A GLSM view on Homological Projective Duality

Given a gauged linear sigma model (GLSM) ${\mathcal{T}}_X$ realizing a projective variety $X$ in one of its phases, i.e. its quantum K"ahler moduli has a maximally unipotent point, we propose an \emph{extended} GLSM ${\mathcal{T}}_{\mathcal{X}}$ realizing the homological projective dual category $\mathcal{C}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of one of its phases. In most of the cases, the models ${\mathcal{T}}_X$ and ${\mathcal{T}}_{\mathcal{X}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the semiorthogonal/Lefschetz decompositions of $\mathcal{C}$ and $D^{b}Coh(X)$. We also study the models ${\mathcal{T}}_{X_L}$ and ${\mathcal{T}}_{{\mathcal{X}}_L}$ that correspond to homological projective duality of linear sections $X_L$ of $X$. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear and Veronese embeddings of ${\mathbb{P}}^n$ and Fano complete intersections in ${\mathbb{P}}^n$. In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Pl"ucker embedding of the Grassmannian $G(k,N)$.