Exact partition functions on ℝℙ² and orientifolds

We consider gauged linear sigma models (GLSM) on ℝℙ², obtained from a parity projection of S². The theories admit squashing deformation, much like GLSM on S², 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 θ-dependence as a relative sign between two holonomy sectors on ℝℙ². We also show how our results are consistent with known topological properties of D-brane and Orientifold plane world-volumes, and discuss what part of the wrapped D-brane/Orientifold central charge should be attributed to the quantum volumes