Relative (pre)-modular categories from special linear Lie superalgebras

TL;DR

This paper explores two m-traces in the representation category of quantum Lie superalgebras related to sl(m|n) at roots of unity, leading to new topological quantum field theories and 3-manifold invariants, with conjectures for broader cases.

Contribution

It introduces two distinct m-traces in the category of quantum sl(m|n) representations and demonstrates their applications to topological invariants and ETQFTs, proposing a general conjecture.

Findings

First m-trace yields new ETQFTs.

Second m-trace on typical modules leads to 3-manifold invariants for sl(2|1).

Conjecture: extends to sl(m|n) for 3-manifold invariants.

Abstract

We examine two different m-traces in the category of representations over the quantum Lie superalgebra associated to $\mathfrak{sl}(m|n)$ at root of unity. The first m-trace is on the ideal of projective modules and leads to new Extended Topological Quantum Field Theories. The second m-trace is on the ideal of perturbative typical modules. We consider the quotient with respect to negligible morphisms coming from this m-trace and show that in the case of $\mathfrak{sl}(2|1)$ this quotient leads to 3-manifolds invariants. We conjecture that the quotient category of perturbatives over quantum $\mathfrak{sl}(m|n)$ leads to 3-manifold invariants and more generally ETQFTs.