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

By D. Jafferis, I. Klebanov, Silviu S. Pufu, B. Safdi. 2011

346 citations recorded by the source corpus.

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 N3/2 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-Simons-matter 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/c7c4eff567c1f50065b6f87d59cc3ee0d68fbd6d

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Citations in the corpus

Supersymmetric field theories on three-manifolds (2012, methodology): “grounds for N = 1 theories in four dimensions [26, 27, 28]. In three-dimensional N = 2 theories with a U(1)R symmetry, it has led to a proof of the F-maximization principle (originally conjectured in [3,32]), the discovery of a novel superconformal anomaly, and the deﬁnition and exact computation of”
Entanglement entropy from a holographic viewpoint (2012, methodology): “[33] D.”
Central charges for BCFTs and holography (2012, background): “cbdy(rB) drB ≤ 0. (5.3) This can be regarded as a three dimensional version of the g-theorem [6]. An analogue of c-theorem for three dimensional CFTs without boundaries has already been formulated in [29, 30, 31] and is called F-theorem. It is straightforward to generalize (5.3) to much higher dimen”
Lectures on localization and matrix models in supersymmetric Chern–Simons-matter theories (2011, methodology): “The extremization property proved in [55] has been applied and tested in various situations [76, 24, 56, 9], although the validity of the F-theorem is still conjectural and possible counterexamples have been discussed in [56, 81].”
Comments on 3d Seiberg-like dualities (2011, background): “ked in several examples that whenever two theories can be connected by a RG ﬂow, the “free energy” deﬁned as F≡ −ln|Z| (6.8) is smaller in the IR theory. This corroborates the conjectured “F-theorem” [20]. Abelian theory at CS level k= 0. Consider a U(1) theory with k= 0, and generic s1, s2 such tha”
Holography, Entanglement Entropy, and Conformal Field Theories with Boundaries or Defects (2013, background): “roportional to the partition function of the Euclidean theory on Sd[12,13]. In d= 3, the current conjecture is that the partition function at the UV xed point of an RG ow is that at the IR xed point [54]. In even dthe universal information is sL, which is propoprtional to a central charge, dened fro”
F-theorem without supersymmetry (2011, methodology): “ were claried in [32]. Also, for large N theories with AdS 4 Y 7 dual descriptions in M-theory, F-maximization is correctly mapped to the minimization of the volume of the Sasaki-Einstein spaces Y 7 [12,19]. Other ideas on how to best dene the number of degrees of freedom in eld theory, this time in”
The gravity dual of supersymmetric gauge theories on a squashed three-sphere (2011, methodology): “Setting b = 1 we precisely recover the results of [30, 31, 32].”