Flowing from $AdS_5$ to $AdS_3$ with $T^{1,1}$

We construct supersymmetric domain wall solutions of type IIB supergravity that interpolate between $AdS_5\times T^{1,1}$ in the UV and $AdS_3\times\mathbb{R}^2\times S^2\times S^3$ solutions in the IR. The $\mathbb{R}^2$ factor can be replaced with a two-torus and then the solution describes a supersymmetric flow across dimensions, similar to wrapped brane solutions. While the domain wall solutions preserve $(0,2)$ supersymmetry, the $AdS_3$ solutions in the IR have an enhanced $(4,2)$ superconformal supersymmetry and are related by two T-dualities to the $AdS_3\times S^3\times S^3\times S^1$ type IIB solutions which preserve a large $(4,4)$ superconformal supersymmetry. The domain wall solutions exist within the $N=4$ $D=5$ gauged supergravity theory that is obtained from a consistent Kaluza-Klein truncation of type IIB supergravity on $T^{1,1}$; a feature driving the flows is that two $D=5$ axion like fields, residing in the $N=4$ Betti multiplet, depend linearly on the two legs of the $\mathbb{R}^2$ factor.