A new duality for categories of B-branes

Given an Artin stack $X$, there is growing evidence that there should be an associated `category of B-branes', which is some subcategory of the derived category of coherent sheaves on $X$. The simplest case is when $X$ is just a vector space modulo a linear action of a reductive group, or `gauged linear sigma model' in physicists' terminology. In this case we know some examples of what the category B-branes should be. Hori has conjectured a physical duality between certain families of GLSMs, which would imply that their B-brane categories are equivalent. We prove this equivalence of categories. As an application, we construct Homological Projective Duality for (non-commutative resolutions of) Pfaffian varieties.