7.3 The Hamiltonian formulation

We shall now indicate why the Chern-Simons Lagrangian is supposed to lead to the Hamiltonian version of the theory which we have been developing in earlier chapters.

To go from the path-integral to the Hamiltonian formula­tion we have to separate out space and time. We therefore consider a 3-manifold of the form 2 x R. To get the Hilbert space of the theory we are supposed to quantize the space of 'classical solutions', i.e. critical points of the Lagrangian. But

these are just gauge equivalence classes of flat connections and so give us the moduli space of flat G-bundles on 2 x R. However, these are the same as the flat G-bundles on 2. These are the moduli spaces we met in Chapter 3 whose quantiz­ations give the (finite-dimensional) Hilbert spaces we have been studying.

The fact that these spaces are independent of the R-variable (time) shows that the Hamiltonian of the theory is zero (i.e. that the theory has no dynamics and is topological).

An alternative derivation is to re-interpret a connection A on 2 x R as a path A, of connections on 2. This comes by using parallel transport in the time direction (R) to identify bundles at different times or, as physicists would say, by working in a gauge in which A0 = 0. This simplifies the Chern- Simons Lagrangian since the cubic term now drops out and we simply get

 

 

This is just the classical formula for the action for a path in the symplectic linear space sd. It follows that our Hilbert space should be the «^-invariant part of the quantum Hilbert space of sis. However, as we have argued formally in Chapters 4 and 5, this should be the same as quantizing the symplectic quotient sdsi.e. the moduli space of flat G-bundles over 2.

We shall conclude this chapter with a few brief comments on the relation between the phase subtleties in the Lagrangian and Hamiltonian approaches. We recall from § 7.2 that (for the limit k -» oo) there was a non-topological term, depending on a background metric. Witten shows in [36] that, by sub­tracting a 'counter-term' (the gravitational Chern-Simons), we can recover a purely topological theory. However, for this one has to pick a framing of the 3-manifold (itself a piece of topological data).

In the Hamiltonian version the corresponding difficulty has to do with the phase ambiguity in our Hilbert spaces: the fact that the curvature of the bundle of Hilbert spaces (over Teichmuller space) is a non-zero scalar.

The relation between these two manifestations of the phase ambiguity depends on earlier ideas of Witten [37], sub­sequently given rigorous formulation and proof by Bismut and Freed [8]. This relates the gravitational r?-invariant of the 3-manifold constructed from /eDiff+(2), with the monodromy of the Quillen determinant line-bundle. We have essentially been ignoring these subtleties so it would make little sense to enter now into an elaborate discussion. We should emphasize, however, that they are a crucial aspect of the theory (related also to the central extensions of loop groups) and refer the reader to the reference above as well as [36]; see also [4].