Non-semisimple link and manifold invariants for symplectic fermions

TL;DR

This paper explores non-semisimple link and 3-manifold invariants derived from symplectic fermion categories, revealing their dependence on simple factors and their ability to distinguish complex objects.

Contribution

It introduces new invariants from symplectic fermion categories, analyzes their dependence on tensor ideals, and demonstrates their distinguishing power for indecomposable objects.

Findings

Lyubashenko-invariant of lens spaces equals the order of the first homology group to the power N.

Invariants depend only on simple composition factors unless passing to proper ideals.

Modified trace invariants can distinguish indecomposable objects with identical composition series.

Abstract

We consider the link and three-manifold invariants from arXiv:1912.02063, which are defined in terms of certain non-semisimple finite ribbon categories $\mathcal{C}$ together with a choice of tensor ideal and modified trace. If the ideal is all of $\mathcal{C}$, these invariants agree with those defined by Lyubashenko in the 90's. We show that in that case the invariants depend on the objects labelling the link only through their simple composition factors, so that in order to detect non-trivial extensions one needs to pass to proper ideals. We compute examples of link and three-manifold invariants for $\mathcal{C}$ being the category of $N$ pairs of symplectic fermions. Using a quasi-Hopf algebra realisation of $\mathcal{C}$, we find that the Lyubashenko-invariant of a lens space is equal to the order of its first homology group to the power $N$, a relation we conjecture to hold for all rational homology spheres. For $N \ge 2$, $\mathcal{C}$ allows for tensor ideals $\mathcal{I}$ with a modified trace which are different from all of $\mathcal{C}$ and from the projective ideal. Using the theory of pull-back traces and symmetrised cointegrals, we show that the link invariant obtained from $\mathcal{I}$ can distinguish a continuum of indecomposable but reducible objects which all have the same composition series.