# Theta, time reversal and temperature

## Abstract

SU(N) gauge theory is time reversal invariant at θ = 0 and θ = π. We show that at θ = π there is a discrete ’t Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at θ = π the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at θ = 0 is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at θ = π, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for SU(2) gauge theory. The underlying symmetry at θ = π is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two O(2)-symmetric fixed points. In conclusion, it may also be that the four-dimensional theory around θ = π is gapless, e.g. a Coulomb phase could match the underlying anomalies.

- Authors:

- Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada)
- California Inst. of Technology (CalTech), Pasadena, CA (United States)
- Weizmann Institute of Science, Rehovot (Israel)
- Institute for Advanced Study, Princeton, NJ (United States)

- Publication Date:

- Research Org.:
- California Institute of Technology, Pasadena, CA (United States), Institute for Advanced Study

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1389778

- Grant/Contract Number:
- SC0011632; SC0009988

- Resource Type:
- Journal Article: 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: 5; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Anomalies in Field and String Theories; Confinement; Spontaneous Symmetry Breaking; Wilson; ’t Hooft and Polyakov loops

### Citation Formats

```
Gaiotto, Davide, Kapustin, Anton, Komargodski, Zohar, and Seiberg, Nathan.
```*Theta, time reversal and temperature*. United States: N. p., 2017.
Web. doi:10.1007/JHEP05(2017)091.

```
Gaiotto, Davide, Kapustin, Anton, Komargodski, Zohar, & Seiberg, Nathan.
```*Theta, time reversal and temperature*. United States. doi:10.1007/JHEP05(2017)091.

```
Gaiotto, Davide, Kapustin, Anton, Komargodski, Zohar, and Seiberg, Nathan. Wed .
"Theta, time reversal and temperature". United States.
doi:10.1007/JHEP05(2017)091. https://www.osti.gov/servlets/purl/1389778.
```

```
@article{osti_1389778,
```

title = {Theta, time reversal and temperature},

author = {Gaiotto, Davide and Kapustin, Anton and Komargodski, Zohar and Seiberg, Nathan},

abstractNote = {SU(N) gauge theory is time reversal invariant at θ = 0 and θ = π. We show that at θ = π there is a discrete ’t Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at θ = π the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at θ = 0 is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at θ = π, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for SU(2) gauge theory. The underlying symmetry at θ = π is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two O(2)-symmetric fixed points. In conclusion, it may also be that the four-dimensional theory around θ = π is gapless, e.g. a Coulomb phase could match the underlying anomalies.},

doi = {10.1007/JHEP05(2017)091},

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

number = 5,

volume = 2017,

place = {United States},

year = {Wed May 17 00:00:00 EDT 2017},

month = {Wed May 17 00:00:00 EDT 2017}

}

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