2
$\begingroup$

Recall that for a given Riemann surface $\Sigma$ Hitchin's self-duality equation consists of a complex rank $r$ vector bundle $E$ (with degree 0 for simplicity), a connection $d_A: \Omega^k(\Sigma, E) \rightarrow \Omega^{k+1}(\Sigma, E)$ along with a Higgs field $\Phi \in \Omega^{(1,0)}(\Sigma, {\rm End}(E))$. The equation itself reads: \begin{equation} \begin{aligned} & F_A + \left[\Phi, \Phi^*\right] = 0\\ & {\bar \partial}_A \Phi = 0 \end{aligned} \end{equation}

There are two ways of viewing this equation:

  • Fix a Hermitian metric $h$ on $E$ and think of the equation as the one for a pair $(A, \Phi)$ modulo complex gauge group $G_{\mathbb{C}}$ transformation; In this way the $(0,1)$ part of the connection under the complex structure $J$ of the Riemann surface gives a holomorphic structure of $E$: ${\bar\partial}_E = {\bar \partial}_A$.
  • Fix the holomorphic structure ${\bar \partial}_E$, then the Hitchin's equation can be viewed as an equation for the Hermitian metric $h$ and Higgs field $\Phi$. Then there is a unique Chern connection, which is unitary and compatible with the holomorphic structure.

Question: what is the transformation between these two conventions explicitly? More specifically, is there any canonical mapping between gauge orbits of solutions in two conventions?

For instance, in the first point of view we take $h$ to be identity matrix (sometimes called unitary gauge), then we have the moduli space of solutions; in the second point of view we have solutions $(h, \Phi)$ where $h$ is in general different from identity. How does one go from one convention to the other by some explicit transformation?

$\endgroup$
1
$\begingroup$

Let me start with the second point of view: you start with a holomorphic structure $\bar\partial^E$ and a (holomorphic) Higgs field $\Phi.$ If this pair is stable (e.g., as defined in Hitchin's original reference), then there exist a unitary metric $h$ on the bundle such that the Chern connection $\nabla$ of $\bar\partial^E$ w.r.t. $h$, $h$ and $\Phi$ satisfy the self-duality equations. If $h_0$ is another fixed unitary metric on your bundle, there is a (compex) gauge transformation $g$ such that $$g^*h=h_0.$$ Then, $g^*\nabla$ is the Chern connection for $g^*\bar\partial^E$ w.r.t. $h_0$ and $$(g^*\nabla,h_0, g^{-1}\circ\Phi\circ g)$$ satisfy the Hitchin equations.

The converse direction goes along the same lines, i.e., you get a solution of the Hitchin equation in the complex gauge orbit of your initial Higgs pair. You can use this gauge transformation, which is unique up to unitary gauges, to get a new unitary metric which yields a solution of the Hitchin equations for you initial Higgs pair.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.