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 ${\mathrm{AdS}}_3$.

中文翻译：

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模空间中测地线的 Ufold

我们利用模域的存在${\mathrm{广告服务}}_3\times S^3\times C{是}_2$， 在哪里$C{是}_2={\mathrm{时间}}^4$或者$K3$，解决IIB型超弦理论，构造一个$U$- 几何折叠解决方案${\mathrm{广告服务}}_2\times S^1\times {\mathrm{S}}^3\times C{是}_2$。这是通过给出模场的非平凡依赖性来实现的$\mathrm{所以}（4,n）/\mathrm{所以}（4）\times \mathrm{所以}（n）$（$n=4$为了$C{是}_2={\mathrm{时间}}^4$和$n=20$为了$C{是}_2=K3$），在坐标上$\eta$紧凑方向的$S^1$沿着边界${\mathrm{广告服务}}_3$，以便这些标量作为函数$\eta$，描述相应模空间上的测地线。这些不断演化的标量对时空的反作用相当于分裂${\mathrm{广告服务}}_3$进入${\mathrm{广告服务}}_2\times S^1$伴随着不平凡的单一性$S^1$由测地线定义。选择单性矩阵$\mathrm{所以}（4,n;\mathbb{Z}）$，这个超引力解被推测为一致的超弦背景。我们从一个未衡量的理论出发概括了这种结构$D=2d$,$d$奇数，描述非最小耦合到 ($d-1$) 形成并具有拓扑特征的解决方案${\mathrm{广告服务}}_d\times S^d$和模标量场。我们表明，在这种一般设置中，赋予模量场对测地线的依赖性$\eta$的坐标$S^1$在边界处${\mathrm{广告服务}}_d$足以将这个空间分成${\mathrm{广告服务}}_{d-1}\times S^1$，伴随着单一性$S^1$由测地线的起点和终点定义。这种机制似乎在已知的情况下起作用$J$- 将解决方案折叠$D=10$IIB 型理论并暗示在 IIB 型理论中存在类似的解$C{是}_2$。我们认为，在这些背景下的全息对偶理论是$1+0$接口上的 CFT$1+1$原始理论的边界${\mathrm{广告服务}}_3$。