We study the spaces of twisted conformal blocks attached to a $\Gamma$-curve $\Sigma$ with marked $\Gamma$-orbits and an action of $\Gamma$ on a simple Lie algebra $\mathfrak {g}$, where $\Gamma$ is a finite group. We prove that if $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak {g}$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $\Gamma$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let $\mathscr {G}$ be the parahoric Bruhat–Tits group scheme on the quotient curve $\Sigma /\Gamma$ obtained via the $\Gamma$-invariance of Weil restriction associated to $\Sigma$ and the simply connected simple algebraic group $G$ with Lie algebra $\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic $\mathscr {G}$-torsors on $\Sigma /\Gamma$ when the level $c$ is divisible by $|\Gamma |$ (establishing a conjecture due to Pappas and Rapoport).

We discuss the Gromov–Witten/Donaldson–Thomas correspondence for 3-folds in both the absolute and relative cases. Descendents in Gromov–Witten theory are conjectured to be equivalent to Chern characters of the universal sheaf in Donaldson–Thomas theory. Relative constraints in Gromov–Witten theory are conjectured to correspond in Donaldson–Thomas theory to cohomology classes of the Hilbert scheme of points of the relative divisor. Independent of the conjectural framework, we prove degree 0 formulas for the absolute and relative Donaldson–Thomas theories of toric varieties.

We conjecture an equivalence between the Gromov–Witten theory of 3-folds and the holomorphic Chern–Simons theory of Donaldson and Thomas. For Calabi–Yau 3-folds, the equivalence is defined by the change of variables $e^{iu}=-q$, where $u$ is the genus parameter of Gromov–Witten theory and $q$ is the Euler characteristic parameter of Donaldson–Thomas theory. The conjecture is proven for local Calabi–Yau toric surfaces.

The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah and Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Künneth components of the Chern classes of the universal bundle.

This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle $K$. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes.

The spaces of Higgs bundles with values in $K(n)$ for $n > 0$ turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes.