We consider compactifications of the Betti, de Rham and Dolbeault
realizations of the character variety. Starting from an example, we
look at what can be said, mostly conjecturally, about the
relationship between these spaces.
We prove that for any $\epsilon > 0$ it is NP-hard to
approximate the non-commutative Grothendieck problem to within a
factor $1/2 + \epsilon$, which matches the approximation ratio of
the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof...
