Integral Lattices of the SU(2)-TQFT-Modules

TL;DR

This paper constructs bases for lattices in SU(2)-TQFT modules over specific rings of integers and demonstrates how certain 3-manifold invariants relate across different TQFT theories.

Contribution

It introduces explicit bases for lattices in SU(2)-TQFT modules and relates the Frohman Kania-Bartoszynska ideal invariants across multiple TQFT frameworks.

Findings

Established bases for lattices over rings of integers in SU(2)-TQFT modules.

Proved the Frohman Kania-Bartoszynska ideal invariant equality across different TQFT theories.

Connected invariants in SU(2), 2', and SO(3)-TQFT theories.

Abstract

We find bases for naturally defined lattices over certain rings of integers in the SU(2)-TQFT-theory modules of surfaces. We consider the TQFT where the Kauffman's A variable is a root of unity of order four times an odd prime. As an application, we show that the Frohman Kania-Bartoszynska ideal invariant for 3-manifolds with boundary using the SU(2)-TQFT-theory is equal to the product of the ideals using the 2^{'}-theory and the SO(3)-TQFT-theory under a certain change of coefficients.