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 SCFTsmore »
Towards the classification of rankr $$ \mathcal{N} $$ = 2 SCFTs. Part I. Twisted partition function and central charge formulae
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.
 Award ID(s):
 1914679
 Publication Date:
 NSFPAR ID:
 10339023
 Journal Name:
 Journal of High Energy Physics
 Volume:
 2020
 Issue:
 12
 ISSN:
 10298479
 Sponsoring Org:
 National Science Foundation
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