VOAs and Rank-Two Instanton SCFTs

Abstract

We analyze the $$\mathcal {N}=2$$ N = 2 superconformal field theories that arise when a pair of D3-branes probe an F-theory 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 fourth-order 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 higher-rank cases.

Free field realizations from the Higgs branch

“…This is actually a more general phenomenon: the associated graded has more generators and more relations compared to the original VOA. An interesting example of this phenomenon that does not involve the stress tensor has been found recently in [30] and further studied in [31]. 34 We hope (and expect) our free field realizations to provide a simplified description of the associated graded algebra.…”

“…The second important non-trivial consistency check of our proposal is that the R-filtration arising from free fields assigns weight R = 1 to the stress tensor. 31 Recall that in the examples we have analyzed, the stress tensor has taken the form 32…”

Schur correlation functions on S3 × S1

“…To preserve the supercharges Q i , the defects should be chosen perpendicular to the plane in which the chiral algebra is defined. Their presence gives rise to a module of the vertex operator algebra [34,[55][56][57][58][59]. Similarly, we can dress our localization computation with defects.…”

Superconformal surfaces in four dimensions

“…It thus counts the same operators as those contributing to the chiral algebra, but refines the counting by the additional fugacity t. Recovering this information from the chiral algebra itself is an open question, as the R grading of the four-dimensional SCFT is lost when passing to the chiral algebra, and so the refinment is by a Cartan not preserved by the chiral algebra. See, however, [128][129][130][131][132] for proposals on recovering the full Macdonald index from the chiral algebra.…”