Geometry and holography of ${\mathrm{A}\mathrm{d}\mathrm{S}}_2\times {\mathrm{S}}^2$: towards the field theory dual

We take a first step in identifying the superconformal quantum mechanics dual to supersymmetric AdS${}_2\times$S${}^2$ solutions. We focus on the Type IIB solutions of the form AdS${}_2\times$S${}^2\times$CY${}_2\times \mathrm{\Sigma }$ that preserve eight supercharges, with $\mathrm{\Sigma }$ a two-dimensional space parametrised by $(x_1,x_2)$, and read off their brane content, a D1-D5-NS5 intersection with D3 and D7 flavour branes. We propose that their dual $\mathcal{N}=4$ quantum mechanics is organised over a two-dimensional lattice of cells in $\mathrm{\Sigma }$. Namely, it consists of copies of the linear quiver along $x_1$ dual to the U(1)-invariant solutions of the class, one for each interval along $x_2$, with ranks given by the D1 and D5 charges of the corresponding cell. Its central charge agrees with the holographic one at leading order. We show that the family of SL$(2,\mathbb{R})$ non-Abelian T-duals of the AdS${}_3\times$S${}^2\times$CY${}_2$ solutions of massive Type IIA supergravity, constructed in previous work, belongs to this class, as does the SL$(2,\mathbb{R})\times$SU$(2)$ non-Abelian T-dual of AdS${}_3\times$S${}^3\times$CY${}_2$, and work both out in detail. For both we propose global completions that introduce additional D3 and D7 branes. Along $x_2$ we interpret the fundamental strings stretched between the colour branes and the additional D3 and D7 branes as baryon vertices. Finally, we extend this proposal from the non-Abelian T-duals to the solutions of the class whose defining functions factorise in $x_1$ and $x_2$, including those in which the AdS${}_2$ and S${}^2$ factors have the same warp factor.