Central charges of N=2 superconformal field theories in four dimensions

Abstract: We present a general method for computing the central charges a and c of N=2 superconformal field theories corresponding to singular points in the moduli space of N=2 gauge theories. Our method relates a and c to the U(1)_R anomalies of the topologically twisted gauge theory. We evaluate these anomalies by studying the holomorphic dependence of the path integral measure on the moduli. We calculate a and c for superconformal points in a variety of gauge theories, including N=4 SU(N), N=2 pure SU(N) Yang-Mills, and USp(2N) with 1 massless antisymmetric and 4 massive fundamental hypermultiplets. In the latter case, we reproduce the conformal and flavor central charges previously calculated using the gravity duals of these gauge theories. For any SCFT in the class under consideration, we derive a previously conjectured expression for 2a-c in terms of the sum of the dimensions of operators parameterizing the Coulomb branch. Finally, we prove that the ratio a/c is bounded above by 5/4 and below by 1/2.

• The paper introduces a novel computational method that relates central charges to R-symmetry anomalies in topologically twisted gauge theories.
• It validates the approach by showing consistent central charge values with both weak-coupling analyses and holographic predictions, including bounds on the a/c ratio.
• The findings extend to strongly-coupled regimes, such as Argyres-Douglas points, deepening insights into the operator dimensions and overall SCFT structure.

Analyzing Central Charges in $N=2$ Superconformal Field Theories

In this paper, the authors, Alfred D. Shapere and Yuji Tachikawa, introduce a comprehensive methodology to compute central charges $a$ and $c$ of $N=2$ superconformal field theories (SCFTs) in four dimensions, focusing on singular points in the moduli space of various $N=2$ gauge theories. Central charges are crucial as they characterize key aspects of four-dimensional conformal field theories, such as the Weyl anomaly and correlation functions of the stress-energy tensor.

Computation Methodology

The authors employ a novel approach that harnesses the relationship between the central charges of SCFTs and the $_R$ anomalies associated with topologically twisted gauge theories. They evaluate the $_R$ anomalies through the holomorphic dependence of the path integral measure on the moduli space parameters. This topological perspective allows for the determination of the scaling dimensions of gauge theory operators relevant to SCFTs. The primary advantage of this method is its general applicability, covering a range of theories including those with dual descriptions via the AdS/CFT correspondence.

Known Results and Theoretical Implications

For known cases like $N=4$ $(N)$ gauge theory, and $N=2$ finite theories with specific matter content, the central charges $a$ and $c$ obtained using this method agree with those calculated from the field theory at weak coupling or using holography. The technique notably extends to inherently strongly-coupled theories such as the Argyres-Douglas points which previously lacked a clear field-theoretic central charge calculation.

The finding that for $N=2$ SCFTs arising from the Coulomb branch of gauge theories, the difference $2a-c$ is directly related to the sum of the dimensions of operators parameterizing the Coulomb branch, proposes a robust formula for central charges in these high-complexity theories. This reinforces conjectures that had been formulated based on indirect evidence and further aligns theoretical predictions with anomalies in the ultraviolet (UV) and infrared (IR) regimes of gauge theories.

Flavor Symmetries and Anomalies

Additionally, the paper explores the flavor symmetries present in these SCFTs. Through the introduction of an external gauge field for these symmetries, the flavor symmetry central charge $k_G$ is directly computed. These results not only synchronize with those found using non-perturbative techniques like S-duality but also adhere closely to predictions made by holography in the large-N limit.

Bounds and Speculation on Future Developments

The paper proposes bounds for the ratio of the central charges $a/c$, positing that for any $N=2$ SCFT, this ratio is confined between $1/2$ and $5/4$. This conjecture is grounded in an analysis of modular properties of the path integral, and, importantly, these bounds align with recent conjectures made regarding positivity constraints from causality and unitarity principles. Future developments may explore whether these bounds can be relaxed or derived in alternative ways, potentially extending insights to theories with less supersymmetry.

Conclusion

The authors of this paper have laid the groundwork for a deepened understanding of central charges in a broad class of $N=2$ SCFTs, bridging the gap between abstract field-theoretic constructs and computational formalism. The outcomes provide pivotal insights into the structure of SCFT anomalies and offer a compelling approach for computing central charges in theories where conventional techniques may fall short. Their work constitutes a meaningful contribution to the theoretical groundwork needed for advancing our understanding of dualities and gauge theories in diverse high-energy physics contexts.