Complete Graphs, Hilbert Series, and the Higgs branch of the 4d N=2 (An,Am)(A_n,A_m) SCFT's

The strongly interacting 4d N=2 SCFT's of type $(A_n,A_m)$ are the simplest examples of models in the $(G,G^\prime)$ class introduced by Cecotti, Neitzke, and Vafa in arXiv:1006.3435. These systems have a known 3d N=4 mirror only if $h(A_n)$ divides $h(A_m)$, where $h$ is the Coxeter number. By 4d/2d correspondence, we show that in this case these systems have a nontrivial global flavor symmetry group and, therefore, a non-trivial Higgs branch. As an application of the methods of arXiv:1309.2657, we then compute the refined Hilbert series of the Coulomb branch of the 3d mirror for the simplest models in the series. This equals the refined Hilbert series of the Higgs branch of the $(A_n,A_m)$ SCFT, providing interesting information about the Higgs branch of these non-lagrangian theories.Comment: 20 page

About the Absence of Exotics and the Coulomb Branch Formula

The absence of exotics is a conjectural property of the spectrum of BPS states of four--dimensional $\mathcal{N}=2$ supersymmetric QFT's. In this letter we revisit the precise statement of this conjecture, and develop a general strategy that, if applicable, entails the absence of exotic BPS states. Our method is based on the Coulomb branch formula and on quiver mutations. In particular, we obtain the absence of exotic BPS states for all pure SYM theories with simple simply--laced gauge group $G$, and, as a corollary, of infinitely many other lagrangian $\mathcal{N}=2$ theories

Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2

We study compactification of 6 dimensional (1,0) theories on T^2. We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d N=2 geometry for a large number of the (1,0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N=2 SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2,Z) duality symmetry inherited from global diffeomorphisms of the T^2. This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T^2. Among the resulting 4d N=2 CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class S with punctures from toroidal compactification of (1,0) SCFTs where the curve of the class S theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class S theories with no punctures on arbitrary genus Riemann surface.Comment: 58 pages, 8 figures, v2: references added, typos fixed, table 2 update

The ALE Partition Functions of M-String Orbifolds

The ALE partition functions of a 6d (1,0) SCFT are interesting observables which are able to detect the global structure of the SCFT. They are defined to be the equivariant partition functions of the SCFT on a background with the topology of a two-dimensional torus times an ALE singularity. In this work, we compute the ALE partition functions of M-string orbifold SCFTs, extending our previous results for the M-string SCFTs. Via geometric engineering, our results about ALE partition functions are connected to the theory of higher-rank Donaldson-Thomas invariants for resolutions of elliptic Calabi-Yau threefold singularities. We predict that their generating functions satisfy interesting modular properties. The partition functions receive contributions from BPS strings probing the ALE singularity, whose worldsheet theories we determine via a chain of string dualities. For this class of backgrounds the BPS strings' worldsheet theories become relative field theories that are sensitive to discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. A novel feature we observe in the case of M-string orbifold SCFTs, which does not arise for the M-string SCFT, is the existence of frozen BPS strings which are pinned at the orbifold singularity and carry fractional instanton charge with respect to the 6d gauge fields.Comment: 69 page

The ALE Partition Functions of M-Strings

We compute the equivariant partition function of the six-dimensional M-string SCFTs on a background with the topology of a product of a two-dimensional torus and an ALE singularity. We determine the result by exploiting BPS strings probing the singularity, whose worldvolume theories we determine via a chain of string dualities. A distinguished feature we observe is that for this class of background the BPS strings' worldsheet theories become relative field theories that are sensitive to finer discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. We test our proposal against a conjectural 6d N = (1,0) generalization of the Nekrasov master formula, as well as against known results on ALE partition functions in four dimensions.Comment: 44 page

Global structures from the infrared

Quantum field theories with identical local dynamics can admit different choices of global structure, leading to different partition functions and spectra of extended operators. Such choices can be reformulated in terms of a topological field theory in one dimension higher, the symmetry TFT. In this paper we show that this TFT can be reconstructed from a careful analysis of the infrared Coulomb-like phases. In particular, the TFT matches between the UV and the IR. This provides a purely field theoretical counterpart of several recent results obtained via geometric engineering in various string/M/F theory setups for theories in four and five dimensions that we confirm and extend.Comment: 26 pages, 4 figure

On the Defect Group of a 6D SCFT

We use the F-theory realization of 6D superconformal field theories (SCFTs) to study the corresponding spectrum of stringlike, i.e. surface defects. On the tensor branch, all of the stringlike excitations pick up a finite tension, and there is a corresponding lattice of string charges, as well as a dual lattice of charges for the surface defects. The defect group is data intrinsic to the SCFT and measures the surface defect charges which are not screened by dynamical strings. When non-trivial, it indicates that the associated theory has a partition vector rather than a partition function. We compute the defect group for all known 6D SCFTs, and find that it is just the abelianization of the discrete subgroup of U(2) which appears in the classification of 6D SCFTs realized in F-theory. We also explain how the defect group specifies defining data in the compactification of a (1,0) SCFT.Comment: 24 page