Multiplets of Superconformal Symmetry in Diverse Dimensions (1612.00809v1)

Abstract: We systematically analyze the operator content of unitary superconformal multiplets in $d > 3$ spacetime dimensions. We present a simple, general, and efficient algorithm that generates all of these multiplets by correctly eliminating possible null states. The algorithm is conjectural, but passes a vast web of consistency checks. We apply it to tabulate a large variety of superconformal multiplets. In particular, we classify and construct all multiplets that contain conserved currents or free fields, which play an important role in superconformal field theories (SCFTs). Some currents that are allowed in conformal field theories cannot be embedded in superconformal multiplets, and hence they are absent in SCFTs. We use the structure of superconformal stress tensor multiplets to show that SCFTs with more than 16 Poincar\'e supercharges cannot arise in $d \geq 4$, even when the corresponding superconformal algebras exist. We also show that such theories do arise in $d = 3$, but are necessarily free.

• The paper introduces an algorithm that systematically classifies superconformal multiplets by decomposing them into irreducible bosonic representations.
• It establishes recombination rules that reveal how long multiplets split into short multiplets at unitarity thresholds.
• The study informs SCFT construction by outlining constraints on conserved currents and paving the way for computational tools in high-energy physics.

Superconformal Multiplets of Various Dimensions and Degrees of Supersymmetry

This essay analyzes the comprehensive work by Cordova, Dumitrescu, and Intriligator on the operator content and structure of superconformal multiplets in spacetime dimensions $d \geq 3$. These multiplets, pertinent to superconformal field theories (SCFTs), are classified according to unitary representations of superconformal algebras arising from the amalgamation of conformal and supersymmetric symmetries.

Summary of Findings

The classification and construction detailed in this work are systematic and algorithmic across various dimensions and degrees of supersymmetry, encapsulating not only theoretical frameworks but also, where applicable, computational approaches to determine the structure of these multiplets.

• Algorithm Development: A conjectural algorithm is introduced to classify superconformal multiplets, focusing on their composition as dictated by their primaries. This involves decomposing these multiplets into irreducible representations of the bosonic $\frak{so}(d,2) \times \frak{R}$ algebra, subjected to the constraints imposed by supercharges and their anticommuting nature.
• Recombination Rules: The paper presents recombination rules which offer insights into how long multiplets can fragment into short multiplets upon reaching certain unitarity bounds. This contributes significantly to understanding shifts in the spectrum of short multiplets upon marginal deformations in SCFTs.
• Unitary Bound Analysis: For different values of $d$ and $\mathcal{N}$, the authors deliver a detailed unitarity analysis. They elaborate on scaling dimensions and constraints faced by the primary state in a multiplet that ensures its presence within the bounds of the superconformal validity.
• Irreducible Representations: As the cornerstone of their methodology, the authors enumerate representations in terms of their highest weight states and explicitly decompose those into a sum of other conformal primaries by leveraging the Racah-Speiser (RS) method, refined to manage the supercharges associated with short multiplets.
• Conserved Currents: By dissecting the operator content of short multiplets, they illustrate the circumstances under which multiplets contain conserved currents or free fields. This sheds light on the phenomenology of SCFTs and the peculiar absence of certain currents that could otherwise exist in purely conformal contexts.

Practical and Theoretical Implications

The algorithmic classification of these multiplets and their content has direct implications for both the explicit construction of SCFT models and broader theoretical understandings:

1. SCFT Construction: The scaffold provided by this work equips researchers with the tools to construct candidate theories, especially in the field of holography and in pursuing conjectured dualities between certain gravity theories in higher dimensions.
2. Recombination Mechanisms: Understanding recombination within multiplets allows researchers to recognize phenomena such as emergent symmetries or accidental symmetry enhancements that may not be overtly present in the base theory.
3. Constraints on Available Currents: By explicitly cataloging which currents are impermissible, this paper stipulates critical constraints that inform on limiting behaviors of certain SCFTs, influencing insights into their high-energy behaviors or conformal phase structures.
4. Software Utilization: The additional mention of a forthcoming Mathematica package suggests a future direction where computational tools will further democratize access to these complex classifications, facilitating deeper experimental cross-verification and theoretical manipulation.

Speculative Outlook

Looking ahead, the framework posed for $d \geq 3$ dimensions primes research into extending such algorithmic classifications to theories within 2-dimensional superconformal algebras, where richer and multifaceted behaviors are expected. Additionally, variants of this classification might illuminate SCFTs that appear as effective theories derived from more fundamental UV completions in high-energy physics.

This work forms a crucial stepping stone, dovetailing algebraic classification with potential computational efficiency to forge pathways in abstract and applied theoretical physics, with significant importance in interpreting physical phenomena through the lens of conformal invariance and supersymmetry.

In conclusion, the paper represents a methodical synthesis of collective insights into superconformal representations, unifying varied phenomena under a singular theoretical and computational framework, and heralding advancements in SCFT studies within mathematical physics.