Truncated affine Rozansky--Witten models as extended defect TQFTs

TL;DR

This work provides a rigorous construction of fully extended defect TQFTs for affine Rozansky--Witten models by applying the cobordism hypothesis with defects to a tractable 2-category ${\mathcal{C}}$ built from matrix factorisations. The authors prove that ${\mathcal{C}}$ is pivotal and that all objects are fully dualisable, enabling explicit evaluation of defect-labeled bordisms via a 3D graphical calculus. They develop extended defect TQFTs valued in ${\mathcal{C}}$, compute bulk and defect state spaces, and analyze symmetry defects and boundaries in detail, including networks that realize background gauge fields and twisted sector line operators. The results yield concrete, calculable state spaces for surfaces with arbitrary defect networks and establish an open-closed TQFT framework with a generalized, categorified pairing that mirrors a Hirzebruch–Riemann–Roch-type structure in this setting. Together, these constructions provide a robust, computable bridge between 3D Rozansky--Witten physics and 2D extended TQFT techniques, with potential applications to gauge theories, boundary VOAs, and categorical representations of defect algebras.

Abstract

We apply the cobordism hypothesis with singularities to the case of affine Rozansky--Witten models, providing a construction of extended TQFTs that includes all line and surface defects. On a technical level, this amounts to proving that the associated homotopy 2-category is pivotal, and to systematically employing its 3-dimensional graphical calculus. This in particular allows us to explicitly calculate state spaces for surfaces with arbitrary defect networks. As specific examples we discuss symmetry defects which can be used to model non-trivial background gauge fields, as well as boundary conditions.

Theorems & Definitions (1)

• Theorem 1.1