Exact two-dimensional superconformal R-symmetry and c-extremization (1211.4030v2)

Abstract: We uncover a general principle, dubbed c-extremization, which determines the exact R-symmetry of a two-dimensional unitary superconformal field theory with N=(0,2) supersymmetry. To illustrate its utility, we study superconformal theories obtained by twisted compactifications of four-dimensional N=4 super-Yang-Mills on Riemann surfaces, and construct their gravity duals.

• The paper introduces c-extremization, a method analogous to 4D a-maximization, to precisely identify the superconformal R-symmetry in two-dimensional N=(0,2) SCFTs by extremizing a trial central charge.
• This method utilizes 't Hooft anomalies, which are renormalization group invariant, to determine the central charge and the R-symmetry, providing a powerful tool even without a Lagrangian formulation.
• The authors demonstrate the method's effectiveness by computing central charges for SCFTs arising from 4D N=4 super-Yang-Mills compactifications, showing agreement with results from holographic duality in type IIB supergravity.

An Essay on "Exact two-dimensional superconformal R-symmetry and $c$-extremization"

The study presented in the paper "Exact two-dimensional superconformal R-symmetry and $c$-extremization" focuses on the intricacies of determining the R-symmetry in two-dimensional unitary superconformal field theories (SCFTs) with $\mathcal{N}=(0,2)$ supersymmetry. The authors, Francesco Benini and Nikolay Bobev, introduce a concept labeled as $c$-extremization, analogous to $a$-maximization in four dimensions, which is used to identify the superconformal R-symmetry by extremizing a trial central charge. This trial central charge, $c_R^\text{tr}$, depends on 't Hooft anomalies, providing a crucial insight into the structure and dynamics of these SCFTs without requiring a Lagrangian formulation.

Key Aspects of $c$-extremization

1. Determination of R-symmetry: In non-conformal $\mathcal{N}=(0,2)$ theories, R-symmetry is not uniquely defined due to possible mixing with other Abelian symmetries. However, at an IR fixed point, the theory's conformal symmetry singles out the unique R-symmetry, which the authors determine through $c$-extremization.
2. Anomaly Constraints and Central Charge: The work establishes a connection between the conformal anomaly $c_R$ and the R-symmetry anomaly of the theory. In particular, the central charge $c_R$ reflects a sum of the R-symmetry and possible flavor symmetries and is linked to the 't Hooft anomalies, which are renormalization group invariant. Thus, these anomalies remain unchanged over energy scales, marking their utility in identifying the superconformal R-symmetry even at a UV level.
3. Constructing Gravity Duals and Holographic Techniques: To demonstrate the utility of their approach, Benini and Bobev apply $c$-extremization to compute central charges for two-dimensional $\mathcal{N}=(0,2)$ SCFTs resulting from twisted compactifications of four-dimensional $\mathcal{N}=4$ super-Yang-Mills theory on Riemann surfaces. The large $N$ limit allows them to construct holographic duals within type IIB supergravity, aligning the computations of central charges obtained via holography with those derived from the field theory using $c$-extremization.

Implications and Future Directions

The implications of this work resonate within both practical applications and theoretical expansions. Practically, the results offer a robust framework through which the superconformal R-symmetry in intricate field theories can be determined from their anomaly structures, streamlining analyses that could otherwise be extremely intricate or computationally intractable.

Theoretically, this paper paves the way for future research in both two-dimensional cases and potentially analogous higher-dimensional quantum field theories. The analogy to $a$-maximization encourages speculation on further extensions and adaptations of this method in broader contexts. Particularly, as quantum field theories and their gravitational duals (in the context of AdS/CFT correspondence) continue to underpin many aspects of theoretical physics, advancements in understanding exactly how symmetries like the R-symmetry can be identified and characterized are extremely valuable.

In summary, the research done by Benini and Bobev provides a mathematically rigorous and conceptually transparent approach to determining the R-symmetry in $\mathcal{N}=(0,2)$ SCFTs, contributing effectively to the dialogue between field theory and its higher-dimensional gravitational demonstrations. The perspective provided by $c$-extremization is an instrument to unifying our understanding of anomaly relations within the framework of duality theories, affirming its relevance and necessity in contemporary theoretical physics pursuits.