I will give an overview of geometric and algebraic techniques used to construct string field theory solutions. Possible applications will be discussed.

I describe four dimensional N=2 supersymmetric gauge theory in the Ω-background with two dimensional N=2 super-Poincare invariance. I explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. This four dimensional gauge theory in its low energy description has a two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. I present the thermodynamic-Bethe-ansatz like formulae for this Yang-Yang function and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. Particular examples of  many-body systems include the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions. Gauge theory gives a complete characterization of the L2-spectrum for these integrable systems.

In three spacetime dimensions there exists an interacting generally covariant and super-renormalizable extension of the Fierz-Pauli theory for a free spin 2 field. This “new massive gravity” theory is a special case of a “general massive gravity” theory that also contains the “topologically massive gravity” theory as a special case. These models and their supergravity extensions will be described, in particular the connection provided by N=2 supersymmetry to models of massive spin 1, and recent work on the maximal N=8 massive supergravity theory.