B-brane Transport and Grade Restriction Rule for Determinantal Varieties

Abstract

We study autoequivalences of $D^{b}Coh(X)$ associated to B-brane transport around loops in the stringy K\"ahler moduli of $X$. We consider the case of $X$ being certain resolutions of determinantal varieties embedded in $\mathbb{P}^{d}\times G(k,n)$. Such resolutions have been modeled, in general, by nonabelian gauged linear sigma models (GLSM). We use the GLSM construction to determine the window categories associated with B-brane transport between different geometric phases using the machinery of grade restriction rule and the hemisphere partition function. In the family of examples analyzed the monodromies around phase boundaries enjoy the interpretation as loop inside link complements. We exploit this interpretation to find a decomposition of autoequivalences into simpler spherical functors and we illustrate this in two examples of Calabi-Yau 3-folds $X$, modeled by an abelian and nonabelian GLSM respectively. In additon we also determine explicitly the action of the autoequivalences on the Grothendieck group $K(X)$ (or equivalently, B-brane charges).