Field Theory Methods in Two-Dimensional and Heterotic String Theories

This thesis has three parts. In the first, we study the Das-Jevicki collective field description of arbitrary classical solutions in the c = 1 matrix model, which are believed to describe nontrivial spacetime backgrounds in 2D string theory. Our analysis naturally includes the case of a Fermi droplet cosmology. We cast the droplet collective field theory in standard coordinates and comment on the form of the interactions. In the second part, we prove the existence of topological rings in (0, 2) theories containing non-anomalous left-moving U(1) currents by which they may be twisted. While the twisted models are not topological, their ground operators form a ring under non-singular OPE which reduces to the (a,c) or (c,c) ring at (2, 2) points and define a quantum sheaf cohomology. In the special case of Calabi-Yau compactifications, these rings are shown to exist globally on the moduli space in many cases. In the third part, we construct worldsheet descriptions of heterotic flux vacua as the IR limits of N = 2 gauge theories. Spacetime torsion is incorporated via a 2D Green-Schwarz mechanism in which a doublet of axions cancels a one-loop gauge anomaly. Manifest (0, 2) supersymmetry and the compactness of the gauge theory instanton moduli space suggest that these models, which include Fu-Yau models, are stable against worldsheet instantons.