# Gauging Fractons and Linearized Gravity

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## Abstract

**:**

## 1. Introduction

- Is the vector gauge condition a good gauge fixing, or, equivalently, do the propagators exist in the pure fractonic limit, possibly without being forced to choose a particular gauge?
- In gauge field theory the gauge fixing condition serves to eliminate the redundant degrees of freedom which render infinite the Green functions’ generating functional $Z\left[J\right]$$$Z\left[J\right]=\int \mathcal{D}{h}_{\mu \nu}{e}^{i{S}_{inv}+\int {J}^{\mu \nu}{h}_{\mu \nu}}.$$

## 2. The Model

## 3. Propagators

- ${g}_{1}={g}_{2}$In this case, after a field redefinition, the invariant action (8) is$${\left.{S}_{inv}({g}_{1},{g}_{2})\right|}_{{g}_{1}={g}_{2}}=\int {d}^{4}p(\tilde{h}{p}^{2}\tilde{h}-2\tilde{h}{p}_{\mu}{p}_{\nu}{\tilde{h}}^{\mu \nu}+{\tilde{h}}^{\mu \nu}{p}^{\rho}{p}_{\mu}{\tilde{h}}_{\nu \rho}).$$$${\tilde{H}}^{\rho}\equiv {p}^{\rho}\tilde{h}-{p}_{\nu}{\tilde{h}}^{\rho \nu},$$$${\left.{S}_{inv}({g}_{1},{g}_{2})\right|}_{{g}_{1}={g}_{2}}=-\int {d}^{4}p{\tilde{H}}^{\rho}{\tilde{H}}_{\rho},$$
- $2{g}_{1}+{g}_{2}=0$In this case the invariant action (8), after a field redefinition, reads$${\left.{S}_{inv}({g}_{1},{g}_{2})\right|}_{2{g}_{1}+{g}_{2}=0}=\int {d}^{4}p\left(\tilde{h}{p}^{2}\tilde{h}-3{\tilde{h}}_{\mu \nu}{p}^{2}{\tilde{h}}^{\mu \nu}-2\tilde{h}{p}_{\mu}{p}_{\nu}{\tilde{h}}^{\mu \nu}+4{\tilde{h}}^{\mu \nu}{p}^{\rho}{p}_{\mu}{\tilde{h}}_{\nu \rho}\right).$$$${\overline{h}}_{\mu \nu}\left(p\right)\equiv {\tilde{h}}_{\mu \nu}\left(p\right)-\frac{1}{4}{\eta}_{\mu \nu}\tilde{h}\left(p\right),$$$$\overline{h}\left(p\right)=0,$$$${\left.{S}_{inv}({g}_{1},{g}_{2})\right|}_{2{g}_{1}+{g}_{2}=0}=\int {d}^{4}p\left(-3{\overline{h}}_{\mu \nu}{p}^{2}{\overline{h}}^{\mu \nu}+4{\overline{h}}^{\mu \nu}{p}^{\rho}{p}_{\mu}{\overline{h}}_{\nu \rho}\right).$$$${S}_{gf}\left(\xi \right)=\int {d}^{4}p\left(-i{\tilde{b}}_{\mu}{p}_{\nu}{\overline{h}}^{\mu \nu}+\frac{\xi}{2}{\tilde{b}}_{\mu}{\tilde{b}}^{\mu}\right).$$$$\begin{array}{cc}\widehat{t}=-\frac{1}{3{p}^{2}}\hfill & \phantom{\rule{50.pt}{0ex}}\widehat{u}=\frac{2\xi}{(2\xi -1){p}^{2}}\hfill \end{array}$$$$\begin{array}{cc}\widehat{v}=-\frac{1}{3{p}^{2}}\hfill & \phantom{\rule{50.pt}{0ex}}\widehat{z}=\frac{-4\xi}{(4\xi +1){p}^{2}}\hfill \end{array}$$$$\begin{array}{cc}\phantom{\rule{-10.pt}{0ex}}\widehat{w}=0\hfill & \phantom{\rule{60.pt}{0ex}}\widehat{f}=\frac{2}{(4\xi +1){p}^{2}}\hfill \end{array}$$$$\begin{array}{cc}\phantom{\rule{-18.pt}{0ex}}\widehat{g}=0\hfill & \phantom{\rule{60.pt}{0ex}}\widehat{l}=\frac{-2}{(2\xi -1){p}^{2}}\hfill \end{array}$$$$\begin{array}{cc}\phantom{\rule{-18.pt}{0ex}}\widehat{r}=\frac{8}{(4\xi +1)}\hfill & \phantom{\rule{43.pt}{0ex}}\widehat{s}=\frac{4}{(2\xi -1)}.\hfill \end{array}$$
- $2\xi {g}_{2}-1=0$We first notice that this singularity is not present in the pure LG case ${g}_{2}=0$, as it is readily seen from the coefficients of the propagators (36), (38) and (41). Then, we remark that ${g}_{2}$ is a physical parameter, hence, cannot depend on $\xi $, which is a gauge, unphysical parameter. This means that $2\xi {g}_{2}-1=0$ should be interpreted as a singularity in $\xi $ as a function of ${g}_{2}$, and not $viceversa$. In other words, we shall not exclude values of ${g}_{2}$ in order to admit a particular gauge, but, rather, the singularity must be read as a condition on the gauge fixing parameter $\xi $:$$\xi \ne \frac{1}{2{g}_{2}}.$$$$\begin{array}{c}{\left.S({g}_{1},{g}_{2};\xi ,\kappa )\right|}_{2\xi {g}_{2}-1=0}=\int {d}^{4}p\left[({g}_{1}-{g}_{2}{\kappa}^{2})\tilde{h}{p}^{2}\tilde{h}-({g}_{1}-{g}_{2}){\tilde{h}}_{\mu \nu}{p}^{2}{\tilde{h}}^{\mu \nu}+\right.\hfill \\ \phantom{\rule{115.pt}{0ex}}\left.-2({g}_{1}+{g}_{2}\kappa )\tilde{h}{p}_{\mu}{p}_{\nu}\tilde{{h}^{\mu \nu}}+2({g}_{1}-{g}_{2}){\tilde{h}}^{\mu \nu}{p}_{\mu}{p}^{\rho}{\tilde{h}}_{\nu \rho}\right].\end{array}$$$${\left.S({g}_{1},{g}_{2};\xi ,\kappa )\right|}_{2\xi {g}_{2}-1=0,\kappa +1=0}=({g}_{1}-{g}_{2}){S}_{LG}.$$

