# Resurgence and holomorphy: From weak to strong coupling

## Abstract

We analyze the resurgence properties of finite-dimensional exponential integrals which are prototypes for partition functions in quantum field theories. In these simple examples, we demonstrate that perturbation theory, even at arbitrarily weak coupling, fails as the argument of the coupling constant is varied. It is well-known that perturbation theory also fails at stronger coupling. We show that these two failures are actually intimately related. The formalism of resurgent transseries, which takes into account global analytic continuation properties, fixes both problems and provides an arbitrarily accurate description of exact result for any value of coupling. This means that strong coupling results can be deduced by using merely weak coupling data. Finally, we give another perspective on resurgence theory by showing that the monodromy properties of the weak coupling results are in precise agreement with the monodromy properties of the strong-coupling expansions, obtained using analysis of the holomorphy structure of Picard-Fuchs equations.

- Authors:

- Fine Theoretical Physics Institute, Department of Physics, University of Minnesota, Minneapolis, Minnesota 55455 (United States)
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L2Y5 (Canada)
- Department of Physics, North Carolina State University, Raleigh, North Carolina 27695 (United States)

- Publication Date:

- OSTI Identifier:
- 22403150

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COUPLING CONSTANTS; PARTITION FUNCTIONS; PERTURBATION THEORY; QUANTUM FIELD THEORY; STRONG-COUPLING MODEL

### Citation Formats

```
Cherman, Aleksey, E-mail: acherman@umn.edu, Koroteev, Peter, E-mail: pkoroteev@perimeterinstitute.ca, and Ünsal, Mithat, E-mail: unsal.mithat@gmail.com.
```*Resurgence and holomorphy: From weak to strong coupling*. United States: N. p., 2015.
Web. doi:10.1063/1.4921155.

```
Cherman, Aleksey, E-mail: acherman@umn.edu, Koroteev, Peter, E-mail: pkoroteev@perimeterinstitute.ca, & Ünsal, Mithat, E-mail: unsal.mithat@gmail.com.
```*Resurgence and holomorphy: From weak to strong coupling*. United States. doi:10.1063/1.4921155.

```
Cherman, Aleksey, E-mail: acherman@umn.edu, Koroteev, Peter, E-mail: pkoroteev@perimeterinstitute.ca, and Ünsal, Mithat, E-mail: unsal.mithat@gmail.com. Fri .
"Resurgence and holomorphy: From weak to strong coupling". United States. doi:10.1063/1.4921155.
```

```
@article{osti_22403150,
```

title = {Resurgence and holomorphy: From weak to strong coupling},

author = {Cherman, Aleksey, E-mail: acherman@umn.edu and Koroteev, Peter, E-mail: pkoroteev@perimeterinstitute.ca and Ünsal, Mithat, E-mail: unsal.mithat@gmail.com},

abstractNote = {We analyze the resurgence properties of finite-dimensional exponential integrals which are prototypes for partition functions in quantum field theories. In these simple examples, we demonstrate that perturbation theory, even at arbitrarily weak coupling, fails as the argument of the coupling constant is varied. It is well-known that perturbation theory also fails at stronger coupling. We show that these two failures are actually intimately related. The formalism of resurgent transseries, which takes into account global analytic continuation properties, fixes both problems and provides an arbitrarily accurate description of exact result for any value of coupling. This means that strong coupling results can be deduced by using merely weak coupling data. Finally, we give another perspective on resurgence theory by showing that the monodromy properties of the weak coupling results are in precise agreement with the monodromy properties of the strong-coupling expansions, obtained using analysis of the holomorphy structure of Picard-Fuchs equations.},

doi = {10.1063/1.4921155},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 5,

volume = 56,

place = {United States},

year = {2015},

month = {5}

}