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 Σ the Chern–Simons functional L(A) of Chapter 7 is not really a complex number (modulo 2πZ). Intrinsically the exponential eiL(A) should be viewed as a vector in the complex line LAΣthe fibre of the standard line-bundle L over the point AΣ in the space AΣ of connections on the boundary. For the special case Y = Σ × I with the boundary

this can be seen as follows.

Using parallel transport in the I-directions we can identify connections on Y with a path At of connections on Σ, 0 ≤ t ≤ 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 At, in AΣ) from the fibre L0 to the fibre L1 Thus

and, raising to the kth power,

We shall now show formally how a 3-manifold Y with ∂ Y = Σ gives rise to a vector

Z(Y)∈ Z

in the Hilbert space Z(Σ), as required by the axioms of Chapter 2. Recall that Z(Σ) is defined, at level k, by a space of sections of the line-bundle LkΣ, where LΣ is the line-bundle on the symplectic quotient AΣ // gΣ.