  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.

High Energy Physics - Theory