Disentangling modular Walker-Wang models via fermionic invertible boundaries

TL;DR

This paper demonstrates the triviality of certain Walker-Wang models in 3+1 dimensions by constructing explicit invertible domain walls and circuits, especially when incorporating fermionic degrees of freedom, revealing their topological triviality.

Contribution

It provides explicit constructions showing the triviality of Walker-Wang models with modular input categories, including fermionic extensions, and discusses boundaries and axiomatization of extended TQFT.

Findings

Walker-Wang models are trivial for modular input categories with explicit invertible domain walls.

Fermionic degrees of freedom extend triviality to a larger class of categories.

Explicit constructions of disentangling circuits and boundaries are provided.

Abstract

Walker-Wang models are fixed-point models of topological order in $3+1$ dimensions constructed from a braided fusion category. For a modular input category $\mathcal M$, the model itself is invertible and is believed to be in a trivial topological phase, whereas its standard boundary is supposed to represent a $2+1$-dimensional chiral phase. In this work we explicitly show triviality of the model by constructing an invertible domain wall to vacuum as well as a disentangling generalized local unitary circuit in the case where $\mathcal M$ is a Drinfeld center. Moreover, we show that if we allow for fermionic (auxiliary) degrees of freedom inside the disentangling domain wall or circuit, the model becomes trivial for a larger class of modular fusion categories, namely those in the Witt classes generated by the Ising UMTC. In the appendices, we also discuss general (non-invertible) boundaries of general Walker-Wang models and describe a simple axiomatization of extended TQFT in terms of tensors.