Truncated affine Rozansky--Witten models as extended TQFTs

TL;DR

This work constructs fully extended 2D TQFTs from truncated affine Rozansky–Witten models with target $T^*oldsymbol{C}^n$ by passing to a nonsemisimple 2-category capital C and applying the cobordism hypothesis. It proves that every object is fully dualisable with trivialisable Serre automorphism, yielding a unique extended oriented TQFT z_n valued in capital C, and computes genus-$g$ state spaces al H_{g} that recover the infinite-dimensional state spaces of the 3D RW theory in a graded setting. The circle and pair-of-pants data give a commutative Frobenius algebra arising from endomorphisms of circle-defect lines, and the construction extends to a graded version capital C^{ ext{gr}} that tracks flavour and R-charges, connecting the Grothendieck ring of matrix factorisations to the TQFT data. Overall, the paper makes the RW construction tractable in a fully extended 2D setting, enabling explicit state-space and Frobenius-algebra computations and clarifying the role of noncompact target spaces in a defect-theory framework.

Abstract

We construct extended TQFTs associated to Rozansky--Witten models with target manifolds $T^*\mathbb{C}^n$. 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 $\mathcal{C}$ 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 $\mathcal{C}$ 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.

Theorems & Definitions (23)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4
• proof
• Proposition 2.5
• proof
• Theorem 2.6
• Theorem 3.1: HSVHVHesse
• Example 3.2