Abstract
For 3-dimensional field theories with \( \mathcal{N} = 2 \) 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 ofsuch large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the super potential. In all our \( \mathcal{N} = 2 \) 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-Simons-matter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N 5/3 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.
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Jafferis, D.L., Klebanov, I.R., Pufu, S.S. et al. Towards the F-theorem: \( \mathcal{N} = 2 \) field theories on the three-sphere. J. High Energ. Phys. 2011, 102 (2011). https://doi.org/10.1007/JHEP06(2011)102
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DOI: https://doi.org/10.1007/JHEP06(2011)102