### Abstract

For 3-dimensional field theories with N = 2 supersymmetry the Euclidean path integrals on the three-sphere can be calculated using the method of localization; they reduce to certain matrix integrals that depend on the R-charges of the matter fields. We solve a number of such large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the superpotential. In all our N = 2 superconformal examples, the local maximization of F yields answers that scale as N ^{3/2} and agree with the dual M-theory backgrounds AdS_{4} × Y , where Y are 7-dimensional Sasaki- Einstein spaces. We also find in toric examples that local F-maximization is equivalent to the minimization of the volume of Y over the space of Sasakian metrics, a procedure also referred to as Z-minimization. Moreover, we find that the functions F and Z are related for any trial R-charges. In the models we study F is positive and decreases along RG flows.We therefore propose the " F-theorem" that we hope applies to all 3-d field theories: the finite part of the free energy on the three-sphere decreases along RG trajectories and is stationary at RG fixed points. We also show that in an infinite class of Chern-Simonsmatter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N5/3 at large N. This non-trivial scaling matches that of the free energy of the gravity duals in type IIA string theory with Romans mass.

Original language | English (US) |
---|---|

Article number | 102 |

Journal | Journal of High Energy Physics |

Volume | 2011 |

Issue number | 6 |

DOIs | |

State | Published - Sep 29 2011 |

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Keywords

- AdS-CFT correspondence
- Matrix models
- Renormalization group
- Strong coupling expansion

## Fingerprint Dive into the research topics of 'Towards the F-theorem: N = 2 field theories on the three-sphere'. Together they form a unique fingerprint.

## Cite this

*Journal of High Energy Physics*,

*2011*(6), [102]. https://doi.org/10.1007/JHEP06(2011)102