  We propose $AdS_2$/CFT$_1$ dualities between exactly solvable topological
quantum mechanics theories with vector or matrix large $N$ limits (on the
boundary) and weakly coupled gauge theories on a fixed $AdS_2$ background (in
the bulk). The boundary theories can be embedded as 1d sectors of 3d ${\cal N}
= 4$ superconformal field theories with holographic duals, from which they can
be obtained using supersymmetric localization. We study a few examples of such
1d theories: theories with vector large $N$ limits that are embedded into 3d
theories of many free massless hypermultiplets with $AdS_4$ higher spin duals;
and a 1d theory with a matrix large $N$ limit embedded into the 3d ABJM theory
at Chern-Simons level $k=1$, which has an $AdS_4$ supergravity dual. We propose
that the $U(N)$ singlet sectors of the 1d vector models are dual to 2d gauge
theories on $AdS_2$ whose gauge algebras are finite dimensional and whose full
non-linear actions we completely determine in some cases. The 1d theory
embedded into ABJM theory has a $\mathbb{Z}_2$-invariant sector dual to a 2d
gauge theory on $AdS_2$ whose gauge algebra is the infinite dimensional algebra
of area preserving diffeomorphisms of a two-sphere. We provide evidence that
the 2d gauge theories on $AdS_2$ can be obtained from localizing the $AdS_4$
duals of the 3d SCFTs mentioned above, and thus argue that our 2d/1d dualities
can be obtained via supersymmetric localization on both sides of their parent
$AdS_4$/CFT$_3$ dualities. We discuss the boundary terms required by
holographic renormalization in the 2d gauge theories on $AdS_2$ and show how
they arise from supersymmetric localization.
