Almost integral TQFTs from simple Lie algebras

Almost integral TQFTs were introduced by Gilmer (G). The aim of this paper is to modify the TQFT of the category of extended 3- cobordisms given by (T) to obtain almost integral TQFT. AMS Classification 57M27, 57R56 Inspired by a 3-dimensional interpretation of the Jones polynomial for knots, Witten predicted that one can define topological invariants for 3-manifolds us- ing simple complex Lie algebras. The first concrete construction was obtained by Reshetikhin and Turaev (RT) for sl2. Soon after, similar invariants were constructed for all simple Lie algebras. They are called the Witten-Reshetikhin- Turaev invariants (WTR-invariants for short) or quantum invariants because Reshetikhin and Turaev's construction is based on the theory of quantum groups. It was also Witten's vision that the WTR-invariants can be extended to the Topological Quantum Field Theory (TQFT) in a rather natural manner. Loosely speaking, a TQFT with base ring K is a functor from the category C to the category of K-modules where the objects in C are surfaces and morphisms in C are 3-dimensional cobordisms. We denote a TQFT by a pair (T , τ) where T and τ are maps between objects and morphisms respectively. Most known TQFTs have anomalies. Anomalies can be resolved by introducing p1-structure as in (BHMV) or by studying the category of extended 3-cobordisms as in (T). In this paper we follow the latter. One of the features of extended 3-cobordisms is that they contain embedded colored ribbon graphs. (See Section 2.2 for more details about TQFT and ribbon graph.) For each simple complex Lie algebra g and certain integer r, one can define a TQFT (T g r , τ g r). In this case, the col- ors of the ribbon graphs come from the representations of the quantum group Uv(g) associated to g with the parameter v equal to a primitive r-th root of unity.