Stringy Hodge numbers of Pfaffian double mirrors and Homological Projective Duality

We study the stringy Hodge numbers of Pfaffian double mirrors, generalizing previous results of Borisov and Libgober. In the even-dimensional cases, we introduce a modified version of stringy $E$-functions and obtain interesting relations between the modified stringy $E$-functions on the two sides. We use them to make numerical predictions on the Lefschetz decompositions of the categorical crepant resolutions of Pfaffian varieties.