  We identify the category of integrable lowest-weight representations of the
loop group LG of a compact Lie group G with the linear category of twisted,
conjugation-equivariant curved Fredholm complexes on the group G: namely, the
twisted, equivariant matrix factorizations of a super-potential built from the
loop rotation action on LG. This lifts the isomorphism of K-groups of [FHT1,2,
3] to an equivalence of categories. The construction uses families of Dirac
operators.
