Index for three dimensional superconformal field theories with general R-charge assignments (1101.0557v3)

Abstract: We derive a general formula of an index for three dimensional N=2 superconformal field theories with general R-charge assignments to chiral multiplets by using the localization method in S^2xS^1 background. As examples we compute the index for theories in a few mirror pairs, and confirm the agreement of the indices in each mirror pair.

• The paper introduces a superconformal index for 3D N=2 SCFTs with non-canonical R-charges using the localization method on S2×S1.
• It refines supersymmetry transformations and deformation terms to reduce the path integral into tractable matrix integrals, successfully matching indices for mirror dual pairs like QEDₙ (N=1,2,3).
• The work paves the way for future research on SCFTs by broadening analytical tools for dualities, including potential applications in AdS/CFT correspondence and Z-extremization.

An Index for Three-Dimensional Superconformal Field Theories with General R-Charge Assignments

The paper provides a comprehensive framework for constructing an index for three-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) with arbitrary R-charge assignments for chiral multiplets, utilizing the localization technique on $S^2 \times S^1$ backgrounds. This work addresses a significant gap in the literature by extending previous results, which were mainly confined to theories with canonical R-charges due to the protective nature of non-abelian R-symmetries found in theories with $\mathcal{N} \geq 3$ supersymmetry. The authors systematically derive a general formula that accommodates theories with chiral multiplets possessing non-canonical R-charges, thereby broadening the class of SCFTs that can be analyzed using this methodology.

Methodological Advancements

The authors utilize the localization method, a powerful technique that transforms the evaluation of the path integral into a manageable computation of matrix integrals. They meticulously redefine the supersymmetry transformations and deformation terms suitable for conformally flat backgrounds. These redefinitions allow for the localization of the path integral for SCFTs on $S^2 \times S^1$, yielding a tractable expression for the superconformal index. The derivation includes detailed calculations of deformation terms necessary for the localization and demonstrates how the path integral reduces in the limit $t \to \infty$.

Application to Mirror Symmetry and Examples

The paper applies the derived index to analyze mirror symmetry in specific theories. It investigates pairs of theories known to be mirror duals, such as $\text{QED}_N$ and its mirror counterparts for $N = 1, 2, 3$. Through a meticulous calculation of indices for these mirror pairs, the agreement of the indices is confirmed, thus affirming the robustness and validity of the proposed index formula. This exercise highlights the utility of the index in verifying dualities in $\mathcal{N}=2$ theories.

Implications and Future Directions

This work lays the groundwork for future research exploring SCFTs with non-canonical R-charges, providing new avenues to test the AdS/CFT correspondence in the field of $\mathcal{N}=2$ theories. The ability to handle large N limits within this framework enables the probing of dualities beyond the simplest gauge groups, thereby offering a more comprehensive understanding of the operator spectrum and BPS states in these theories. Additionally, the paper suggests the potential integration of these findings with the Z-extremization method, which proposes a novel approach for determining R-charge assignments at infrared fixed points.

The implications of this work are far-reaching, contributing not only to the mathematical formalism of SCFTs but also offering tools for practical computation in theoretical physics. As the authors suggest, future research could extend these methodologies to more complex gauge theories with richer symmetries and a larger number of supersymmetries, potentially exploring new aspects of the intricate connection between gauge theories and their gravitational duals. These explorations could further elucidate the landscape of three-dimensional SCFTs, paving the way for new insights into the underlying principles governing quantum field theories and their holographic descriptions.