A bstract We derive explicit formulae to compute the a and c central charges of four dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) directly from Coulomb branch related quantities. The formulae apply at arbitrary rank. We also discover general properties of the lowenergy limit behavior of the flavor symmetry of $$ \mathcal{N} $$ N = 2 SCFTs which culminate with our $$ \mathcal{N} $$ N = 2 UVIR simple flavor condition . This is done by determining precisely the relation between the integrand of the partition function of the topologically twisted version of the 4d $$ \mathcal{N} $$ N = 2 SCFTs and the singular locus of their Coulomb branches. The techniques developed here are extensively applied to many rank2 SCFTs, including new ones, in a companion paper. This manuscript is dedicated to the memory of Rayshard Brooks, George Floyd, Breonna Taylor and the countless black lives taken by US police forces and still awaiting justice. Our hearts are with our colleagues of color who suffer daily the consequences of this racist world.
Needles in a haystack. An algorithmic approach to the classification of 4d $$ \mathcal{N} $$ = 2 SCFTs
A bstract There is a wellknown map from 4d $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds to the VOA vacuum character, and must be a solution with integral coefficients of a modular differential equation. This suggests a classification program for 4d $$ \mathcal{N} $$ N = 2 SCFTs that starts with modular differential equations and proceeds by imposing all known constraints that follow from the 4d → 2d map. This program becomes fully algorithmic once one specifies the order of the modular differential equation and the rank (complex dimension of the Coulomb branch) of the $$ \mathcal{N} $$ N = 2 theory. As a proof of concept, we apply the algorithm to the study of ranktwo $$ \mathcal{N} $$ N = 2 SCFTs whose Schur indices satisfy a fourthorder untwisted modular differential equation. Scanning over a large number of putative cases, only 15 satisfy all of the constraints imposed by our algorithm, six of which correspond to known 4d SCFTs. More sophisticated constraints can be used to argue against the existence of the remaining nine cases. Altogether, this indicates that our knowledge of such ranktwo SCFTs more »
 Award ID(s):
 1915093
 Publication Date:
 NSFPAR ID:
 10376818
 Journal Name:
 Journal of High Energy Physics
 Volume:
 2022
 Issue:
 3
 ISSN:
 10298479
 Sponsoring Org:
 National Science Foundation
More Like this


A bstract We study the stratification of the singular locus of four dimensional $$ \mathcal{N} $$ N = 2 Coulomb branches. We present a set of selfconsistency conditions on this stratification which can be used to extend the classification of scaleinvariant 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 selfconsistency 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 [1] by one of the authors. We illustrate the use of these conditions by reanalyzing many previouslyknown 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.

We analyze the N = 2 superconformal field theories that arise when a pair of D3branes probe an Ftheory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourthorder linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number.We comment briefly on expectations for the still higherrank cases.

A bstract We introduce a systematic approach to constructing $$ \mathcal{N} $$ N = 1 Lagrangians for a class of interacting $$ \mathcal{N} $$ N = 2 SCFTs. We analyse in detail the simplest case of the construction, arising from placing branes at an orientifolded ℂ 2 / ℤ 2 singularity. In this way we obtain Lagrangian descriptions for all the R 2 ,k theories. The rank one theories in this class are the E 6 MinahanNemeschansky theory and the C 2 × U(1) ArgyresWittig theory. The Lagrangians that arise from our brane construction manifestly exhibit either the entire expected flavour symmetry group of the SCFT (for even k ) or a fullrank subgroup thereof (for odd k ), so we can compute the full superconformal index of the $$ \mathcal{N} $$ N = 2 SCFTs, and also systematically identify the Higgsings associated to partial closing of punctures.

A bstract We extend the anomaly inflow methods developed in Mtheory to SCFTs engineered via D3branes in type IIB. We show that the ’t Hooft anomalies of such SCFTs can be computed systematically from their geometric definition. Our procedure is tested in several 4d examples and applied to 2d theories obtained by wrapping D3branes on a Riemann surface. In particular, we show how to analyze halfBPS regular punctures for 4d $$ \mathcal{N} $$ N = 4 SYM on a Riemann surface. We discuss generalizations of this formalism to type IIB configurations with F 3 , H 3 fluxes, as well as to Ftheory setups.