  We construct a family of 3d quantum field theories $\mathcal T_{n,k}^A$ that
conjecturally provide a physical realization -- and derived generalization --
of non-semisimple mathematical TQFT's based on the modules for the quantum
group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(i\pi/k)$.
The theories $\mathcal T_{n,k}^A$ are defined as topological twists of certain
3d $\mathcal N=4$ Chern-Simons-matter theories, which also admit
string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$
Chern-Simons theories, coupled to a twisted $\mathcal N=4$ matter sector (the
source of non-semisimplicity). We show that $\mathcal T_{n,k}^A$ admits
holomorphic boundary conditions supporting two different logarithmic vertex
operator algebras, one of which is an $\mathfrak{sl}_n$-type Feigin-Tipunin
algebra; and we conjecture that these two vertex operator algebras are related
by a novel logarithmic level-rank duality. (We perform detailed computations to
support the conjecture.) We thus relate the category of line operators in
$\mathcal T_{n,k}^A$ to the derived category of modules for a boundary
Feigin-Tipunin algebra, and -- using a logarithmic Kazhdan-Lusztig-like
correspondence that has been established for $n=2$ and expected for general $n$
-- to the derived category of $U_q(\mathfrak{sl}_n)$ modules. We analyze many
other key features of $\mathcal T_{n,k}^A$ and match them from quantum-group
and VOA perspectives, including deformations by flat $PSL(n,\mathbb C)$
connections, one-form symmetries, and indices of (derived) genus-$g$ state
spaces.
