# Quantum geometry of resurgent perturbative/nonperturbative relations

## Abstract

For a wide variety of quantum potentials, including the textbook ‘instanton’ examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain N = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and ‘special geometry’. These systems inherit a natural modular structure corresponding to Ramanujan’s theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Lastly, our approach is very elementary, using basic classical geometry combined with all-orders WKB.

- Authors:

- Univ. of Maryland, College Park, MD (United States)
- Univ. of Connecticut, Storrs, CT (United States)
- North Carolina State Univ., Raleigh, NC (United States)

- Publication Date:

- Research Org.:
- Univ. of Connecticut, Storrs, CT (United States); Univ. of Maryland, College Park, MD (United States); North Carolina State Univ., Raleigh, NC (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

- OSTI Identifier:
- 1389787

- Grant/Contract Number:
- SC0010339; FG02-93ER40762; SC0013036

- 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; Nonperturbative Effects; Solitons Monopoles and Instantons; Topological Strings

### Citation Formats

```
Basar, Gokce, Dunne, Gerald V., and Unsal, Mithat.
```*Quantum geometry of resurgent perturbative/nonperturbative relations*. United States: N. p., 2017.
Web. doi:10.1007/JHEP05(2017)087.

```
Basar, Gokce, Dunne, Gerald V., & Unsal, Mithat.
```*Quantum geometry of resurgent perturbative/nonperturbative relations*. United States. doi:10.1007/JHEP05(2017)087.

```
Basar, Gokce, Dunne, Gerald V., and Unsal, Mithat. Tue .
"Quantum geometry of resurgent perturbative/nonperturbative relations". United States.
doi:10.1007/JHEP05(2017)087. https://www.osti.gov/servlets/purl/1389787.
```

```
@article{osti_1389787,
```

title = {Quantum geometry of resurgent perturbative/nonperturbative relations},

author = {Basar, Gokce and Dunne, Gerald V. and Unsal, Mithat},

abstractNote = {For a wide variety of quantum potentials, including the textbook ‘instanton’ examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain N = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and ‘special geometry’. These systems inherit a natural modular structure corresponding to Ramanujan’s theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Lastly, our approach is very elementary, using basic classical geometry combined with all-orders WKB.},

doi = {10.1007/JHEP05(2017)087},

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

number = 5,

volume = 2017,

place = {United States},

year = {Tue May 16 00:00:00 EDT 2017},

month = {Tue May 16 00:00:00 EDT 2017}

}

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