There is evidence that one can compute tree-level super Yang-Mills amplitudes using either connected or completely disconnected curves in twistor space. We give a partial explanation of the equivalence between the two computations, by showing that they could both be reduced to the same integral over a moduli space of singular curves, subject to some assumptions about the choices of integration contours. We also formulate a class of new “intermediate” prescriptions to calculate the same amplitudes.

We define topological Landau-Ginzburg models on a world-sheet foam, that is, on a collection of 2-dimensional surfaces whose boundaries are sewn together along the edges of a graph. We use the matrix factorizations in order to formulate the boundary conditions at these edges and then produce a formula for the correlators. Finally, we present the gluing formulas, which correspond to various ways in which the pieces of a world-sheet foam can be joined together.

We study the topological sector of $N$ = 2 sigma-models with $H$-flux. It has been known for a long time that the target-space geometry of these theories is not Kähler and can be described in terms of a pair of complex structures, which do not commute, in general, and are parallel with respect to two different connections with torsion. Recently an alternative description of this geometry was found, which involves a pair of commuting twisted generalized complex structures on the target space. In this paper, we define and study the analogs of A and B-models for $N$ = 2 sigma-models with $H$-flux and show that the results are naturally expressed in the language of twisted generalized complex geometry. For example, the space of topological observables is given by the cohomology of a Lie algebroid associated to one of the two twisted generalized complex structures. We determine the topological scalar product, which endows the algebra of observables with the structure of a Frobenius algebra. We also discuss mirror symmetry for twisted generalized Calabi-Yau manifolds.