7.1 The Chern-Simons Lagrangian
So far we have presented the Jones-Witten theory from the Hamiltonian point of view. This gave functors surface 2 finite dimensional vector space Z(S)
3-manifold Y with I = d Y vector Z(Y)eZ(I)
starting from the data of a compact Lie group G and an integer, k, the level.
This Hamiltonian approach is mathematically rigorous, although it is not yet entirely developed.
In this chapter we shall present Witten's Feynman path- integral approach. It is not mathematically rigorous, but it is conceptually simple, and provides a natural starting point for the theory.
Fix a compact Lie group, G, which for simplicity we take to be SU(n). In the Feynman approach, one uses the Chern- Simons Lagrangian. Let Y be a closed oriented 3-manifold. Consider sd, the space of all G-connections on the trivial G-bundle over Y.
For any connection A its curvature FA is a Lie-algebra- valued 2-form. In three dimensions, the dual to a 2-form is a 1-form, i.e. *Fa is a 1-form. However, si is an affine space, and so its tangent space at any point consists of Lie-algebra- valued 1-forms.
Thus F is a 1-form on si. Its value on a tangent vector to si is given by multiplying by FA and integrating over Y, contracting on the Lie algebra variables. Let
Clearly $ acts on si; and F is ^-invariant. Moreover in the fibration (with singularities)
F vanishes in the vertical (fibre) direction, and thus comes from the base. So F is a well-defined 1-form on sifS.
Also dF = 0, i.e. F is a closed 1-form. Thus one would expect that F can be expressed in the form
for some function f where / is a ^-invariant scalar-valued function on si determined up to a constant. One can fix this constant by requiring that on the trivial connection.
This works if sd/^ is simply connected. Otherwise, one can only expect / to be locally defined and, globally, it will be multi-valued. In fact, sd/^ is not simply connected, and / is well defined only up to integral multiples of some constant. This / is the Chern-Simons functional. It is well defined modulo integers, and is ^-invariant.
where Ae si. Here L is a multiple of /: the notation L has been used to be consistent with Witten . One now verifies that L is invariant under the subgroup
given by the connected component of 'ê containing the identity. Here 'S, % differ by a copy of Z; and, under a generator of <§/%, L is not invariant: it picks up a multiple of 2ir.
Thus c'kUA) is a well-defined function of A, for keZ. Witten's invariant of 3-manifolds is now defined formally as the 'partition function1:
This is a very elegant definition provided one believes that the integral makes sense! More generally, we consider a closed oriented curve
and fix an irreducible representation, A, of G, in addition to the data required previously: G, k.
Explicit formula Define
A connection A on Fthen defines a parallel transport along any curve in Y. In particular, around C, one obtains a monodromy element Monc (A). Then evaluated by taking the trace in the representation A. Here WC(A) is known as a Wilson line. Define
This is a generalization of Z( Y). In physicists' language,
where (,) denotes the (unnormalized) expectation value.
i.e. A is a flat connection and thus corresponds to a representation of iTii Y):
Of course, one can similarly deal with several components Ci,..., Cr, to each one associating a different irreducible representation of G. Then
It is important to notice that the above definitions involve no metrics or volumes. This is an indication that we have defined topological invariants.