Non-semisimple link and manifold invariants for symplectic fermions

Abstract

We consider the link and three-manifold invariants in De Renzi et al. [Selecta Math. 28 (2022), article no. 42], which are defined in terms of certain non-semisimple ribbon categories $\mathcal{C}$ (locally finite for link invariants, finite for manifold invariants) 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. 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 three-manifolds with finite first homology group. 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 pullback 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.