5.1 Connections on Riemann surfaces

We now come to the infinite-dimensional case of symplectic quotients and their quantization which is relevant to the Jones-Witten theory. This case was studied, for other purposes, in [5] and we refer the reader there for other details.

is just the curvature /a( A) = FA. Note that FA, a Lie-algebra-

Given a compact oriented surface I, and a compact Lie group G we consider the infinite-dimensional affine space si of G-connections on the trivial bundle over X Note that, if G is simply connected, for example SU(n), all G-bundles over Z are trivial. The space si has a natural symplectic structure. If A e si, a tangent vector at A is a Lie-algebra- valued 1-form a. Hence, for two such tangent vectors a, /3, we can define the skew pairing

 

 

Here we have written the formula in the case G = SU(n). In general we replace -Trace by the fixed G-invariant inner product on Lie G.

The group ^ of gauge transformations, i.e. the group of smooth maps I. -* G, acts naturally on si preserving its sym­plectic structure. An elementary calculation [5; p. 587] shows that the moment map

 

valued 2-form, gives a linear function on Lie the space of Lie-algebra-valued 0-forms, by using the inner product on Lie(^) and then integration over 2,. Hence the symplectic quotient

 

 

is the moduli space of flat G-connections on S, or equivalently the moduli space of homomorphisms (up to conjugacy) G. This was the moduli space denoted by Hx(2, G) in Chapter 3.

For an irreducible homomorphism ir^l)-* G the only gauge automorphisms that preserve it are the constant central automorphisms arising from the centre of G.

For semi-simple G this is finite so that we are, at least formally, in the 'good' case for the G-action. An abelian factor in G causes only minor differences because of the first Chern class.

We now turn to the holomorphic view-point by fixing a complex structure r on X This induces a natural complex structure on si so that we have an infinite-dimensional analogue of the linear situation discussed in Chapter 2. Moreover by taking the (0,1) part d^ of the covariant deriva­tive dA of a connection A we can identify si with the space

of holomorphic structures on the trivial bundle 2xGc [5; Chapter 7]. Also the complexification 'Sc of % given by smooth maps of Z -* Gc, acts naturally on e€. The moduli space MT of holomorphic Gc-bundles over ST is the analogue of the Mumford quotient of Chapter 4. It contains as an open set the subspace parametrizing stable Gc-bundles.

In fact we can do a little better. There is actually a holomor­phic line-bundle i? with connection over ^ whose curvature is —2tt'\ times the Kahler form, and acts naturally on if with preserving its metric and connection. This line-bundle is the Quillen line-bundle whose fibre at Ae si = is

 

where Ea is (for G= U(n) or SU(n)) the holomorphic rank n vector bundle defined by dA and det denotes the highest exterior power. The metric on .TA is defined by regularized determinants of Laplacians and Quillen [25] proved that this gives the right curvature. For general G there is a Quillen line-bundle for each representation (pull back from U(n) by G-> U(n)), and in particular for the adjoint representation. In general these will give powers of the desired line-bundle if.

The constant centre of G acts trivially on si and its action on the fibres of !£A is given, for G = SU(n), by the scalar action of nth roots of 1 on the sheaf cohomology of EA. But, since the first Chern class is zero,

 

 

so that the scalar actions cancel. Thus 'S acts on (si, !£) through the quotient ^ by the constant centre. This shows that the line-bundle i? descends (without resorting to powers) to give a line-bundle L on the moduli space M, at least on the stable part.

A more careful examination shows that the line-bundle L extends to the whole of MT, the essential point being that the semi-stable bundles which are identified to give a single point of MT differ by extensions so that the determinant line Z£A is the same for all.

Just as for the finite-dimensional case discussed in Chapter 4, we now have a map

 

 

which we expect, by analogy, to be a homeomorphism. This is the content of the Narasimhan-Seshadri theorem [22] already mentioned in Chapter 3. There is a direct proof by Donaldson [11] which is more in the spirit of our present context.

So far we have just described the classical picture leading to moduli spaces as quotients. We now take their quantiz­ations, which is what we are really after.

We want to consider the quantization of the symplectic space si and then take its ^-invariant part. We expect this to be the same as the quantization of the symplectic quotient

 

 

To define this quantization we will pick a complex structure r on 2 and use the Narasimhan-Seshadri theorem to replace H\2, G) by the moduli space MT. Now we quantize this, at level k, by taking the space of holomorphic sections of Lk over MT. We expect this to be projectively independent of r.

In the next chapter we shall discuss the various methods that can be used to establish this key result. At this stage we merely note that si is too infinite-dimensional to have a genuine quantization of the right kind. This is why we make the reduction to the finite-dimensional (and compact) quotients.