Twisted 6d $(2,0)$ SCFTs on a Circle

TL;DR

This paper investigates twisted circle compactifications of 6d (2,0) SCFTs to 5d gauge theories with non-simply-laced groups, providing new methods to compute BPS partition functions and revealing universal behaviors.

Contribution

It introduces two approaches—blowup equations and modular bootstrap—for calculating twisted elliptic genera and BPS invariants, advancing understanding of twisted 6d SCFT compactifications.

Findings

Explicit one-instanton contributions for all simple Lie groups.

A novel modular ansatz for twisted elliptic genera under $ ext{SL}(2, ext{Z})$ subgroups.

Universal behavior of Cardy formulas across all simple Lie groups.

Abstract

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 $\Gamma_0(N)$ of $\text{SL}(2,\mathbb{Z})$. 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.