Twisted Elliptic Genera

We study the twisted elliptic genera of 2d $(0,4)$ SCFTs associated with the BPS strings in the twisted circle compactification of 6d rank-one $(1,0)$ SCFTs. Such objects can arise when the 6d gauge algebra allows outer automorphism, thus are classified by twisted affine Lie algebras. We study several fascinating aspects of the twisted elliptic genera including 2d localization, twisted elliptic blowup equations, Higgsing and spectral flow symmetry. We derive a recursion formula with respect to the number of strings to exactly compute the twisted elliptic genera. We also investigate the modular bootstrap of twisted one-string elliptic genera and find the modularity of congruence subgroups $\Gamma_1(N)$ naturally appears with possible $N=2,3,4$. Geometrically, our study solves the refined BPS partition of the underlying genus-one fibered Calabi-Yau threefolds with $N$-section.