  Defining complexity in quantum field theory is a difficult task, and the main
challenge concerns going beyond free models and associated Gaussian states and
operations. One take on this issue is to consider conformal field theories in
1+1 dimensions and our work is a comprehensive study of state and operator
complexity in the universal sector of their energy-momentum tensor. The
unifying conceptual ideas are Euler-Arnold equations and their
integro-differential generalization, which guarantee well-posedness of the
optimization problem between two generic states or transformations of interest.
The present work provides an in-depth discussion of the results reported in
arXiv:2005.02415 and techniques used in their derivation. Among the most
important topics we cover are usage of differential regularization, solution of
the integro-differential equation describing Fubini-Study state complexity and
probing the underlying geometry.
