  In this paper, we complete the nonabelian Hodge theory (NAHT) triangle of
isomorphisms for stacks between the Borel-Moore homologies of the Dolbeault,
Betti, and de Rham moduli stacks. We first explain how to realise the category
of connections on a smooth projective curve as a subcategory of a 2-Calabi-Yau
dg-category satisfying some appropriate geometric conditions. Then, we define a
cohomological Hall algebra (CoHA) product on the Borel-Moore homology of the
stack of connections on a smooth projective curve. This allows us to not only
compare the Borel-Moore homologies of the stacks at the relative and absolute
levels for the three sides of NAHT, but also to compare their CoHA structures:
they all coincide. To compare the Dolbeault and de Rham sides, we define a CoHA
for the Hodge-Deligne moduli space parametrising $\lambda$-connections. The
Betti and de Rham sides are compared using the (derived) Riemann-Hilbert
correspondence. The comparison of the Borel-Moore homologies of the Dolbeault
and Betti moduli stacks was previously considered by the author with Davison
and Schlegel Mejia (without taking the cohomological Hall algebra structures
into account). This paper completes this study and provides a CoHA enhancement
of the classical NAHT for curves.

Algebraic Geometry