A GLSM view on homological projective duality

Zhuo Chen YMSC, Tsinghua University Jirui Guo YMSC, Tsinghua University Mauricio Romo YMSC, Tsinghua University

Category Theory Mathematical Physics Algebraic Geometry mathscidoc:2205.04001

Communications in Mathematical Physics, 2022.5
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 geometric 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 $G(k,N)$.
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@inproceedings{zhuo2022a,
  title={A GLSM view on homological projective duality},
  author={Zhuo Chen, Jirui Guo, and Mauricio Romo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519154918793954281},
  booktitle={Communications in Mathematical Physics},
  year={2022},
}
Zhuo Chen, Jirui Guo, and Mauricio Romo. A GLSM view on homological projective duality. 2022. In Communications in Mathematical Physics. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519154918793954281.
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