Orientifolds and Mirror Symmetry

We study parity symmetries and crosscap states in classes of N=2 supersymmetric quantum field theories in 1+1 dimensions, including non-linear sigma models, gauged WZW models, Landau-Ginzburg models, and linear sigma models. The parity anomaly and its cancellation play important roles in many of them. The case of the N=2 minimal model are studied in complete detail, from all three realizations -- gauged WZW model, abstract RCFT, and LG models. We also identify mirror pairs of orientifolds, extending the correspondence between symplectic geometry and algebraic geometry by including unorientable worldsheets. Through the analysis in various models and comparison in the overlapping regimes, we obtain a global picture of orientifolds and D-branes.