Exact Partition Functions on RP2 and Orientifolds

Understand

We consider gauged linear sigma models (GLSM) on $\mathbb{RP}^2$, obtained from a parity projection of $S^2$.

• The theories admit squashing deformation, much like GLSM on $S^2$, which allows us to interpret the partition function as the overlap amplitude between the vacuum state and crosscap states.
• From these, we extract the central charge of Orientifold planes, and observe that the Gamma class makes a prominent appearance as in the recent D-brane counterpart.
• We also repeat the computation for the mirror Landau-Ginzburg theory, which naturally brings out the $\theta$-dependence as a relative sign between two holonomy sectors on $\mathbb{RP}^2$.