State sum invariants of 3-manifolds and quantum \(6j\)-symbols.

*(English)*Zbl 0779.57009The authors show how some algebraic input (over a commutative ring \(K\) with 1) gives rise to (unoriented) topological quantum field theories (TQFT). A TQFT here is a functor from the category of surfaces (with homomorphisms being 3-dimensional bordisms) into the category of finite- dimensional \(K\)-modules. Such a TQFT in particular assigns to each closed unoriented (maybe non-orientable) 3-manifold \(M\) an element \(| M | \in K\).

The basic idea for the construction is that of admissable colorings of triangulated surfaces and 3-manifolds. Fix some finite set \(I\) and a distinguished subset of unordered triples of elements of \(I\), called admissable. Then an admissable coloring of a triangulated surface (resp. 3-manifold) is a function from the corresponding set of edges into \(I\), such that the colors of the boundary of each 2-simplex determine an admissable triple. The algebraic input now allows to assign to each compact triangulated 3-manifold \(M\) (possibly with boundary) and admissable coloring \(\varphi\) an element \(| M |_ \varphi \in K\). Using this and assuming independence of triangulations, the TQFT is constructed in a purely formal way. The \(K\)-module associated to a surface is essentially the free module on admissable colorings of \(F\). The invariant of a closed 3-manifold \(M\) is the sum \(\sum| M |_ \varphi\) over all colorings \(\varphi\) of \(M\). In order to prove that \(| M | _ \varphi\) does not depend on the triangulation the authors define equivalent notions of coloring and \(|\;|_ \varphi\) for a category of simple 2-complexes (dual to triangulations), and then translate the Alexander moves (moves relating different triangulations of a dimensionally homogeneous polyhedron) into this dual framework. This is necessary to reduce to a finite set of moves. It should be mentioned that the authors prove a relative version of the Alexander theorem. The relation to simple spines of manifolds, and a resulting method of computation of \(| M|_ \varphi\) from Heegard diagrams is discussed.

It is shown that for each \(r\)-th root of unity \(q\), \(r>2\), the quantum \(6j\)-symbols (associated with the quantized enveloping algebra \(U_ q(\text{sl}_ 2(\mathbb{C})))\) provide the necessary algebraic input. Thus for each such \(q\) the authors have constructed a TQFT. For these TQFTs explicit calculations of \(| M |\) for \(S^ 3\), \(\mathbb{R} P^ 3\), \(L (3,1)\) and \(S^ 2 \times S^ 1\) and topological interpretations (Betti numbers) of \(| M |\) for \(r=3\) and arbitrary 3-manifolds are given.

The basic idea for the construction is that of admissable colorings of triangulated surfaces and 3-manifolds. Fix some finite set \(I\) and a distinguished subset of unordered triples of elements of \(I\), called admissable. Then an admissable coloring of a triangulated surface (resp. 3-manifold) is a function from the corresponding set of edges into \(I\), such that the colors of the boundary of each 2-simplex determine an admissable triple. The algebraic input now allows to assign to each compact triangulated 3-manifold \(M\) (possibly with boundary) and admissable coloring \(\varphi\) an element \(| M |_ \varphi \in K\). Using this and assuming independence of triangulations, the TQFT is constructed in a purely formal way. The \(K\)-module associated to a surface is essentially the free module on admissable colorings of \(F\). The invariant of a closed 3-manifold \(M\) is the sum \(\sum| M |_ \varphi\) over all colorings \(\varphi\) of \(M\). In order to prove that \(| M | _ \varphi\) does not depend on the triangulation the authors define equivalent notions of coloring and \(|\;|_ \varphi\) for a category of simple 2-complexes (dual to triangulations), and then translate the Alexander moves (moves relating different triangulations of a dimensionally homogeneous polyhedron) into this dual framework. This is necessary to reduce to a finite set of moves. It should be mentioned that the authors prove a relative version of the Alexander theorem. The relation to simple spines of manifolds, and a resulting method of computation of \(| M|_ \varphi\) from Heegard diagrams is discussed.

It is shown that for each \(r\)-th root of unity \(q\), \(r>2\), the quantum \(6j\)-symbols (associated with the quantized enveloping algebra \(U_ q(\text{sl}_ 2(\mathbb{C})))\) provide the necessary algebraic input. Thus for each such \(q\) the authors have constructed a TQFT. For these TQFTs explicit calculations of \(| M |\) for \(S^ 3\), \(\mathbb{R} P^ 3\), \(L (3,1)\) and \(S^ 2 \times S^ 1\) and topological interpretations (Betti numbers) of \(| M |\) for \(r=3\) and arbitrary 3-manifolds are given.

Reviewer: U.Kaiser (Siegen)

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

81T25 | Quantum field theory on lattices |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |