  We initiate quantitative studies of complexity in (1+1)-dimensional conformal
field theories with a view that they provide the simplest setting to find a
gravity dual to complexity. Our work pursues a geometric understanding of
complexity of conformal transformations and embeds Fubini-Study state
complexity and direct counting of stress tensor insertion in the relevant
circuits in a unified mathematical language. In the former case, we iteratively
solve the emerging integro-differential equation for sample optimal circuits
and discuss the sectional curvature of the underlying geometry. In the latter
case, we recognize that optimal circuits are governed by Euler-Arnold type
equations and discuss relevant results for three well-known equations of this
type in the context of complexity.
