Towards the F-Theorem: N=2 Field Theories on the Three-Sphere

Abstract: For 3-dimensional field theories with {\cal 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 {\cal 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 x 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.

• The paper demonstrates that the finite free energy F decreases along RG flows, supporting the F-theorem in three-dimensional field theories.
• The authors employ localization techniques to reduce path integrals to matrix models, explicitly revealing N^(3/2) scaling consistent with M-theory predictions.
• The study reinforces the holographic duality by aligning computed free energy with expectations from AdS₄ backgrounds and various gauge theory models.

Analysis of the $F$-Theorem and Free Energy Calculation in $\mathcal{N}=2$ Field Theories

The paper under review presents a comprehensive examination of large $N$ matrix models within the context of $\mathcal{N}=2$ supersymmetric field theories on a three-dimensional sphere. This exploration is fundamentally centered around the evaluation of the free energy, $F$, which is proposed to play a significant role analogous to the $a$-coefficient in four-dimensional conformal field theories (CFTs).

Key Contributions

The authors contribute significantly to the analysis of $F$ in various gauge theories, providing evidence for the so-called "$F$-theorem." The $F$-theorem posits that the finite part of the free energy on $S^3$ decreases along Renormalization Group (RG) flows and remains stationary at RG fixed points. This conjecture holds promising parallels to the $a$-theorem in four dimensions, which states that the $a$-anomaly decreases along RG flows.

Methodology and Results

The paper leverages the localization technique in supersymmetric theories, which reduces the computational complexity of the path integrals on the three-sphere to solving matrix models. Through this approach, the authors provide explicit computations for the free energy $F$ as a function of trial R-charges consistent with the constraints of the superpotential's marginality.

Several critical models are solved, exhibiting the $N^{3/2}$ scaling of free energy, which aligns with the expectations from dual M-theory backgrounds such as $AdS_4\times Y$, where $Y$ are Sasaki-Einstein spaces. The results underscore a strong agreement between the localized free energy calculations and the M-theory predictions, highlighting the robustness of the AdS/CFT duality in three dimensions.

Furthermore, the authors address specific cases, such as the $\mathcal{N}=3$ necklace quiver gauge theories and their deformation, leading to proposals like $AdS_4\times V_{5,2}$ dualities. The systematic exploration of various RG flows demonstrates that $F$ behaves predictively under these transformations.

Implications and Future Directions

The implications of these findings are profound, suggesting that the $F$-theorem could be an essential tool for understanding the dynamics in three-dimensional CFTs, including non-supersymmetric setups. Practically, this insight opens new avenues for testing theoretical models against holographic predictions, offering potential insights into quantum gravity and condensed matter systems.

Looking forward, additional investigations are needed to solidify the $F$-theorem through rigorous proofs in non-supersymmetric contexts and to explore its ramifications in holographic dualities beyond the scenarios presented. The interplay between field theories with chiral bifundamental fields and their potential resolution remains an intriguing open problem.

In bringing to light a calculable analogue to the $a$-coefficient, the groundwork laid in this paper marks a pivotal advance in three-dimensional field theory, providing a new lens through which to examine fundamental aspects of holography and quantum field theory.