invariant) 2-form. The restriction of ω on the space F therefore is

closed as well.

Now we need to introduce the gauge group G acting on the space A.

This group is the group of all sections of all smooth maps X ’ G via the

pointwise multiplication. Let us consider g ∈ G and A ∈ A1 (X, ˜) A.

g

One can easily check that the action of g on A is given by g —¦ A =

’1

+ gAg ’1 . Let us prove

ρg · g

LEMMA 5.7.2 The curvature of g —¦ A is equal to gKg ’1 , where

1

K= ρA + [A, A]

2

is the curvature of A.

236 CHAPTER 5. FAMILIES AND MODULI SPACES

Proof. All we have to check is that the equation

1

(g —¦ A) + [g —¦ A, g —¦ A] = gKg ’1

ρ

2

is satis¬ed. It can be done, since we know that

· g ’1 + gAg ’1 .

g—¦A= ρg

We leave the details of this computation to the reader as an exercise.

COROLLARY 5.7.3 The subset F ‚ A of ¬‚at connections is pre-

served by the action of G.

LEMMA 5.7.4 The restriction of the 2-form ω to F is G-invariant

Proof. Another simple exercise for the reader.

Let F irr be the subset of F consisting of those connections which

give rise to irreducible ¬‚at bundles, meaning that no proper subbundle

is preserved by the ¬‚at connection. We need here to say that F irr is

a manifold, that G-action on F irr is free, and that the quotient F irr /G

is also a manifold. Next, we would like to show that the tangent space

to F irr /G at the class of ρ is equal to H 1 (X, ˜). For this we need

g

to establish that the space B 1 (X, ˜) of 1-coboundaries of the complex

g

5.7.6 viewed as a subspace of T ρ A = A1 (X, ˜) is exactly the tangent

g

space to the G-orbit of the point ρ . To see this, let us consider the

map G ’ A given by g ’ g —¦ ρ and compute the di¬erential of this

map. Using the identi¬cation A A1 (X, ˜) and the explicit expression

g

for the G-action on A we see that the linear part of an in¬nitesimal

action of G is precisely the di¬erential ρ : A0 (X, ˜) ’ A1 (X, ˜).

g g

LEMMA 5.7.5 The form ω|F is vertical, meaning that if s ∈ “(X, ˜)

g

and f ∈ Z 1 (X, ˜) then ω ( ρ s, f ) = 0.

g

Proof. We have the Leibnitz rule:

dB s, f = B ρ s, f + B s, ρf .

5.8. VERLINDE FORMULA 237

The second term in the right hand side vanishes, because f is a cocycle.

Therefore we have

ω( ρ s, f ) = B ρ s, f = d(B s, f ) = 0

X X

by the Stokes™ theorem.

These three Lemmas allow us to make the following important con-

clusion: the 2-form ω descends to a closed two-form ω on the quotient

F irr /G.

The basic correspondence between ¬‚at bundles and representations

of π1 (X) amounts to the fact that two spaces M and F irr /G can be

identi¬ed and the tangent spaces H 1 (π1 (X), g) and H 1 (X, ˜) at points

g

[ρ] and [ ρ ] are canonically isomorphic. Under this identi¬cation, the

forms ω and ω correspond to one another. We refer to a paper by

Karshon [37], where explicit computations purely in terms of group

cohomology show the closedness of ω. In fact, this work of Karshon

gives a K¨hler structure to the smooth locus of the moduli space of

a

representations of the fundamental group of any compact K¨hler man-

a

ifold generalizing Goldman™s theorem to higher dimensions (when the

conditions of the Goldman-Millson theorem stated above are satis¬ed).

