We study twisted circle compactification of 6d (2, 0) SCFTs to 5d \( \mathcal{N} \) = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ₀(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

我们研究了 6d (2 , 0) SCFT 到 5d \( \mathcal{N} \) = 2 具有非简单排列规范群的超对称规范理论的扭曲圆紧缩。我们为 BPS 分区函数提供了两种互补的方法，分别反映了 5d 和 6d 的观点。第一个是基于瞬子配分函数的爆破方程，特别是我们从中明确地确定了所有简单李群的单瞬子贡献。第二种是基于模块化引导程序，并且我们提出了下全等子组Γ该变换扭曲椭圆属新颖的模块化拟设₀（Ñ SL）（2 ，ℤ)。我们推测一属的精制 Gopakumar-Vafa 不变量的消失界是纤维化的 Calabi-Yau 三重，根据它可以递归地确定扭曲椭圆属。我们使用我们的结果来获得 6d Cardy 公式并找到所有简单李群的通用行为。此外，一旦考虑了紧圆的归一化，Cardy 公式在扭曲下保持不变。