# 5.2 Marked points

As we mentioned in Chapters 2 and 3 it is necessary for the Jones-Witten theory to generalize to the case of sur­faces 2 with marked points. The situation of the previous section has a natural generalization as follows.

Given the moment map defined by the curvature we can pick other orbits than zero in Lie (&)*. In particular, given a point P on I we have an evaluation eP: G and hence, dually, an embedding

The image consists of delta functions on P with values in L(G)*. In particular a G-orbit MA in L(G)* defines a G-orbit SP(MK) in Lie(9)*.

Now fix points Pi,... ,Pr on 2 and integral orbits (or G-representations) A,,..., Ar. This defines the G-orbit

We can therefore, for each integer k, look at the generalized symplectic quotient

(5.2.1)

This consists of connections which are flat outside the Pj and have appropriate 5-function curvatures at the Pj. A local model for a connection near Pj, with Pj as origin of polar coordinates (r, 0), is A} A0 where A} is in the conjugacy class of the G-orbit (l/fc)MAj (and we identify L(G) = L(G)* using our invariant metric). The monodromy around Pj of such a connection is just exp (2iri Aj) and is a fcth root of unity. Thus our symplectic quotient (5.2.1) is just the moduli space of representations which we denoted in Chapter 3 by H1(2,P; G, C), the Q being conjugacy classes of order k in G.

Once we pick a complex structure r on 2 we again have the identification of this space of representations with a moduli space of holomorphic bundles. This was described in Chapter 3. Again a proof along the lines of Donaldson [11] would be most natural.

As explained in § 4.4 we can replace the generalized sym­plectic quotients by the usual ones. Thus consider the product where kstf stands for si with its symplectic form multiplied by k, or equivalently if replaced by !£k. The symplectic quotient                can then be identified with (k times) the quotient in (5.2.1). Moreover the quantization of should pick out that part of the quantization of ksi which transforms according to the representation

of <£To summarize, for each k, we have a moduli space Mk (depending on k) with a line-bundle Lk. The space H°(Mk, Lk) is the 'multiplicity space' for the representation ©jCp/Aj-) of ^ in the quantization at level k of the space si.

Thus even if the quantization of si is not too well defined we have given a meaning to that part of it that transforms according to 'evaluation representations' of Of course we still have to investigate the role of the complex structure chosen, but this question is postponed until the next chapter.