Abstract
We propose novel functional equations for the BPS partition functions of 6d (1, 0) SCFTs, which can be regarded as an elliptic version of Göttsche-Nakajima-Yoshioka’s K-theoretic blowup equations. From the viewpoint of geometric engineering, these are the generalized blowup equations for refined topological strings on certain local elliptic CalabiYau threefolds. We derive recursion formulas for elliptic genera of self-dual strings on the tensor branch from these functional equations and in this way obtain a universal approach for determining refined BPS invariants. As examples, we study in detail the minimal 6d SCFTs with SU(3) and SO(8) gauge symmetry. In companion papers, we will study the elliptic blowup equations for all other non-Higgsable clusters.
Highlights
Quantum field theories with the highest amount of symmetries, namely supersymmetry as well as conformal symmetry, in the highest possible dimension are 6d superconformal field theories
The geometry of the base B of the Calabi-Yau manifold directly translates into the tensor multiplet sector of the 6d SCFTs where the number of tensor multiplets is given by the dimension of H1,1(B, Z) and the intersection form on B gives the couplings of these tensor multiplets to each other
We demonstrate the validity of the generalized blowup equations through the already well-studied cases of n = 3 and n = 4 minimal SCFTs in the present paper, and illustrate their power by computing the elliptic genera as well as the BPS invariants with them
Summary
Quantum field theories with the highest amount of symmetries, namely supersymmetry as well as conformal symmetry, in the highest possible dimension are 6d superconformal field theories. As we will see in the subsection, the form of the generalized blowup equations is simple and universal, and it does not put any constraints on the target space of the topological string except that it has to be non-compact to allow for U(1) isometry crucial for the preservation of supersymmetry in the presence of the Omega background This naturally poses the question of the validity of the generalized blowup equations beyond 5d SU(N ) SYM engineered by the XN,m geometries. What is fascinating is that the generalized blowup equations may even be valid for 6d SCFTs as the topological string theory on non-compact elliptic Calabi-Yau threefolds used in F-theory compactifications computes precisely Z6d of these 6d SCFTs on the Omega background. In the subsection we give a quick overview of the generalized blowup equations
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