  In one dimensional quantum gases there is a well known "duality" between hard
core bosons and non-interacting fermions. However, at the field theory level,
no exact duality connecting strongly interacting bosons to weakly interacting
fermions is known. Here we propose a solution to this long standing problem.
Our derivation relies on regularizing the only point-like interaction between
fermions in 1D that induces a discontinuity in the wave function proportional
to its derivative. In contrast to all known regularisations our potential is
weak for small interaction strengths. Crucially, this allows one to apply
standard methods of diagrammatic perturbation theory to strongly interacting
bosons. As a first application we compute the finite temperature spectral
function of the Cheon-Shigehara model, the fermionic model dual to the
celebrated Lieb-Liniger model.