## 4. Degrees of Freedom

#### 4.1. Case $2{g}_{1}+{g}_{2}\ne 0$

#### 4.2. Case $2{g}_{1}+{g}_{2}=0$

## 5. Summary and Discussion

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Basis for the Ω-Tensors

- decomposition of the rank-4 tensor identity ${\mathcal{I}}_{\mu \nu ,\alpha \beta}$:

- idempotency:

- orthogonality of A, B, C and D:

- contractions with E:

## Appendix B. Calculation of the Propagators

#### Appendix B.1. 2g_{1} + g_{2} ≠ 0

#### Appendix B.2. 2g_{1} + g_{2} = 0

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${\mathit{g}}_{1}$, ${\mathit{g}}_{2}$ | Vectorial Gauge Fixing | Scalar Gauge Fixing | ||
---|---|---|---|---|

Degrees of Freedom | Forbidden Gauges | Degrees of Freedom | Forbidden Gauges | |

${g}_{1}\ne {g}_{2}\ne 0$, $2{g}_{1}+{g}_{2}\ne 0$ | 6 | $\xi =\frac{1}{2{g}_{2}}$, $\kappa =-1$ | 6 | $\xi \ne 0$ |

${g}_{2}=0$ (LG) | 6 | $\kappa =-1$ | not defined | |

$2{g}_{1}+{g}_{2}=0$ | 5 | $\xi =\left\{\frac{1}{2},-\frac{1}{4}\right\}$ | 5 | $\xi \ne 0$ |

${g}_{1}={g}_{2}$ | trivial |

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**MDPI and ACS Style**

Bertolini, E.; Blasi, A.; Damonte, A.; Maggiore, N.
Gauging Fractons and Linearized Gravity. *Symmetry* **2023**, *15*, 945.
https://doi.org/10.3390/sym15040945

**AMA Style**

Bertolini E, Blasi A, Damonte A, Maggiore N.
Gauging Fractons and Linearized Gravity. *Symmetry*. 2023; 15(4):945.
https://doi.org/10.3390/sym15040945

**Chicago/Turabian Style**

Bertolini, Erica, Alberto Blasi, Andrea Damonte, and Nicola Maggiore.
2023. "Gauging Fractons and Linearized Gravity" *Symmetry* 15, no. 4: 945.
https://doi.org/10.3390/sym15040945