Orientifolds and Mirror Symmetry

TL;DR

Brunner and Hori provide a comprehensive framework for parity symmetries, crosscaps, and orientifolds across ${\cal N}=2$ theories in 1+1 dimensions, unifying nonlinear sigma models, gauged WZW models, LG models, and linear sigma models. They classify A- and B-parities, derive their action on fields, and confront parity anomalies with cancellation mechanisms such as $(-1)^F_R$ and $B$-fields, yielding consistent crosscap constructions and Klein bottle amplitudes. The authors conduct a thorough, multi-realization study of ${\cal N}=2$ minimal models (via RCFT, gauged WZW, and LG), obtaining exact crosscap data and parity actions on D-branes, then extend to LG orientifolds and their twisted Witten indices, matching results across UV/IR phases. They further develop mirrors between linear sigma models and LG orientifolds, deriving how parity data and orientifold types translate under mirror symmetry and illustrating with concrete examples including compact Calabi–Yau scenarios. The work provides a global picture linking orientifolds, D-branes, and mirror symmetry across diverse realizations, offering a robust toolbox for constructing and analyzing supersymmetric orientifolds in Calabi–Yau and related geometries.

Abstract

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.

Figures (11)

• Figure 1: Crosscap $\langle\Sigma|\Scr{C}_P\rangle$
• Figure 2: Crosscap $\langle\Scr{C}_P|\Sigma \rangle$
• Figure 3: The semi-infinite ${{\mathbb R}{\mathbb P}}^2$
• Figure 4: The Bilinear Identities (\ref{['biIP']}), (\ref{['biIPa']}) and (\ref{['biIab']}).
• Figure 5: Applying parity to the cylinder