Elliptic blowup equations for 6d SCFTs. Part IV. Matters

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.

Novel wall-crossing behaviour in rank one $$ \mathcal{N} $$ = 2* gauge theory

We study the BPS spectrum of four-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric Yang-Mills theory with gauge group SU(2) and a massive adjoint hypermultiplet, which has an extremely intricate structure with infinite spectrum in all chambers of its Coulomb moduli space, and is not well understood. We build on previous results by employing the BPS quiver description of the spectrum, and explore the qualitative features in detail using numerical techniques. We find novel and unexpected behaviour in the form of wall-crossings involving interactions between BPS particles with negative electric-magnetic pairings, which we interpret in terms of the reverse orderings of the usual wall-crossing formulas for rank one $$ \mathcal{N} $$ N = 2 field theories. This identifies new a priori unrelated states in the spectrum.

A note on discrete dynamical systems in theories of class S

In this note we consider the set of line operators in theories of class S. We show that this set carries the action of a natural discrete dynamical system associated with the BPS spectrum. We discuss several applications of this perspective; the relation with global properties of the theory; the set of constraints imposed on the spectrum generator, in particular for the case of SU(2) $$ \mathcal{N} $$ N = 2*; and the relation between line defects and certain spherical Double Affine Hecke Algebras.

Bootstrapping BPS spectra of 5d/6d field theories

We propose a systematic approach to computing the BPS spectrum of any 5d/6d supersymmetric quantum field theory in Coulomb phases, which admits either gauge theory descriptions or geometric descriptions, based on the Nakajima-Yoshioka’s blowup equations. We provide a significant generalization of the blowup equation approach in terms of both properly quantized magnetic fluxes on the blowup $$ \hat{\mathrm{\mathbb{C}}} $$ ℂ ̂ 2 and the effective prepotential for 5d/6d field theories on the Omega background which is uniquely determined by the Chern-Simons couplings on their Coulomb branches. We employ our method to compute BPS spectra of all rank-1 and rank-2 5d Kaluza-Klein (KK) theories descending from 6d $$ \mathcal{N} $$ N = (1, 0) superconformal field theories (SCFTs) compactified on a circle with/without twist. We also discuss various 5d SCFTs and KK theories of higher ranks, which include a few exotic cases such as new rank-1 and rank-2 5d SCFTs engineered with frozen singularity as well as the 5d SU(3)8 gauge theory currently having neither a brane web nor a smooth shrinkable geometric description. The results serve as non-trivial checks for a large class of non-trivial dualities among 5d theories and also as independent evidences for the existence of certain exotic theories.

Quivers for 3-manifolds: the correspondence, BPS states, and 3d $$ \mathcal{N} $$ = 2 theories

We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as FK or $$ \hat{Z} $$ Z ̂ ). Apart from assigning quivers to complements of T(2,2p+1) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $$ \mathcal{N} $$ N = 2 theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a t-deformation of all objects mentioned above.