U-Folds From Geodesics in Moduli Space

We exploit the presence of moduli fields in the ${\rm AdS}_3\times { S}^3\times CY_2$, where $CY_2=T^4$ or $K3$, solution to Type IIB superstring theory, to construct a U-fold solution with geometry ${\rm AdS}_2\times S^1\times {\rm S}^3\times CY_2$. This is achieved by giving a non-trivial dependence of the moduli fields in ${\rm SO}(4,n)/{\rm SO}(4)\times {\rm SO}(n)$ ($n=4$ for $CY_2=T^4$ and $n=20$ for $CY_2=K3$ ), on the coordinate $η$ of a compact direction $S^1$ along the boundary of ${\rm AdS}_3$, so that these scalars, as functions of $η$, describe a geodesic on the corresponding moduli space. The back-reaction of these evolving scalars on spacetime amounts to a splitting of ${\rm AdS}_3$ into ${\rm AdS}_2\times S^1$ with a non-trivial monodromy along $S^1$ defined by the geodesic. Choosing the monodromy matrix in ${\rm SO}(4,n;\,\mathbb{Z})$, 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 non-minimally coupled to $(d-1)$-forms and featuring solutions with topology ${\rm AdS}_d\times S^d$, and moduli scalar fields. We show, in this general setting, that giving the moduli fields a geodesic dependence on the $η$ coordinate of an $S^1$ at the boundary of ${\rm AdS}_d$ is sufficient to split this space into ${\rm AdS}_{d-1}\times S^1$, 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 towards the existence of similar solutions in the Type IIB theory compactified on $CY_2$. 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 ${\rm AdS}_3$.