Yang-Mills equations over Riemann surfaces

In algebraic geometry it is well known that for any degree $d$ the set of isomorphism classes of holomorphic line bundles of degree $d$ over a Riemann surface $M$ of genus $g$ forms a smooth projective variety which is topologically a torus of dimension $g$ . This space is called the moduli space of holomorphic line bundles of degree $d$ over $M$ .

For holomorphic vector bundles of rank $k \ge 2$ , the situation is far more complicated. First, in order to have an algebraic structure on the moduli space, it is necessary to discard the so-called ``unstable'' bundles in the sense of Mumford. It is then known that for $k$ and $d$ relatively prime, the isomorphism classes of the remaining bundles, called ``semistable bundles,'' form a smooth projective variety $N(k,d)$ .

In [N] Newstead computed the Poincaré polynomial of $N(2,1)$ . Apart from this, the topology of $N(k,d)$ remained mysterious.

In [81] Atiyah and Bott introduced the new and powerful method of equivariant Morse theory to study the topology of these moduli spaces.

Let $P= M \times U(n)$ be the trivial principal $U(n)$ -bundle over the Riemann surface $M$ , $\mathcal{A}=\mathcal{A}(P)$ the affine space of connections on $P$ , and $\mathcal{G}=\mathcal{G}(P)$ the gauge group, i.e., the group of automorphisms of $P$ that cover the identity. Then the gauge group $\mathcal{G}(P)$ acts on the space $\mathcal{A}(P)$ of connections and there is a Yang-Mills functional $L$ on $\mathcal{A}(P)$ invariant under the action of the gauge group.

Equivariant Morse theory harks back to Bott's extension of classical Morse theory to nondegenerate critical manifolds three decades earlier. The key result of Atiyah and Bott is that the Yang-Mills functional $L$ is a perfect equivariant Morse function on $\mathcal{A}(P)$ . This means the equivariant Poincaré series of $\mathcal{A}(P)$ is equal to the equivariant Morse series of $L$ :

$$P_t^{\mathcal{G}} (\mathcal{A}(P)) = \mathcal{M}_t^{\mathcal{G}}(L).$$ (5)

Once one unravels the definition, the left-hand side of (5) is simply the Poincaré series of the classifying space of $\mathcal{G}(P)$ , which is computable from homotopy considerations. The right-hand side of (5) is the sum of contributions from all the critical sets of $L$ . By the work of Narasimhan and Seshadri, the minimum of $L$ is precisely the moduli space $N(k,d)$ . It contributes its Poincaré polynomial to the equivariant Morse series of $L$ . By an inductive procedure, Atiyah and Bott were able to compute the contributions of all the higher critical sets. They then solved (5) for the Poincaré polynomial of $N(k,d)$ .