从 $\mathcal N=1$ 规范理论涌现的 $\mathcal N=3$ 与冻结 $\mathcal N=2$ 超共形场论

中文摘要 AI 辅助

四维 $\mathcal{N}=3$ 超共形场论（SCFT）是一类奇异的量子场论，它们没有传统的拉格朗日描述。在本信中，我们构造了一个 $\mathcal{N}=1$ 超对称规范理论，并论证其红外不动点表现出超对称增强为 $\mathcal{N}=3$ SCFT。紫外理论具有规范群 $SU(2)^3$、一组手性多重态以及合适的超势。通过执行 $a$-最大化，我们得到红外中心荷 $a=c=5/4$，与秩一 $\mathcal{N}=3$ $\mathbb{Z}_3$ S-fold SCFT 的中心荷匹配。紫外规范理论描述使我们能够计算具有四个独立 fugacity 的完整超共形指标。然后我们考察完整指标的各种极限，并进一步提出 Macdonald 指标的闭式表达式，重现所提出的 $\mathcal{N}=3$ 理论的所有已知结果。更引人注目的是，我们论证进一步规范 $\mathcal{N}=3$ 理论会产生从 Seiberg--Witten 几何推测的冻结秩一 $\mathcal{N}=2$ SCFT，为其存在提供了强有力的证据，并解锁了其先前无法获得的完整超共形指标。因此，我们的构造为使用传统 $\mathcal{N}=1$ 拉格朗日技术探索这些奇异、本质强耦合理论的精细受保护结构开辟了一条系统途径。

英文摘要

Four-dimensional $\mathcal{N}=3$ superconformal field theories (SCFTs) are an exotic class of quantum field theories that admit no conventional Lagrangian description. In this Letter, we construct an $\mathcal{N}=1$ supersymmetric gauge theory and argue that its infrared fixed point exhibits supersymmetry enhancement to an $\mathcal{N}=3$ SCFT. The ultraviolet theory has gauge group $SU(2)^3$, a set of chiral multiplets, and a suitable superpotential. By performing $a$-maximization, we obtain infrared central charges $a=c=5/4$, matching those of the rank-one $\mathcal{N}=3$ $\mathbb{Z}_3$ S-fold SCFT. The ultraviolet gauge-theory description allows us to compute the full superconformal index with four independent fugacities. We then examine various limits of the full index and further propose a closed-form expression for the Macdonald index, reproducing all known results for the proposed $\mathcal{N}=3$ theory. More remarkably, we argue that further gauging the $\mathcal{N}=3$ theory yields the frozen rank-one $\mathcal{N}=2$ SCFT conjectured from Seiberg--Witten geometry, providing strong evidence for its existence and unlocking its previously inaccessible full superconformal index. Our construction thus opens a systematic route to exploring the refined protected structure of these exotic, intrinsically strongly coupled theories using conventional $\mathcal{N}=1$ Lagrangian techniques.

发表机构

• Fudan University（复旦大学）
• Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）

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