---
date: 2022-08-06 16:04
modification date: Saturday 6th August 2022 16:04:11
title: "GLSM"
aliases: [GLSM, gauged linear sigma model, Landau-Ginzberg model, sigma model, LGM]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #mirror-symmetry
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# LGMs

![](attachments/2023-03-05.png)

![](attachments/2023-03-05lgm-1.png)

In the case of $\mathbb{P}^n, \check{X}$ is the algebraic torus $\left(\mathbb{C}^*\right)^n$ with coordinates $x_1 \ldots x_n$, and the superpotential is $W=x_1+\ldots+x_n+\frac{q}{x_1 \ldots x_n}$. This means that the quantum cohomology $Q H^*\left(\mathbb{P}^n, \mathbb{C}\right)$ of $\mathbb{P}^n$ is isomorphic to the Jacobi ring of $W$.

# Sigma models

![](attachments/2023-02-08sigma.png)

In [QFT](Unsorted/TQFT.md):

![](attachments/2023-03-11sigma.png)
# GLSM

![](attachments/Pasted%20image%2020220806160412.png)
![](attachments/Pasted%20image%2020220806160453.png)

See <https://arxiv.org/pdf/1412.1268.pdf#page=40&zoom=160,-89,655>.

# Landau Ginzberg

Landau–Ginzburg model, by which we mean a pair of a [stack](Unsorted/stacks%20MOC.md) and a section of a line bundle on it.

![](attachments/Pasted%20image%2020220514193925.png)

![](attachments/Pasted%20image%2020220806160522.png)