## Stacky GKM Graphs and Orbifold Gromov-Witten Theory

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### Standard

Stacky GKM Graphs and Orbifold Gromov-Witten Theory. / Liu, Chiu-Chu Melissa; Sheshmani, Artan.

I: arXiv preprint, 16.07.2018.

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avisTidsskriftartikelForskning

### Author

Liu, Chiu-Chu Melissa ; Sheshmani, Artan. / Stacky GKM Graphs and Orbifold Gromov-Witten Theory. I: arXiv preprint. 2018.

### Bibtex

@article{375faa34d4ec477189b60c9af93a7602,
title = "Stacky GKM Graphs and Orbifold Gromov-Witten Theory",
abstract = " Following Zong (arXiv:1604.07270), we define an algebraic GKM orbifold $\mathcal{X}$ to be a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. The 1-skeleton of $\mathcal{X}$ is the union of its zero-dimensional and one-dimensional $T$-orbits; its formal neighborhood $\hat{\mathcal{X}}$ in $\mathcal{X}$ determines a decorated graph, called the stacky GKM graph of $\mathcal{X}$. The $T$-equivariant orbifold Gromov-Witten (GW) invariants of $\mathcal{X}$ can be computed by localization and depend only on the stacky GKM graph of $\mathcal{X}$ with the $T$-action. We also introduce abstract stacky GKM graphs and define their formal equivariant orbifold GW invariants. Formal equivariant orbifold GW invariants of the stacky GKM graph of an algebraic GKM orbifold $\mathcal{X}$ are refinements of $T$-equivariant orbifold GW invariants of $\mathcal{X}$. ",
keywords = "math.AG",
author = "Liu, {Chiu-Chu Melissa} and Artan Sheshmani",
note = "40 pages; generalization of arXiv:1407.1370; Section 6 extends Section 9 of arXiv:1107.4712",
year = "2018",
month = jul,
day = "16",
language = "English",
journal = "arXiv preprint",

}

### RIS

TY - JOUR

T1 - Stacky GKM Graphs and Orbifold Gromov-Witten Theory

AU - Liu, Chiu-Chu Melissa

AU - Sheshmani, Artan

N1 - 40 pages; generalization of arXiv:1407.1370; Section 6 extends Section 9 of arXiv:1107.4712

PY - 2018/7/16

Y1 - 2018/7/16

N2 - Following Zong (arXiv:1604.07270), we define an algebraic GKM orbifold $\mathcal{X}$ to be a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. The 1-skeleton of $\mathcal{X}$ is the union of its zero-dimensional and one-dimensional $T$-orbits; its formal neighborhood $\hat{\mathcal{X}}$ in $\mathcal{X}$ determines a decorated graph, called the stacky GKM graph of $\mathcal{X}$. The $T$-equivariant orbifold Gromov-Witten (GW) invariants of $\mathcal{X}$ can be computed by localization and depend only on the stacky GKM graph of $\mathcal{X}$ with the $T$-action. We also introduce abstract stacky GKM graphs and define their formal equivariant orbifold GW invariants. Formal equivariant orbifold GW invariants of the stacky GKM graph of an algebraic GKM orbifold $\mathcal{X}$ are refinements of $T$-equivariant orbifold GW invariants of $\mathcal{X}$.

AB - Following Zong (arXiv:1604.07270), we define an algebraic GKM orbifold $\mathcal{X}$ to be a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. The 1-skeleton of $\mathcal{X}$ is the union of its zero-dimensional and one-dimensional $T$-orbits; its formal neighborhood $\hat{\mathcal{X}}$ in $\mathcal{X}$ determines a decorated graph, called the stacky GKM graph of $\mathcal{X}$. The $T$-equivariant orbifold Gromov-Witten (GW) invariants of $\mathcal{X}$ can be computed by localization and depend only on the stacky GKM graph of $\mathcal{X}$ with the $T$-action. We also introduce abstract stacky GKM graphs and define their formal equivariant orbifold GW invariants. Formal equivariant orbifold GW invariants of the stacky GKM graph of an algebraic GKM orbifold $\mathcal{X}$ are refinements of $T$-equivariant orbifold GW invariants of $\mathcal{X}$.

KW - math.AG

M3 - Journal article

JO - arXiv preprint

JF - arXiv preprint

ER -