We discuss a simple example of an F-theory compactification on a Calabi--Yau fourfold where background fluxes, non-perturbative effects from Euclidean D3-instantons, and gauge dynamics on D7-branes allow us to fix all closed and open string moduli. We explicitly check that the known higher-order corrections to the potential, which we neglect in our leading approximation, only shift the results by a small amount. In our exploration of the model, we encounter interesting new phenomena, including examples of transitions where D7-branes absorb O3-planes, while changing topology to preserve the net D3 charge.

It was recently shown by Witten that B-type open topological string theory with the supertwistor space $\mathbb{C}P^{3|4}$ as a target space is equivalent to holomorphic Chern--Simons (hCS) theory on the same space. This hCS theory in turn is equivalent to self-dual $\mathcal{N}=4$ super-Yang–Mills (SYM) theory in four dimensions. We review the supertwistor description of self-dual and anti-self-dual $\mathcal{N}$-extended SYM theory as the integrability of SYM fields on complex $(2|\mathcal{N})$-dimensional superplanes and demonstrate the equivalence of this description to Witten's formulation. The equivalence of the field equations of hCS theory on an open subset of $\mathbb{C}P^{3|\mathcal{N}}$ to the field equations of self-dual $\mathcal{N}$-extended SYM theory in four dimensions is made explicit. Furthermore, we extend the picture to the full $\mathcal{N}=4$ SYM theory and, by using the known supertwistor description of this case, we show that the corresponding constraint equations are (gauge) equivalent to the field equations of hCS theory on a quadric in $\mathbb{C}P^{3|3}\times \mathbb{C}P^{3|3}$.

Using only the Fukaya category and the monodromy around large complex structure, we reconstruct the mirror map in the case of a symplectic torus. This realizes an idea described by Paul Seidel.

We describe a topological field theory that studies the moduli space of solutions of the symplectic vortex equations. It contains as special cases the topological sigma-model and topological Yang-Mills over Kähler surfaces. The correlation functions of the theory are closely related to the recently introduced Hamiltonian Gromov--Witten invariants.