5.8 Verlinde formula

The moduli space M M (0, r) is a singular projective variety in gen-

eral and therefore one will encounter many di¬culties while trying to

use the standard di¬erential geometry apparatus in attempts to give an

exhaustive description to this space. In fact, the singularities have not

yet benn fully analyzed. It was mentioned earlier that in the case when

r and d are coprime integer numbers, the moduli space M (r, d) of holo-

morphic bundles of degree d and rank r and ¬xed determinant is smooth

(projective) manifold. To show this, we need to appeal to another cor-

respondence between holomorphic bundles and unitary representations

of the fundamental group. But in this situation we have to consider

the open Riemann surface S := X \ {x0 } obtained from a complete

Riemann surface X of genus g by removing one point (making a punc-

ture). The fundamental group π1 (S) is a free group in 2g generators

238 CHAPTER 5. FAMILIES AND MODULI SPACES

A1 , ..., Ag , B1 , ..., Bg and admits a natural surjective map onto π1 (X)

with the kernel generated by C = [A1 , A2 , · · · Ag , B1 , B2 , · · · Bg ], where

the generalized commutator [., ., ., ...] was de¬ned in the preceding sec-

tion. As before we take G = SU (r) and let us consider the space of rep-

resentations HomC (π1 (S), G) such that for each ρ ∈ HomC (π1 (S), G)

we have √

2π ’1d

ρ(C) = exp( )Id =: ±.

r

In other words, we require that such a representation ρ will send the

class of a speci¬c loop l encircling the puncture x0 to the central element

± ∈ G.

As before, let us consider the moduli space of such representations

MC := HomC (π1 (S), G)/G, because the group G acts by conjugation

on the set of representations and we do not want to distinguish between

two conjugate representations. We notice that since ± is central, the

conjugation will not a¬ect the condition ρ(C) = ±. We also notice that

when r and d are coprime, the element ± is equal to a primitive root

of unity times the identity of G. The space HomC (π1 (S), G) lies inside

the cartesian product of 2g copies of G and is given by the equation

[X1 , X2 , ..., X2g ] = ±,

where Xi is the coordinate on the i-th multiple in G2g . The group G

acts by conjugation on each factor and the quotient with respect to this

action is the moduli space MC .

LEMMA 5.8.1 The moduli space MC is smooth.

Proof. To begin with, let us notice that the group

P G = G/{center of G}

acts freely on HomC (π1 (S), G). Indeed, let g ∈ G stabilize an element

y ∈ HomC (π1 (S), G), meaning that g commutes with elements

A1 , ..., Ag , B1 , ..., Bg ∈ G

which obey the relation

[A1 , ..., Ag , B1 , ..., Bg ] = C.

5.8. VERLINDE FORMULA 239

Let » be an eigenvalue of g and let W ‚ Cr be the kernel of g ’».Id (we

consider Cr as the standard representation of G). It immediately follows

that all 2g elements A1 , ..., Ag , B1 , ..., Bg should stabilize W too. There-

fore the product of eigenvalues of [A1 , ..., Ag , B1 , ..., Bg ] corresponding

to W is equal to 1. This means that W = Cr and g is a central el-

ement of G. Therefore the group P G acts freely on HomC (π1 (S), G)

with closed compact equidimensional orbits and thus the quotient is a

compact manifold.

It turns out that the compact manifold MC has a natural K¨hler a

structure. Let us identify the tangent space to it in terms of group

cohomology. Let “ Z be the cyclic group generated by the class

C ∈ π1 (S) of the loop l. If we had made no restriction upon the

image ρ(C) then the method explored in the previous section gives us

T[ρ] M = H 1 (π1 (S), g), where M := Hom(π1 (S), G)/G is the moduli

space of those unrestricted representations. As before, g = Lie(G) is a

π1 (S)-module via the adjoint action followed by ρ. For any g ∈ G we

identify the tangent space Tg G to g via the action of the left translation

by g. This allows us to identify the tangent space to the conjugacy class

Cg of g in G as the subspace of g given by the range of Ad(g)’Id : g ’ g.

We leave to the reader to check this simple statement in the full detail.

Let us notice that the space MC lies inside M and, in fact, M is

strati¬ed by the moduli spaces of this kind, when one ¬xes the conju-

gacy class in G - the target for the image ρ(C). In general this conju-

gacy class will be non-central. Therefore, the tangent space T[ρ] MC is

a linear subspace of H 1 (π1 (S), g). In fact, we have

LEMMA 5.8.2

T[ρ] MC = Ker(H 1 (π1 (S), g) ’ H 1 (“, g)).

Proof. The map in parentheses is induced by the natural inclusion

“ ’ π1 (S). Let us show that in our situation, actually, H 1 (“, g) = g.

Indeed, ρ is such that ρ(C) is in the center and thus every 1-boundary

b ∈ B 1 (“, g) vanishes:

b(C) = Ad(ρ(C))x ’ x = x ’ x = 0, x ∈ g.

240 CHAPTER 5. FAMILIES AND MODULI SPACES

Now, let us look at the space of 1-cocycles; let z ∈ Z 1 (π1 (S), g). The

cocycle condition amounts to

z(γ1 γ2 ) = z(γ1 ) + Ad(ρ(γ1 ))z(γ2 ) = z(γ1 ) + z(γ2 )

since the group “ is cyclic and ρ is a representation. Thus we only have

to know the value of z(C), which can be any element of g.

Our discussion amounts to the fact the as a subspace of H 1 (π1 (S), g),

the tangent space T[ρ] MC is given as the kernel of the evaluation map

H 1 (π1 (S), g) ’ g : φ ’ φ(C).

Although this map seems to be de¬ned on cocycles only, it is easy

to check that it vanishes on coboundaries and therefore descends to

cohomology.

The space Ker(H 1 (π1 (S), g) ’ H 1 (“, g)) is called the parabolic co-

homology of X with coe¬cients in the local system ˜ on S associated

g

to the action of π1 (S) on g.

Now we shall wait no further to explain our interest in the space

MC (see e.g. Atiyah-Bott [3]):

FACT. The moduli space M (r, d) of holomorphic bundles of rank r

and degree d and ¬xed determinant on the complete Riemann surface

X identi¬es with the space MC .

Mumford™s theorem 5.5.6 then endows the space MC with the struc-

ture of projective manifold. Let us explicitly show the symplectic struc-

ture on the space MC . This will require the notion of the mapping cone.

We start with the consideration of the restriction map

M ap(π1 (S), g) ’ M ap(“, g)

and we denote by C • (π1 (S), g) the complex of group cochains for the

π1 (S)-module g. Then the relative cochain complex is de¬ned as the

cone of the map of complexes R : C • (π, g) ’ C • (“, g). By de¬nition,

Conek (R) = C k (π, g) • C k’1 (“, g)

g

5.8. VERLINDE FORMULA 241

with the di¬erential (’‚, R + ‚).

Consider the exact sequence

· · · ’ H 1 (Cone• ) ’ H 1 (π1 (S), g) ’ •i H 1 (“i , g) ’ H 2 (Cone• ) ’ 0,

g g

where

Cone• is · · · ’ Conek’1 ’ Conek ’ Conek+1 ’ · · · .

g g g g

Similarly one can de¬ne Cone• , Cone• , etc. If we apply the bilinear

Z R

form B , together with the pairing in cohomology then we get a map

H i (Coneg ) — H j (Coneg ) ’ H i+j (ConeR ).

LEMMA 5.8.3 H 2 (Cone• ) Z.

Z

Proof. Let ∆ be a small disk centered at the point x0 . Then ∆— be

obtained from ∆ by removing the marked point. We have the class

ˇ

γ ∈ H 1 (“, Z) H 1 (∆— , Z), which can be de¬ned as follows. In terms

of group cohomology it corresponds to the map φ : “ ’ Z sending C

to 1.

Our point is that

±

H 2 (Cone• ) = Coker(H 1 (π1 (S), Z) ’ H 1 (“, Z));

Z

the cokernel of the map ± is Z and is generated by the class of the

element

γ ∈ H 1 (∆— , Z) H 1 (“, Z).

This lemma gives us the idea how to construct a symplectic form on