Towards the F-theorem: N = 2 field theories on the three-sphere

Daniel L. Jafferis, Igor R. Klebanov, Silviu S. Pufu, Benjamin R. Safdi

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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 AdS4 × 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 languageEnglish (US)
Article number102
JournalJournal of High Energy Physics
Issue number6
StatePublished - 2011

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics


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


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