A connection between lattice and surgery constructions of three-dimensional topological field theories

We study the relation between lattice construction and surgery construction of threedimensional topological field theories. We show that a class of the Chung-Fukuma-Shapere theory on the lattice has representation theoretic reformulation which is closely related to the Altschuler-Coste theory constructed by surgery. There is a similar relation between the Turaev-Viro theory and the Reshetikhin-Turaev theory. E-mail address: [email protected] E-mail address: [email protected] Various three-dimensional topological field theories which satisfy Atiyah’s axiom [1] have been constructed by now [2–4]. Relations among them are, however, still not clear. It is important to investigate them more with a view to systematic classifications of threedimensional topological field theories . In the following, we study several topological field theories and establish a connection among them. There are two principal methods to construct three-dimensional topological field theories with mathematical rigor. One is the lattice or ‘state-sum’ method in which we represent a manifold by a simplicial complex and consider a statistical model on it. The other method employs the surgery representation of three-manifolds. One can construct arbitrary closed three-manifold M(L,f) by a surgery of S 3 along a framed link (L, f) in it. Therefore a class of framed link invariants, actually those which are invariant under Kirby moves, give rise to invariants of three-manifolds. Sometimes this construction is lifted to that of a topological field theory whose partition function is the three-manifold invariant. Though these two methods differ a lot in nature, surprisingly topological field theories defined by the two methods sometimes reveal close relationship. The most famous example is the relation between the lattice theory by Turaev and Viro [3] (TV) and the ReshetikhinTuraev (or Chern-Simons [7]) theory [4] (RT) defined by surgery: Z = |Z | [8]. There is also a suggested relation [9, 10] between the Dijkgraaf-Witten [2] or Chung-FukumaShapere [6] theory on the lattice, and the Altschuler-Coste theory [9] constructed using the surgery representation. It is desirable to have a more general statement which relates the lattice construction and the surgery construction. In this note, we study whether a functor used in the surgery construction of topological field theories can induce a topological field theory on lattice. We take the Altschuler-Coste functor F employed in the surgery construction in ref. [9] as an example and show that it is possible. We are motivated by a remark in ref. [11] on the relation between the Turaev-Viro theory and the Reshetikhin-Turaev theory. In ref. [11], an attempt was made to rewrite the Turaev-Viro invariant in the form of the partition function of a ‘three-dimensional q-deformed lattice gauge theory,’ defined making use of the functor F Uq(sl(2,C)). It is a functor from the category of colored ribbon graphs to the category of representations of a ribbon Hopf algebra and was originally used to define the Reshetikhin-Turaev theory. 3 Note that two-dimensional unitary topological field theories are classified completely [5].