ${\mathrm{AdS}}_{d-1}\times S^1$, with a monodromy along $S^1$ 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 $C{Y}_2$. 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 ${\mathrm{AdS}}_3$.