  We further advance the study of the notion of computational complexity for 2d
CFTs based on a gate set built out of conformal symmetry transformations.
Previously, it was shown that by choosing a suitable cost function, the
resulting complexity functional is equivalent to geometric (group) actions on
coadjoint orbits of the Virasoro group, up to a term that originates from the
central extension. We show that this term can be recovered by modifying the
cost function, making the equivalence exact. Moreover, we generalize our
approach to Kac-Moody symmetry groups, finding again an exact equivalence
between complexity functionals and geometric actions. We then determine the
optimal circuits for these complexity measures and calculate the corresponding
costs for several examples of optimal transformations. In the Virasoro case, we
find that for all choices of reference state except for the vacuum state, the
complexity only measures the cost associated to phase changes, while assigning
zero cost to the non-phase changing part of the transformation. For Kac-Moody
groups in contrast, there do exist non-trivial optimal transformations beyond
phase changes that contribute to the complexity, yielding a finite gauge
invariant result. Furthermore, we also show that the alternative complexity
proposal of path integral optimization is equivalent to the Virasoro proposal
studied here. Finally, we sketch a new proposal for a complexity definition for
the Virasoro group that measures the cost associated to non-trivial
transformations beyond phase changes. This proposal is based on a cost function
given by a metric on the Lie group of conformal transformations. The
minimization of the corresponding complexity functional is achieved using the
Euler-Arnold method yielding the Korteweg-de Vries equation as equation of
motion.
