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In an earlier article we introduced a new definition for the `quantization'of a Hamiltonian loop group space $\mathcal{M}$, involving the equivariant$L^2$-index of a Dirac-type operator $\mathscr{D}$ on a non-compact finitedimensional submanifold $\mathcal{Y}$ of $\mathcal{M}$. In this article westudy a Witten-type deformation of this operator, similar to the work ofTian-Zhang and Ma-Zhang. We obtain a formula for the index with infinitely manynon-trivial contributions, indexed by the components of the critical set of thenorm-square of the moment map. This is the main part of a new proof of the$[Q,R]=0$ theorem for Hamiltonian loop group spaces.

Norm-square localization and the quantization of Hamiltonian loop group spaces