8 - Final comments

Summary

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 e^iL(A) should be viewed as a vector in the complex line L_AΣthe fibre of the standard line-bundle L over the point A_Σ in the space A_Σ 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 A_t 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 A_t, in A_Σ) 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 L^k_Σ, where L_Σ is the line-bundle on the symplectic quotient A_Σ // g_Σ.