  We show that a class of $L$-loop conformal ladder graphs correspond to
twisted partition functions of free massive complex scalars in $d=2L+1$
dimensions. The graphs arise as four-point functions in certain two- and
four-dimensional conformal fishnet models. The twisted thermal two-point
function of the scalars is a generator of such conformal graphs for all loops.
We argue that this correspondence is seeded by a system of two decoupled
harmonic oscillators twisted by an imaginary chemical potential. We find a
number of algebraic and differential relations among the conformal graphs which
mirror the underlying free dynamics.
