论文标题
关于同源投影二重性的GLSM观点
A GLSM view on Homological Projective Duality
论文作者
论文摘要
给定计量的线性Sigma模型（GLSM）$ \ MATHCAL {t} _ {X} $在其一个阶段中实现一个投射品种$ x $，即其QuantumKählerOduli具有最大单位的点$ \ MATHCAL {t} _ {\ MATHCAL {X}} $实现同源射击双类别$ \ Mathcal {C} $ to $ d^{b} coh（x）$作为其阶段的Higgs分支的B-Branes类别。在大多数情况下，模型$ \ Mathcal {t} _ {x} $和$ \ Mathcal {t} _ {\ Mathcal {X {X}} $是异常的，并且分析了他们的Coulomb和Coolomb-Higgs分支，并在半lefschetz decompoots $ compoots $ compoots $ compoottion $ compoots} $ d^{b} coh（x）$。我们还研究了模型$ \ Mathcal {t} _ {x_ {l}} $和$ \ Mathcal {t} _ {\ Mathcal {x} _ {l}} $，与$ x_ {l} $ x $ x $的线性截面$ x_ {l} $的同源射击双重性相对应。这解释了为什么在许多情况下，GLSM的两个阶段与同源二元性有关。我们主要研究Abelian示例：$ \ Mathbb {p}^{n} $的线性和Veronese嵌入$，而Fano在$ \ Mathbb {p}^{n} $中完成了交集。在这种情况下，我们能够复制已知结果并产生一些新的猜想。此外，我们对HPD的构建给Nonabelian GLSM，以嵌入Grassmannian $ g（k，n）$的plücker嵌入。
Given a gauged linear sigma model (GLSM) $\mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum Kähler 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 Plücker embedding of the Grassmannian $G(k,N)$.
