# Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory

## Abstract

The Hilbert space of a quantum system with internal global symmetry *G* decomposes into sectors labelled by irreducible representations of *G*. If the system is chaotic, the energies in each sector should separately resemble ordinary random matrix theory. We show that such “sector-wise” random matrix ensembles arise as the boundary dual of two- dimensional gravity with a *G* gauge field in the bulk. Within each sector, the eigenvalue density is enhanced by a nontrivial factor of the dimension of the representation, and the ground state energy is determined by the quadratic Casimir. We study the consequences of ’t Hooft anomalies in the matrix ensembles, which are incorporated by adding specific topological terms to the gauge theory action. The effect is to introduce projective representations into the decomposition of the Hilbert space. Finally, we consider ensembles with *G* symmetry and time reversal symmetry, and analyze a simple case of a mixed anomaly between time reversal and an internal $$\mathbb{Z}$$_{2} symmetry.

- Publication Date:

- Research Org.:
- Institute for Advanced Study, Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1631983

- Grant/Contract Number:
- SC0009988

- 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: 2020; Journal Issue: 4; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 2D Gravity; AdS-CFT Correspondence; Anomalies in Field and String Theories; Field Theories in Lower Dimensions; High Energy Physics - Theory; Condensed Matter - Strongly Correlated Electrons

### Citation Formats

```
Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory. United States: N. p., 2020.
Web. https://doi.org/10.1007/JHEP04(2020)186.
```

```
Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory. United States. https://doi.org/10.1007/JHEP04(2020)186
```

```
Tue .
"Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory". United States. https://doi.org/10.1007/JHEP04(2020)186. https://www.osti.gov/servlets/purl/1631983.
```

```
@article{osti_1631983,
```

title = {Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory},

author = {None, None},

abstractNote = {The Hilbert space of a quantum system with internal global symmetry G decomposes into sectors labelled by irreducible representations of G. If the system is chaotic, the energies in each sector should separately resemble ordinary random matrix theory. We show that such “sector-wise” random matrix ensembles arise as the boundary dual of two- dimensional gravity with a G gauge field in the bulk. Within each sector, the eigenvalue density is enhanced by a nontrivial factor of the dimension of the representation, and the ground state energy is determined by the quadratic Casimir. We study the consequences of ’t Hooft anomalies in the matrix ensembles, which are incorporated by adding specific topological terms to the gauge theory action. The effect is to introduce projective representations into the decomposition of the Hilbert space. Finally, we consider ensembles with G symmetry and time reversal symmetry, and analyze a simple case of a mixed anomaly between time reversal and an internal $\mathbb{Z}$2 symmetry.},

doi = {10.1007/JHEP04(2020)186},

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

number = 4,

volume = 2020,

place = {United States},

year = {2020},

month = {4}

}

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