Anomaly polynomial of Hitchin system $\mathcal{N}=2$ 4d SQFT

Question

I would like to ask about mathematical background of this object. So, I am trying to puzzle out with 4d $\mathcal{N}=2$ SQFT. As far as I can gather this theory can be described in terms of generalized Hitchin integrable system. Thus we live in the module space of Higgs pairs $\mathcal{M}$ and point of this space (in fact algebraic variety under some assumptions) is a class of pair $[(E,\varphi)]$ where $E$ is a fibered bundle associated to principal $G$-bundle on compact smooth curve $\Sigma$, $\varphi$ is a global section of $\mathrm{End}(E)\otimes K(D)$, i.e. global meromorphic form with poles along $D:= p_1+...p_k$, $(p_i \ne p_j$, if $ i \ne j)$. We also can imply further contributions on $\varphi$ such as $\mathrm{Res}_{p_i}\varphi = A_i$ where $A_i \in \mathcal{O}_i$, $\mathcal{O}_i \subset \mathfrak{g}:=Lie(G)$ coadjoint orbit of $G$. Whether or no we have Hitchin map $$h:\mathcal{M} \to \mathcal{B}:= \oplus_i H^{0}(\Sigma, K(D)^{d_i}), \qquad h(E,\varphi) = char(\varphi).$$

1. Could you possibly explain (or give reference) as aforesaid what anomaly polynomial is?

2. And what is the relation between polynomial and prepotential (if it exists).

This post imported from StackExchange Physics at 2015-08-08 15:43 (UTC), posted by SE-user quantum