  We prove a formula for twisted Duistermaat-Heckman distributions associated
to a Hamiltonian $LG$-space. The terms of the formula are localized at the
critical points of the norm-square of the moment map, and can be computed in
cross-sections. Our main tools are the theory of quasi-Hamiltonian $G$-spaces,
as well as the Hamiltonian cobordism approach to norm-square localization
introduced recently by Harada and Karshon.
