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The Geometry of Quivers^{ †}

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## Abstract

**:**

## 1. A First Look at Quivers

#### 1.1. Quiver Representations

#### 1.2. Examples and Basic Notions

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 1.3. Double Quivers

#### 1.4. Quiver Varieties

## 2. The Electric Side

#### 2.1. Physical Theories and Quivers

- A gauge group, which encodes the fundamental forces (electromagnetism, weak and strong forces). This is a Lie group whose complexified Lie algebra is ${\mathfrak{g}}_{\mathrm{SM}}={\mathfrak{sl}}_{3}\oplus {\mathfrak{sl}}_{2}\oplus {\mathfrak{gl}}_{1}$. The forces are mediated by massless gauge bosons${A}_{\mu}$, which are vector fields valued in the adjoint representation of that algebra.
- Matter constituents, which are fermion fields $\psi $ valued in bifundamental (a bifundamental representation of a semisimple Lie algebra $\mathfrak{g}$ is the product of a fundamental representation of a simple summand of $\mathfrak{g}$ with the antifundamental representation of a summand of $\mathfrak{g}$.) representations of ${\mathfrak{g}}_{\mathrm{SM}}$. This matter content can be encoded in a quiver where the vertices are the simple summands of ${\mathfrak{g}}_{\mathrm{SM}}$ and the arrows are the matter fields.

#### 2.2. Supersymmetric Quiver Gauge Theories

- The gauge bosons are part of vector multiplets, which contain spin $\frac{1}{2}$ gauginos and one complex scalar field $\varphi $ in the adjoint representation of the gauge algebra.
- The matter fermions are part of hypermultiplets, which contain a pair of complex scalar fields $(H,\overline{H})$ that transforms in a representation of the form $R\oplus \overline{R}$ of the gauge algebra.

#### 2.3. Higgs Branches, Quiver Varieties, and Beyond

## 3. The Magnetic Side

#### 3.1. Three-Dimensional Coulomb Branches and Magnetic Quivers

#### 3.2. The Scope of Magnetic Quivers

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Five examples of quivers. (

**a**) Trivial quiver. (

**b**) ${A}_{2}$ quiver. (

**c**) Jordan quiver. (

**e**) is the double of (

**d**).

**Figure 2.**Three graphs with the corresponding dimension vector $\mathbf{v}$, indicated as integers next to each node. Below each graph we write the Higgs and 3d Coulomb branches (see Section 3).

**Figure 3.**A very schematic depiction of a portion of the matter content of the standard model of particle physics (right-handed leptons and the Higgs field are not represented). The nodes correspond to the gauge groups for the fundamental interactions (gauge bosons), and the edges correspond to matter fields (chiral fermions) charged under these forces. SU(3) is the strong force; SU(2) the weak isospin; and U(1) a combination of weak hypercharge and the baryonic charge, which makes the charge of the left-handed quarks vanish.

**Figure 4.**Hasse diagram of nilpotent orbits of $\mathfrak{sl}(4,\mathbb{C})$ obtained via quiver subtraction, with the complex dimension of each leaf. The quivers at each step are shown on the right, while the pieces that are subtracted to move from one variety to its singular subvariety are drawn in red.

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Bourget, A. The Geometry of Quivers. *Phys. Sci. Forum* **2022**, *5*, 42.
https://doi.org/10.3390/psf2022005042

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Bourget A. The Geometry of Quivers. *Physical Sciences Forum*. 2022; 5(1):42.
https://doi.org/10.3390/psf2022005042

**Chicago/Turabian Style**

Bourget, Antoine. 2022. "The Geometry of Quivers" *Physical Sciences Forum* 5, no. 1: 42.
https://doi.org/10.3390/psf2022005042