Given a gauged linear sigma model (GLSM) $mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum Kahler 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 Plucker embedding of the Grassmannian $G(k,N)$.
تحميل البحث