Elliptic Blowup Equations for 6d SCFTs. IV: Matters

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.

164 pages, 3 figures, 24 tables, typos corrected, refs added

References in corpus (21)

• The Refined Topological Vertex
• 6d Conformal Matter
• Anomaly polynomial of general 6d SCFTs
• The Higgs Mechanism -- Hasse Diagrams for Symplectic Singularities
• LieART 2.0 -- A Mathematica Application for Lie Algebras and Representation Theory
• Fibers add Flavor, Part II: 5d SCFTs, Gauge Theories, and Dualities
• Twisted Circle Compactifications of 6d SCFTs
• Anomaly of strings of 6d $\mathcal{N}=(1,0)$ theories
• E + E $\to$ H
• On certain aspects of string theory/gauge theory correspondence
• Landscaping with fluxes and the E8 Yukawa Point in F-theory
• Equivalence of several descriptions for 6d SCFT
• Rank-3 antisymmetric matter on 5-brane webs
• General Prescription for Global U(1)'s in 6D SCFTs
• 6d strings from new chiral gauge theories
• Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains
• Instantons from Blow-up
• Higgsing and Twisting of 6d $D_N$ gauge theories
• Testing 5d-6d dualities with fractional D-branes
• Modularity from Monodromy
• $\mathcal{N}=(1,0)$ Anomaly Multiplet Relations in Six Dimensions

Cited by in corpus (9)

• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Elliptic Quantum Curves of 6d SO(N) theories
• Topological strings and Wilson loops
• Refined topological vertex with ON-planes
• Blowup Equations for Little Strings
• Affine characters at negative level and elliptic genera of non-critical strings
• Refined BPS numbers on compact Calabi-Yau threefolds from Wilson loops
• Probing Quantum Curves and Transitions in 5d SQFTs via Defects and Blowup Equations
• More on 5d Wilson Loops in Higher-Rank Theories and Blowup Equations