# High-temperature asymptotics of supersymmetric partition functions

## Abstract

We study the supersymmetric partition function of 4d supersymmetric gauge theories with a U(1) R-symmetry on Euclidean S ^{3} × S _{β} ^{1}, with S ^{3} the unit-radius squashed three-sphere, and β the circumference of the circle. For superconformal theories, this partition function coincides (up to a Casimir energy factor) with the 4d superconformal index. The partition function can be computed exactly using the supersymmetric localization of the gauge theory path-integral. It takes the form of an elliptic hypergeometric integral, which may be viewed as a matrix-integral over the moduli space of the holonomies of the gauge fields around S _{β} ^{1}. At high temperatures (β → 0, corresponding to the hyperbolic limit of the elliptic hypergeometric integral) we obtain from the matrix-integral a quantum effective potential for the holonomies. The effective potential is proportional to the temperature. Therefore the high-temperature limit further localizes the matrix-integral to the locus of the minima of the potential. If the effective potential is positive semi-definite, the leading high-temperature asymptotics of the partition function is given by the formula of Di Pietro and Komargodski, and the subleading asymptotics is connected to the Coulomb branch dynamics on R ^{3} × S ^{1}. In theories wheremore »

- Authors:

- Univ. of Michigan, Ann Arbor, MI (United States)

- Publication Date:

- Research Org.:
- Univ. of Michigan, Ann Arbor, MI (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1326942

- Grant/Contract Number:
- SC0007859

- 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: 2016; Journal Issue: 7; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Matrix Models; Supersymmetric gauge theory; Supersymmetry and Duality

### Citation Formats

```
Ardehali, Arash Arabi.
```*High-temperature asymptotics of supersymmetric partition functions*. United States: N. p., 2016.
Web. doi:10.1007/JHEP07(2016)025.

```
Ardehali, Arash Arabi.
```*High-temperature asymptotics of supersymmetric partition functions*. United States. doi:10.1007/JHEP07(2016)025.

```
Ardehali, Arash Arabi. Tue .
"High-temperature asymptotics of supersymmetric partition functions". United States.
doi:10.1007/JHEP07(2016)025. https://www.osti.gov/servlets/purl/1326942.
```

```
@article{osti_1326942,
```

title = {High-temperature asymptotics of supersymmetric partition functions},

author = {Ardehali, Arash Arabi},

abstractNote = {We study the supersymmetric partition function of 4d supersymmetric gauge theories with a U(1) R-symmetry on Euclidean S3 × Sβ1, with S3 the unit-radius squashed three-sphere, and β the circumference of the circle. For superconformal theories, this partition function coincides (up to a Casimir energy factor) with the 4d superconformal index. The partition function can be computed exactly using the supersymmetric localization of the gauge theory path-integral. It takes the form of an elliptic hypergeometric integral, which may be viewed as a matrix-integral over the moduli space of the holonomies of the gauge fields around Sβ1. At high temperatures (β → 0, corresponding to the hyperbolic limit of the elliptic hypergeometric integral) we obtain from the matrix-integral a quantum effective potential for the holonomies. The effective potential is proportional to the temperature. Therefore the high-temperature limit further localizes the matrix-integral to the locus of the minima of the potential. If the effective potential is positive semi-definite, the leading high-temperature asymptotics of the partition function is given by the formula of Di Pietro and Komargodski, and the subleading asymptotics is connected to the Coulomb branch dynamics on R3 × S1. In theories where the effective potential is not positive semi-definite, the Di Pietro-Komargodski formula needs to be modified. In particular, this modification occurs in the SU(2) theory of Intriligator-Seiberg-Shenker, and the SO(N) theory of Brodie-Cho-Intriligator, both believed to exhibit “misleading” anomaly matchings, and both believed to yield interacting superconformal field theories with c < a. Lastly, two new simple tests for dualities between 4d supersymmetric gauge theories emerge as byproducts of our analysis.},

doi = {10.1007/JHEP07(2016)025},

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

number = 7,

volume = 2016,

place = {United States},

year = {Tue Jul 05 00:00:00 EDT 2016},

month = {Tue Jul 05 00:00:00 EDT 2016}

}

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