Modified Turaev-Viro Invariants from quantum sl(2|1)

TL;DR

This paper develops a modified quantum invariant for the superalgebra U_q(sl(2|1)) by replacing quantum dimensions with modified quantum dimensions, enabling the construction of finite semi-simple categories and 3-manifold invariants.

Contribution

It introduces a new approach using modified quantum dimensions to construct semi-simple categories from U_q(sl(2|1)), overcoming issues with vanishing quantum dimensions.

Findings

Constructed a finite semi-simple category from U_q(sl(2|1)) using modified quantum dimensions.

Established that the resulting categories are relative G-spherical categories.

Derived new 3-manifold invariants from these categories.

Abstract

The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: 1) specialize q to a root of unity; this forces some modules to have zero quantum dimension, 2) quotient by morphisms of modules with zero quantum dimension, 3) show the resulting category is finite and semi-simple. In this paper we show an analogous construction works in the context of U_q(sl(2|1)) by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize q to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative G-spherical categories. As a consequence we obtain invariants of 3-manifold with additional structures.