Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories (1205.4142v2)

Abstract: We consider three-dimensional N=2 superconformal field theories on a three-sphere and analyze their free energy F as a function of background gauge and supergravity fields. A crucial role is played by certain local terms in these background fields, including several Chern-Simons terms. The presence of these terms clarifies a number of subtle properties of F. This understanding allows us to prove the F-maximization principle. It also explains why computing F via localization leads to a complex answer, even though we expect it to be real in unitary theories. We discuss several corollaries of our results and comment on the relation to the F-theorem.

• The paper introduces the F-maximization principle, demonstrating that the correct R-symmetry maximizes the real part of the free energy on a three-sphere.
• It employs localization techniques to reveal complex free energy contributions due to contact and Chern-Simons terms, challenging conventional unitarity expectations.
• The study provides new methodologies to extract conformal data such as τ_ab and κ_ab, advancing our understanding of SCFT dualities and RG flows.

Analysis of F-Maximization in Three-Dimensional Superconformal Theories

This essay explores the intricate findings presented in the paper titled "Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories" by Cyril Closset et al. The research focuses on the profound implications of the free energy function, $F$, in three-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs), primarily when placed on a three-sphere. This setup is particularly pertinent due to its circumvention of infrared ambiguities and allows for a nuanced investigation into various characteristics of superconformal theories, such as F-maximization and its relationship to unitary theories.

Key Insights and Results

1. F-Maximization Principle: The paper establishes the F-maximization principle, analogous to a-maximization in four dimensions. This principle asserts that the superconformal R-symmetry maximizes the real part of the free energy $F(t)$, where $t$ denotes trial parameters for the R-symmetry. Specifically, the paper demonstrates that the real part of $F(t)$ on the three-sphere is maximized when the correct R-symmetry of the SCFT is chosen, thereby constituting a local maximum. This result significantly impacts how renormalization group flows in three-dimensional theories are understood.
2. Complex Nature of Free Energy: Localization techniques reveal a complex-valued free energy $F$, despite expectations of real values in unitary theories. The authors attribute this discrepancy to certain Chern-Simons (CS) terms in the background fields, which induce imaginary components to the computations. These findings emphasize the nuanced role of background gauge and supergravity fields in SCFTs on curved manifolds.
3. Implication of Contact Terms and Chern-Simons Terms: The paper meticulously explores how contact terms in correlation functions, manifested as odd-dimensional CS terms, influence the overall behavior of the SCFT. The presence of specific CS terms clarifies why F computed through localization contradicts typical unitary expectations, and the CS levels reveal new observables derivable from SCFT. The paper provides a detailed formalism for extracting quantities like $\tau_{ab}$ and $\kappa_{ab}$, representing global flavor symmetries, using localization, thus extending the toolkit for SCFT analysis.
4. Theoretical and Practical Implications: The implications of these findings are broad, offering new tools for evaluating the conformal data of SCFTs and asserting the validity of the F-theorem, which posits that the free energy must decrease along the RG flow in unitary theories. This theorem mirrors the four-dimensional a-theorem, encompassing scenarios involving supersymmetric and potentially non-supersymmetric flows while offering practical computational strategies for exact results via localization.
5. AdS/CFT Correspondence: The paper also investigates scenarios where three-dimensional SCFTs exhibit dualities with AdS4 supergravity theories. Through detailed examples, such as the flavored conifold quiver dual to M2-branes, the authors demonstrate how F-maximization coherently links with volume minimization procedures in the context of AdS/CFT correspondence, asserting the universality of these principles across dimensional borders.

Conclusion

The comprehensive analysis of $\mathcal{N}=2$ SCFTs on a three-sphere with background gauge and supergravity fields presented by Closset et al. enriches our understanding of F-maximization. The robust framework they introduce for analyzing the nuanced calculations via localization and the resolution of apparent discrepancies in conformal invariance versus gauge invariance will undoubtedly have far-reaching ramifications, guiding future research in theoretical physics. As the field continues to evolve, the implications of these findings will facilitate deeper insights into three-dimensional SCFT, contributing significantly to high-energy theoretical physics and quantum field theory research domains.