  On the configuration space of projections in a noncommutative algebra, and
for an automorphism of the algebra, we use a twisted Hochschild cocycle for an
action functional, and a twisted cyclic cocycle for a topological term. The
latter is Hochschild-cohomologous to the former and positivity in twisted
Hochschild cohomology results into a lower bound for the action functional.
While the equations for the critical points are rather complicate, the use of
the positivity and the bound by the topological term leads to self-duality
equations (thus yielding twisted noncommutative sigma-model solitons, or
instantons). We present explicit non trivial solutions on the quantum
projective line.
