# Anomalies in the space of coupling constants and their dynamical applications I

## Abstract

It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincaré symmetry) to background gauge fields (and a metric for the Poincaré symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects ’t Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of ’t Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary ’t Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized ’t Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to thismore »

- Authors:

- Institute for Advanced Study, Princeton University
- The University of Texas at Austin
- Princeton University

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1581204

- Grant/Contract Number:
- SC0009988

- Resource Type:
- Published Article

- Journal Name:
- SciPost Physics Proceedings

- Additional Journal Information:
- Journal Name: SciPost Physics Proceedings Journal Volume: 8 Journal Issue: 1; Journal ID: ISSN 2542-4653

- Publisher:
- Stichting SciPost

- Country of Publication:
- Netherlands

- Language:
- English

### Citation Formats

```
Cordova, Clay, Freed, Daniel, Lam, Ho Tat, and Seiberg, Nathan. Anomalies in the space of coupling constants and their dynamical applications I. Netherlands: N. p., 2020.
Web. doi:10.21468/SciPostPhys.8.1.001.
```

```
Cordova, Clay, Freed, Daniel, Lam, Ho Tat, & Seiberg, Nathan. Anomalies in the space of coupling constants and their dynamical applications I. Netherlands. doi:10.21468/SciPostPhys.8.1.001.
```

```
Cordova, Clay, Freed, Daniel, Lam, Ho Tat, and Seiberg, Nathan. Mon .
"Anomalies in the space of coupling constants and their dynamical applications I". Netherlands. doi:10.21468/SciPostPhys.8.1.001.
```

```
@article{osti_1581204,
```

title = {Anomalies in the space of coupling constants and their dynamical applications I},

author = {Cordova, Clay and Freed, Daniel and Lam, Ho Tat and Seiberg, Nathan},

abstractNote = {It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincaré symmetry) to background gauge fields (and a metric for the Poincaré symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects ’t Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of ’t Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary ’t Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized ’t Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to this perspective. Also, we use Quillen’s superconnections to derive the anomaly for a free spinor field with variable mass. In a companion paper we will study four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.},

doi = {10.21468/SciPostPhys.8.1.001},

journal = {SciPost Physics Proceedings},

number = 1,

volume = 8,

place = {Netherlands},

year = {2020},

month = {1}

}

DOI: 10.21468/SciPostPhys.8.1.001

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