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Register a new userTowards a classification of rank $r$ $mathcal{N}=2$ SCFTs Part II: special Kahler stratification of the Coulomb branch
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We study the stratification of the singular locus of four dimensional $mathcal{N}=2$ Coulomb branches. We present a set of self-consistency conditions on this stratification which can be used to extend the classification of scale-invariant rank 1 Coulomb branch geometries to two complex dimensions, and beyond. The calculational simplicity of the arguments presented here stems from the fact that the main ingredients needed -- the rank 1 deformation patterns and the pattern of inclusions of rank 2 strata -- are discrete topological data which satisfy strong self-consistency conditions through their relationship to the central charges of the SCFT. This relationship of the stratification data to the central charges is used here, but is derived and explained in a companion paper by one of the authors. We illustrate the use of these conditions by re-analyzing many previously-known examples of rank 2 SCFTs, and also by finding examples of new theories. The power of these conditions stems from the fact that for Coulomb branch stratifications a conjecturally complete list of physically allowed elementary slices is known. By contrast, constraining the possible elementary slices of symplectic singularities relevant for Higgs branch stratifications remains an open problem.
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We initiate a systematic analysis of moduli spaces of vacua of four dimensional $mathcal{N}=3$ SCFTs. Our analysis is based on the one hand on the properties of $mathcal{N}=3$ chiral rings --- which we review in detail and contrast with chiral rings of theories with less supersymmetry --- and on the other hand on constraints coming from low-energy supersymmetry. This leads us to introduce a new type of geometric structure, which characterizes $mathcal{N}=3$ SCFT moduli spaces, and that we call $triple special Kahler$ (TSK). A rank-$n$ TSK moduli space has complex dimension $3n$, and is singular at complex co-dimension 3 subspaces where charged states become massless. The structure of singularities defines a stratification of the TSK space in terms of lower-dimensional TSK manifolds.
We construct several novel examples of 3d $mathcal{N}=2$ models whose free energy scales as $N^{3/2}$ at large $N$. This is the first step towards the identification of field theories with an M-theory dual. Furthermore, we match the volumes extracted from the free energy with the ones computed from the Hilbert series. We perform a similar analysis for the 4d parents of the 3d models, matching the volume extracted from the $a$ conformal anomaly to that obtained from the Hilbert series. For some of the 4d models, we show the existence of a Sasaki-Einstein metric on the internal space of the candidate type IIB gravity dual.
S-folds are a non-perturbative generalization of orientifold 3-planes which figure prominently in the construction of 4D $mathcal{N} = 3$ SCFTs and have also recently been used to realize examples of 4D $mathcal{N} = 2$ SCFTs. In this paper we develop a general procedure for reading off the flavor symmetry experienced by D3-branes probing 7-branes in the presence of an S-fold. We develop an S-fold generalization of orientifold projection which applies to non-perturbative string junctions. This procedure leads to a different 4D flavor symmetry algebra depending on whether the S-fold supports discrete torsion. We also show that this same procedure allows us to read off admissible representations of the flavor symmetry in the associated 4D $mathcal{N} = 2$ SCFTs. Furthermore this provides a prescription for how to define F-theory in the presence of S-folds with discrete torsion.
We classify orbifold geometries which can be interpreted as moduli spaces of four-dimensional $mathcal{N}geq 3$ superconformal field theories up to rank 2 (complex dimension 6). The large majority of the geometries we find correspond to moduli spaces of known theories or discretely gauged version of them. Remarkably, we find 6 geometries which are not realized by any known theory, of which 3 have an $mathcal{N}=2$ Coulomb branch slice with a non-freely generated coordinate ring, suggesting the existence of new, exotic $mathcal{N}=3$ theories.
We study a set of four-dimensional $mathcal{N}=2$ superconformal field theories (SCFTs) $widehat{Gamma}(G)$ labeled by a pair of simply-laced Lie groups $Gamma$ and $G$. They are constructed out of gauging a number of $mathcal{D}_p(G)$ and $(G, G)$ conformal matter SCFTs; therefore they do not have Lagrangian descriptions in general. For $Gamma = D_4, E_6, E_7, E_8$ and some special choices of $G$, the resulting theories have identical central charges $(a=c)$ without taking any large $N$ limit. Moreover, we find that the Schur indices for such theories can be written in terms of that of $mathcal{N}=4$ super Yang-Mills theory upon rescaling fugacities. Especially, we find that the Schur index of $widehat{D}_4(SU(N))$ theory for $N$ odd is written in terms of MacMahons generalized sum-of-divisor function, which is quasi-modular. For generic choices of $Gamma$ and $G$, it can be regarded as a generalization of the affine quiver gauge theory obtained from $D3$-branes probing an ALE singularity of type $Gamma$. We also comment on a tantalizing connection regarding the theories labeled by $Gamma$ in the Deligne-Cvitanovic exceptional series.
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