Truncated Affine Rozansky–Witten Models as Extended TQFTs

We construct extended TQFTs associated to Rozansky–Witten models with target manifolds T^*ℂ^n . The starting point of the construction is the 3-category whose objects are such Rozansky–Witten models, and whose morphisms are defects of all codimensions. By truncation, we obtain a (non-semisimple) 2-category 𝒞 of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in 𝒞 for every affine Rozansky–Witten model. By evaluating this TQFT on closed surfaces we obtain the infinite-dimensional state spaces (graded by flavour and R-charges) of the initial 3-dimensional theory. Furthermore, we explicitly compute the commutative Frobenius algebras that classify the restrictions of the extended theories to circles and bordisms between them.