moduli space of flat connections $SU(N)$ on an elliptic curve is $\mathbb{C}P^{N-1}$

Question

A physics paper [1] states the moduli space of flat $SU(N)$ connections on an elliptic curve $E$ is the projective space $\mathbb{C}P^{N-1}$. However, I need some clarifications:

• the paper vaguely states some sort of "holomorphic" correspondence between flat $SU(N)$ connections on $E$ and $\mathbb{C}P^{N-1}$
• the paper has a disclaimer that the equivalence is only at the level of algebraic varieties, that that metric on the space of connections is not the Fubini-Study metric
• the paper gives an explicit map between $\mathcal{M}_{flat}$ and $\mathbb{C}P^{N-1}$ using theta functions

I am having trouble picturing these flat $SU(N)$ structures. Can they be parameterized by Abelian differentials? or mapped into a Euclidean space?

Answer 1

They can almost be parameterized by Abelian differentials: Take a flat $SU(n)$-connection and its monodromy along two generators of the first fundamental group of the torus. These are two commuting $SU(n)$ matrices and therefore they diagonalize simultaneously. Their common eigenlines define parallel line subbundles (take the parallel transport of those eigenlines), hence every flat $SU(n)$-connection is gauge equivalent to the direct sum of flat unitary line bundle connections on that torus. (Their product is the trivial connection on the determinant bundle.) Every unitary flat line bundle connection on the torus $\mathbb C/\Gamma$ is gauge equivalent to some \begin{equation} (1)\;\;\;\;\;\nabla=d+\alpha dz-\bar\alpha d\bar z\end{equation} for some $\alpha.$ Hence, your flat $SU(n)$ connection is determinend by $n$ abelian differentials $$\alpha_1 dz,..,\alpha_n dz$$ satisfying the additional determinant condition: $$\sum \alpha_i=0.$$

As the parallel line bundles have no natural ordering, you should factor out the Weyl group. Additionally, there is a lattice $\tilde\Gamma\subset\mathbb C$ with the property that two connections of the form (1) for $\alpha_1,\alpha_2\in\mathbb C$ are gauge equivalent if and only if their difference is in $\tilde\Gamma.$

It should be noted that most people (as the authors of [1]) seem to prefer to parametrize the flat unitary line bundles in terms of the underlying holomorphic structure, hence by the jacobian of the torus.

Answer 2

Sebastian has described an isomorphism between this moduli space and the space of unordered $n$-tuples of elements on a certain two-dimensional torus ($\mathbb C/ \tilde{\Gamma}$) that sum to $0$.

For any two-dimensional torus, the space of unordered $n$-tuples that sum to $0$ is isomorphic to $\mathbb C P^{n-1}$. One can see this by viewing the torus as an elliptic curve $E$ (which in fact you can take to be canonically isomorphic to your original $E$) so unordered $n$-tuples that sum to $0$ are the same as degree $n$ divisors in the divisor class $n[0]$, and hence are the same as nonzero sections in $H^0(E, \mathcal O(n[0]))$ up to scaling. Because $H^0(E, \mathcal O(n[0]))$ is an $n$-dimensional complex vector space, nonzero sections up to scaling are an $n-1$-dimensional complex projective space.

In the case $n=2$, the isomorphism can be written $E / (\pm 1) = \mathbb C P^1$ and is described in the picture you give. However, I prefer a different picture where you take a torus and skewer it on a line that punctures it twice, passing through the center, and quotient by a rotation by angle $\pi$ through the line. A fundamental domain for this action is a half-torus, which is homeomorphic to a cylinder, and you can see that hte quotient closes each hole of the cylinder, producing a sphere.