8.4 Outstanding problems

Since Witten's theory involves the heuristic Feynman integral, it may be helpful to review here how much of the theory is on a rigorous basis, and what outstanding problems remain.

As we have more or less indicated the construction of the vector spaces Z(2) associated to framed surfaces 2 (with or without marked points) can be done quite rigorously. The difficult part of the theory is to construct the vectors Z( Y) e Z(2) associated to framed 3-manifolds Y with boundary 2. However, the axioms in Chapter 2 give rules governing these vectors. These rules can be used to evaluate them. The only difficulty is that the rules might not be consistent. One has therefore to check consistency.

For the Jones polynomials this was essentially the original approach of Jones. For the new invariants of 3-manifolds consistency has been verified by Reshetikhin and Turaev [27].

Essentially the consistency of the Witten axioms (or rules) involves understanding how the Hilbert spaces Z(2) change as a surface 2 acquires a double point. The formulation ofTsuchiya and Yamada in terms of the compactified moduli space Mg appears to incorporate the relevant properties, but it would be desirable to elucidate the situation. See also [19].

Defining the vectors Z(Y)e Z(I) is, as we explained in §8.1, equivalent to computing certain Feynman integrals. Since the Chern-Simons Lagrangian is purely topological there are no real local difficulties of analytical nature in the Feynman integral. We can therefore view the surgery methods indicated above as an effective way of computing our Feynman integral. After all, an integral is simply a linear functional with certain additive local properties, and the consistency verification we have alluded to could be construed as checking these properties.

It would of course be even better if one could define some purely combinatorial version of the Chern-Simons Lagrangian as in lattice-gauge theories. Some encouragement comes from the fact that Reidemeister torsion has such a definition and this enters into the stationary-phase calculation for the Chern-Simons Lagrangian described in Chapter 7. However, this may be too ambitious and we may have to settle for the surgery approach.

In addition to the Hamiltonian approach using the Hilbert space Z(2) the stationary-phase calculations of Chapter 7 also lead to rigorous formulae. Although we only gave the leading term it should be possible to proceed further and develop a fully rigorous series expansion in fe It is then a challenging problem to show that this does in fact give the expansion of the Witten invariant computed by Hamiltonian methods. As yet this problem is very much open. Moreover, it is not clear what kind of function of k we get in general from Witten's theory. For links in S3 the Jones invariant is a polynomial in t = exp (27n/(fc + 2)), but for general 3-mani- folds the situation is more complicated. In particular it is not obvious that the Witten invariant will always be determined by its kexpansion.