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Abstract

Given the recent geometrical classification of 6d $(1,0)$ SCFTs, a major
question is how to compute for this large class their elliptic genera. The
latter encode the refined BPS spectrum of the SCFTs, which determines geometric
invariants of the associated elliptic non-compact Calabi-Yau threefolds. In
this paper we establish for all 6d $(1,0)$ SCFTs in the atomic classification
blowup equations that fix these elliptic genera to large extent. The latter
fall into two types: the unity- and the vanishing blowup equations. For almost
all rank one theories, we find unity blowup equations which determine the
elliptic genera completely. We develop several techniques to compute elliptic
genera and BPS invariants from the blowup equations, including a recursion
formula with respect to the number of strings, a Weyl orbit expansion, a
refined BPS expansion and an $\epsilon_1,\epsilon_2$ expansion. For higher-rank
theories, we propose a gluing rule to obtain all their blowup equations based
on those of rank one theories. For example, we explicitly give the elliptic
blowup equations for the three higher-rank non-Higgsable clusters, ADE chain of
$-2$ curves and conformal matter theories. We also give the toric construction
for many elliptic non-compact Calabi-Yau threefolds which engineer 6d $(1,0)$
SCFTs with various matter representations.