We propose a new exact quantization condition for a class of quantum mechanical systems derived from local toric Calabi-Yau threefolds. Our proposal includes all contributions to the energy spectrum which are nonperturbative in the Planck constant, and is much simpler than the available quantization condition in the literature. We check that our proposal is consistent with previous works and implies nontrivial relations among the topological Gopakumar-Vafa invariants of the toric Calabi-Yau geometries. Together with the recent developments, our proposal opens a new avenue in the long investigations at the interface of geometry, topology and quantum mechanics.

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 Calabi-Yau 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.

Göttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional N = 1 supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on SU (N ) geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the r fields introduced in our previous paper. These blowup equations are in fact the functional equations for the partition function and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other t han the refined topological vertex in the A-model and the refined holomorphic anomaly equations in the B-model. We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity r fields from the polynomial part of refined topological string at large radius point. We also find that certain form of blowup equations exist at generic loci of the moduli space. To Sheldon Katz on his 60th anniversary arXiv:1711.09884v2 [hep-th] 9 Nov 2018 7 Blowup equations at generic points of moduli space 70 7.1 Modular transformation 70 7.2 Conifold point 70 7.3 Orbifold point 73 -i -8 Outlook 75 References 79

The building blocks of 6d (1, 0) SCFTs include certain rank one theories with gauge group G = SU (3), SO(8), F 4 , E 6,7,8 . In this paper, we propose a universal recursion formula for the elliptic genera of all such theories. This formula is solved from the elliptic blowup equations introduced in our previous paper. We explicitly compute the elliptic genera and refined BPS invariants, which recover all previous results from topological string theory, modular bootstrap, Hilbert series, 2d quiver gauge theories and 4d N = 2 superconformal H G theories. We also observe an intriguing relation between the k-string elliptic genus and the Schur indices of rank k H G SCFTs, as a generalization of Lockhart-Zotto's conjecture at the rank one cases. In a subsequent paper, we deal with all other non-Higgsable clusters with matters. arXiv:1905.00864v2 [hep-th] 12 May 2019 E Relation with modular ansatz 85 F Elliptic genera 89 G Refined BPS invariants 971 IntroductionSix is the highest dimension in which representation theory allows for interacting superconformal quantum theories [1]. Limits of non-perturbative string theory compactifications [2] and in particular the decoupling of gravity in F-theory compactifications to 6d provided the first examples [3,4] and lead recently to a complete classification of geometrically engineered 6d superconformal quantum field theories [5][6][7]. Such a classification in 6d is highly desirable, as it might lead by further compactifications, to an exhaustive classification of superconformal theories.The 6d geometry is the one of an -in general desingularised -elliptic fibration with a contractable configuration of desingularised elliptic surfaces fibred over a configuration of curves in the base. In the decoupling limit the volume outside of the configuration of elliptic surfaces is scaled to infinite size, leaving us with an, in general reducible, configuration of complex desingularised elliptic surfaces that can be contracted within a non-compact Calabi-Yau threefold. Because compact components can be contracted such geometries are sometimes called local Calabi-Yau spaces. We will call the above specific ones for short elliptic non-compact Calabi-Yau geometries X and describe them in more detail in section 2.1.The full topological string partition function on these elliptic non-compact CY geometries has received much attention as it contains important information about protected states of the 6d superconformal theories [3,8]. Solving the topological string partition function on compact Calabi-Yau manifolds is currently an open problem. On non-compact Calabi-Yau spaces with an U (1) R isometry a refined topological string partition function Z(t, 1 , 2 ), which depends on the Kähler parameters t and two Ω background parameters 1 , 2 is defined as generating function of refined stable pair invariants. 1 The refinement of the stable pair invariants [9,10] and the relation to the refined BPS invariants N β j l ,jr ∈ N was given in [11,12]. Here β ∈ H 2 (X, Z) is the degree and the half integer...

We establish the elliptic blowup equations for E-strings and M-strings and solve elliptic genera and refined BPS invariants from them. Such elliptic blowup equations can be derived from a path integral interpretation. We provide toric hypersurface construction for the Calabi-Yau geometries of M-strings and those of E-strings with up to three mass parameters turned on, as well as an approach to derive the perturbative prepotential directly from the local description of the Calabi-Yau threefolds. We also demonstrate how to systematically obtain blowup equations for all rank one 5d SCFTs from E-string by blow-down operations. Finally, we present blowup equations for E-M and M string chains.