# Weyl versus conformal invariance in quantum field theory

## Abstract

We argue that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions d ≤ 10. We also study possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. We find possible ‘anomalous’ Weyl transformations proportional to the Weyl (Cotton) tensor for d > 3 (d = 3). The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify possible local anomalies. The arguments can be straightforwardly extended to larger operator dimensions and higher d with additional algebraic complexity.

- Authors:

- Univ. of California, Davis, CA (United States). Center for Quantum Mathematics and Physics (QMAP), Dept. of Physics

- Publication Date:

- Research Org.:
- Univ. of California, Davis, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1505198

- Grant/Contract Number:
- FG02-91ER40674

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 10; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal Field Theory; Effective Field Theories; Space-Time Symmetries

### Citation Formats

```
Farnsworth, Kara, Luty, Markus A., and Prilepina, Valentina. Weyl versus conformal invariance in quantum field theory. United States: N. p., 2017.
Web. doi:10.1007/jhep10(2017)170.
```

```
Farnsworth, Kara, Luty, Markus A., & Prilepina, Valentina. Weyl versus conformal invariance in quantum field theory. United States. doi:10.1007/jhep10(2017)170.
```

```
Farnsworth, Kara, Luty, Markus A., and Prilepina, Valentina. Tue .
"Weyl versus conformal invariance in quantum field theory". United States. doi:10.1007/jhep10(2017)170. https://www.osti.gov/servlets/purl/1505198.
```

```
@article{osti_1505198,
```

title = {Weyl versus conformal invariance in quantum field theory},

author = {Farnsworth, Kara and Luty, Markus A. and Prilepina, Valentina},

abstractNote = {We argue that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions d ≤ 10. We also study possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. We find possible ‘anomalous’ Weyl transformations proportional to the Weyl (Cotton) tensor for d > 3 (d = 3). The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify possible local anomalies. The arguments can be straightforwardly extended to larger operator dimensions and higher d with additional algebraic complexity.},

doi = {10.1007/jhep10(2017)170},

journal = {Journal of High Energy Physics (Online)},

number = 10,

volume = 2017,

place = {United States},

year = {2017},

month = {10}

}

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