5.3 Boundary components
Working on Riemann surfaces with marked points is the natural algebro-geometric approach to the subject. Having marked points allows 'poles'. There is another approach which involves working on Riemann surfaces with boundary. This requires the use of complex analysis and associated boundary- value problems. We can pass from a surface with marked points to a surface with boundary simply by cutting out small discs around the marked points.
Each method has its own advantages. Thus surfaces with boundary can be glued together along a common boundary and this is an important operation in the theory. The analogue for marked points is to allow an algebraic curve to acquire singularities (double points). These questions will be taken up later.
Naturally the theory for surfaces with boundary requires a preliminary investigation of gauge theory on a circle. In fact this is the way conformal field theory enters the picture, and the representation theory plays an important role. We shall begin therefore by a rapid review of some basic aspects of the subject, referring for fuller details to .
Let S be the standard circle and let sis denote the affine space of G-connections for the trivial bundle over S. The is the loop group. It acts affinely on sis with orbits of finite codimension. The orbits are determined by the monodromy of the connection around S: this is a conjugacy class in G.
LG has (for each integral k S: 1) a central extension (by the circle) LG. The co-adjoint action of LG on the dual of its Lie algebra preserves hyperplanes (codimension 1) and s£s, together with its LG- action, can be identified with one of these. Thus the orbits of LG in are co-adjoint orbits, and 'integral' orbits lead, by quantization, to irreducible representations of LG (i.e. projective representations of LG).
Combined with the hyperplane inclusion
Now let us consider a surface 2 with boundary S (for simplicity we discuss only one boundary component, but the results are quite general). As before we consider the space s&z of G-connections on 2 and the group ^ of gauge transformations on 2. The symplectic structure on is defined as for closed surfaces and again we have a moment map
This time, however, the moment map picks up a boundary term and one finds the formula
Here As is the restriction of A to S in the following refined sense. The restriction homomorphism actually lifts naturally to the central extension LG = of LG = &s■ Passing to the Lie algebras and dualizing gives a map this associates to any the element of (Lie we have denoted simply by As.
Now pick a G-orbit Wa in s4s, corresponding to the conjugacy class Ca of G. We can then form the generalized symplectic quotient
In view of the formula (5.3.1) this symplectic quotient can be identified with the moduli space of representations introduced in Chapter 3. It parametrizes representations of ir, (2— P) whose monodromy around P lies in the conjugacy class Ca.
As for the case of marked points we can now restrict Ca to consist of fcth roots of 1 and then quantize Xa at level k. (Note that the natural line-bundle Lk on Xa gives k times the standard symplectic form.)
As before we can, at least formally, interpret the resulting space as a 'multiplicity space'. It gives the part of the quantum Hilbert space of ksis which transforms according to the representation ef(A) of Here es: ^ -> #s is the lift of the restriction or evaluation map and A denotes the irreducible level k representation of (Ss parametrized by the relevant orbit.
Although we shall not pursue this boundary case any farther we should point out it is, in a sense, slightly 'less singular' than the case of marked points. Analytically, taking a boundary value on a codimension one circle is less singular than evaluation at a point. This can have technical advantages.