# Analytic continuation of dimensions in supersymmetric localization

## Abstract

We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension d ≤ 5, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with d ≤ 3. The results are valid for non-integer d as well. We further propose an analytic continuation from d = 3 to d = 4 that gives the perturbative partition function for an N =1 gauge theory. The results are consistent with the free multiplets and the one-loop β-functions for general N = 1 gauge theories. We also consider the analytic continuation of an N = 1 preserving mass deformation of the maximally supersymmetric gauge theory and compare to recent holographic results for N = 1* super Yang-Mills. We find that the general structure for the real part of the free energy coming from the analytic continuation is consistent with the holographic results.

- Authors:

- Uppsala Univ. (Sweden). Dept. of Physics and Astronomy
- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Center for Theoretical Physics

- Publication Date:

- Research Org.:
- Massachusetts Institute of Technology, Cambridge, MA (United States)

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

- OSTI Identifier:
- 1498698

- Grant/Contract Number:
- SC0012567

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Volume: 2018; Journal Issue: 2; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Extended Supersymmetry; Matrix Models; Supersymmetric Gauge Theory

### Citation Formats

```
Gorantis, Anastasios, Minahan, Joseph A., and Naseer, Usman.
```*Analytic continuation of dimensions in supersymmetric localization*. United States: N. p., 2018.
Web. doi:10.1007/jhep02(2018)070.

```
Gorantis, Anastasios, Minahan, Joseph A., & Naseer, Usman.
```*Analytic continuation of dimensions in supersymmetric localization*. United States. doi:10.1007/jhep02(2018)070.

```
Gorantis, Anastasios, Minahan, Joseph A., and Naseer, Usman. Mon .
"Analytic continuation of dimensions in supersymmetric localization". United States. doi:10.1007/jhep02(2018)070. https://www.osti.gov/servlets/purl/1498698.
```

```
@article{osti_1498698,
```

title = {Analytic continuation of dimensions in supersymmetric localization},

author = {Gorantis, Anastasios and Minahan, Joseph A. and Naseer, Usman},

abstractNote = {We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension d ≤ 5, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with d ≤ 3. The results are valid for non-integer d as well. We further propose an analytic continuation from d = 3 to d = 4 that gives the perturbative partition function for an N =1 gauge theory. The results are consistent with the free multiplets and the one-loop β-functions for general N = 1 gauge theories. We also consider the analytic continuation of an N = 1 preserving mass deformation of the maximally supersymmetric gauge theory and compare to recent holographic results for N = 1* super Yang-Mills. We find that the general structure for the real part of the free energy coming from the analytic continuation is consistent with the holographic results.},

doi = {10.1007/jhep02(2018)070},

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

issn = {1029-8479},

number = 2,

volume = 2018,

place = {United States},

year = {2018},

month = {2}

}