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Towards the F-theorem: \( \mathcal{N} = 2 \) field theories on the three-sphere

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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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Authors and Affiliations

  1. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, U.S.A.

    Daniel L. Jafferis

  2. Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA, 02138, U.S.A.

    Daniel L. Jafferis

  3. Joseph Henry Laboratories, Princeton University, Princeton, NJ, 08544, U.S.A.

    Igor R. Klebanov, Silviu S. Pufu & Benjamin R. Safdi

  4. Center for Theoretical Science, Princeton University, Princeton, NJ, 08544, U.S.A.

    Igor R. Klebanov

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  1. Daniel L. Jafferis
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  2. Igor R. Klebanov
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  3. Silviu S. Pufu
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  4. Benjamin R. Safdi
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Correspondence to Silviu S. Pufu.

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ArXiv ePrint: 1103.1181

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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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