## Counting RG flows

## Abstract

Here, interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different "topological sectors" for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) dowain walls? Analyzing these questions in a wide variety of contexts -- from counting RG walls to AdS/CFT correspondence -- will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott's version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and "topological obstructions" for the strong form of the C-theorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong C-theorem happens and predicts "phase transitions" along the RG flow when the topological obstruction is non-trivial. Along the way, we also find new conformal manifolds in well-known 4d CFT's and point out connections withmore »

- Authors:

- California Inst. of Technology (CalTech), Pasadena, CA (United States); Max-Planck-Institut fur Mathematik, Bonn (Germany)

- Publication Date:

- Research Org.:
- California Inst. of Technology (CalTech), Pasadena, CA (United States)

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

- OSTI Identifier:
- 1327329

- Grant/Contract Number:
- SC0011632

- 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: 2016; Journal Issue: 1; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; supersymmetric gauge theory; renormalization group

### Citation Formats

```
Gukov, Sergei. Counting RG flows. United States: N. p., 2016.
Web. doi:10.1007/JHEP01(2016)020.
```

```
Gukov, Sergei. Counting RG flows. United States. doi:10.1007/JHEP01(2016)020.
```

```
Gukov, Sergei. Tue .
"Counting RG flows". United States. doi:10.1007/JHEP01(2016)020. https://www.osti.gov/servlets/purl/1327329.
```

```
@article{osti_1327329,
```

title = {Counting RG flows},

author = {Gukov, Sergei},

abstractNote = {Here, interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different "topological sectors" for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) dowain walls? Analyzing these questions in a wide variety of contexts -- from counting RG walls to AdS/CFT correspondence -- will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott's version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and "topological obstructions" for the strong form of the C-theorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong C-theorem happens and predicts "phase transitions" along the RG flow when the topological obstruction is non-trivial. Along the way, we also find new conformal manifolds in well-known 4d CFT's and point out connections with the superconformal index and classifying spaces of global symmetry groups.},

doi = {10.1007/JHEP01(2016)020},

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

number = 1,

volume = 2016,

place = {United States},

year = {2016},

month = {1}

}

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