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5.4 Topological field theories of Turaev-Viro type
We now discuss the construction of a three-dimensional topological field theory which can be
seen as a three-dimensional cousin of the two-dimensional topological field theory constructed
in example 4.1.14 on 2-manifolds with a triangulation. Our exposition closely follows [BK1].
We work with a three-layered structure involving manifolds of dimensions 1,2 and 3. It
comprises in particular a three-dimensional topological field theory
Z T V :
Cob(3, 2) → vect(C)
in the sense of definition 4.1.5. In particular, we have to achieve the following goals:
• To closed oriented three-manifolds, we want to assign an invariant with values in End C (I).
This invariant should be a topological invariant.
• To closed oriented two-manifolds, we want to assign a finite-dimensional complex vector
space. To a three-manifold M with boundary representing a cobordism ∂ − M −→ M
∂ + M,
we want to assign a linear map. This expresses a locality property of our invariants.
• We want to go one step further and assign categories to closed oriented 1-manifolds.
Thus we have as a zeroth layer categories associated to closed oriented 1-manifolds. At
the first layer, we will have not only closed surfaces, but also 2-manifolds with boundary.
After having chosen an object for each boundary circle, we get a vector space which depends
functorially on the choice of objects. On the third level, we have three-manifolds with corners
relating the 2-manifolds with boundaries. In particular, we obtain invariants of knots and
links in three-manifolds generalizing the constructions of the previous subsection and thus
to representations of braid groups.
This can be formalized by using bicategories, defining a bicategory Cob(1, 2, 3) of 1-2-3
cobordisms, leading to the notion of an extended topological field theory. For an account, see
[L, Section 1.2] and for an informal account see [NS].
We stick here to a low key approach.
Definition 5.4.1
Let K be an algebraically closed field. A fusion category is a rigid, K-linear semisimple tensor
category with finitely many isomorphism classes of simple objects and finite dimensional spaces
of morphisms. We assume that End C (I) = Kid I .
We restrict to the ground field K = C. The following proposition will be used:
Proposition 5.4.2. [ENO, Theorem 2.3]
If C is a spherical fusion category over the field C, then the so-called global dimension of C is
non-zero:
D 2 := ∑ (dim V i ) 2 ≠ 0 .
i∈I
(We do not suppose that a square root D of the right hand side has been chosen; the notation
will just be convenient later.)
Observation 5.4.3.
1. All manifolds are compact, oriented and piecewise linear. We fix as a combinatorial datum
a polytope decomposition Δ, in which we allow individual cells to be arbitrary polytopes
(rather than just simplices). Moreover, we allow the attaching maps to identify some of
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