8.1 Vacuum vectors

In this final chapter we shall deal rather briefly with other aspects of the Jones-Witten theory. First of all we want to discuss how the functional integral, at least formally, gives the extra data required for a topological quantum field theory, as axiomatized in Chapter 2.

For a 3-manifold Y with boundary 2 the Chern-Simons functional L(A) of Chapter 7 is not really a complex number (modulo 2ttZ). Intrinsically the exponential elL<A) should be viewed as a vector in the complex line the fibre of the standard line-bundle i? over the space of connections on the boundary 2. For the special case Y = 2xl with the boundary this can be seen as follows.

Using parallel transport in the /-directions we can identify connections on Y with a path A, of connections on 2, Os 1. As noted in Chapter 7 the Chern-Simons functional then becomes the classical action for paths on a symplectic mani­fold, and its exponential therefore gives the parallel transport (along the path A, in sd£) from the fibre to the fibre . Thusin the Hilbert space Z(2), as required by the axioms of Chapter 2. Recall that Z(X) is defined, at level k, by a space of sections of the line-bundle L|, where Ls is the line-bundle on the symplectic quotient siz//^. We then define Z( Y) as the linear function on Z(2*) = Z(Z)* given by assigning to the section <f> of if vfc the Feynman integral

 

 

This Feynman integral is over all connections A on Y which are flat (and equal to B) on X Intuitively the measure 3)A involves the symplectic measure on si*together with a measure coming from the interior.

We can make more rigorous sense of this procedure, in the large k limit, by applying stationary-phase approximation as in Chapter 7. This reduces the problem to the relevant critical points which are the flat connections on Y. For example, when Y is a 'handlebody', H'(Y, G) can be identified with a Lagrangian sub-manifold of Hl(I,, G). For a Heegard split­ting of a closed 3-manifold along a surface X the two Lagrangian sub-manifolds obtained from the two halves inter­sect at points corresponding to representations 7Ti (Y) -* G. This brings us back to the stationary-phase calculations made in Chapter 7, and the situation is formally similar to that of the Casson invariant which is the invariant of another topo­logical quantum field theory (cf. [2]).