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 ϵ1, ϵ2 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.
Highlights
In the absence of a Lagrangian description, six dimensional theories with (2, 0) and (1, 0) superconformal symmetries are generally difficult to study
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 1, 2 expansion
We demonstrate that for almost all rank one theories, with the exception of 12 theories, the BPS spectrum associated to the non-critical strings or equivalently the elliptic genera can be computed from the elliptic blowup equations
Summary
In the absence of a Lagrangian description, six dimensional theories with (2, 0) and (1, 0) superconformal symmetries are generally difficult to study. The purpose of this paper is to generalise the last approach to generic elliptic non-compact Calabi-Yau spaces, which geometrically engineer six dimensional superconformal field theories with geometrically realisable gauge symmetries and matters in generic flavour representations. We write down elliptic blowup equations for all the rank one 6d superconformal theories with one tensor multiplet These theories are geometrically realised as non-compact elliptic fibrations with an isolated curve of self intersection −n in the base. In [19] a generalisation of the K-theoretic blowup equations of Nakajima and Yoshioka was proposed that calculates the refined BPS invariants for those non-compact toric Calabi-Yau spaces that do not by themselves define gauge theories like local P2. The BPS strings become tensionless precisely at the origin of the tensor branch where all compact base curves are blown down
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