  We lay out an approach to derive the closed string dual to the simplest
possible gauge theory, a single hermitian matrix integral, in the conventional
't Hooft large $N$ limit. In this first installment of three papers, we propose
and verify an explicit correspondence with a (mirror) pair of closed
topological string theories. On the A-model side, this is a supersymmetric
$SL(2, \mathbb{R})_1/U(1)$ Kazama-Suzuki coset (with background momentum modes
turned on). The mirror B-model description is in terms of a Landau-Ginzburg
theory with superpotential $W(Z)=\frac{1}{Z}+t_2Z$ and its deformations. We
arrive at these duals through an "open-closed-open triality". This is the
notion that two open string descriptions ought to exist for the same closed
string theory depending on how closed strings manifest themselves from open
string modes. Applying this idea to the hermitian matrix model gives an exact
mapping to the Imbimbo-Mukhi matrix model. The latter model is known to capture
the physical correlators of the $c=1$ string theory at self-dual radius, which,
in turn, has the equivalent topological string descriptions given above. This
enables us to establish the equality of correlators, to all genus, between
single trace operators in our original matrix model and those of the dual
closed strings. Finally, we comment on how this simplest of dualities might be
fruitfully viewed in terms of an embedding into the full AdS/CFT
correspondence.
