Abstract
We analyze the mathcal {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.
Highlights
The possible choices of singularity follows the Kodaira classification, with the resulting interacting theories being labeled H0, H1, H2, D4, E6, E7, or E8. Their flavor symmetries include as simple factors the corresponding simple Lie groups
We aim to study the higher-rank generalizations of these SCFTs, mainly from the viewpoint of the associated vertex operator algebra introduced in [19]
A standard entry of the SCFT/vertex operator algebra (VOA) correspondence states that the moment map operators give rise to affine currents in the associated VOA
Summary
Branes probing a singular fiber of an elliptic K 3 surface in F-theory on which the dilaton is constant. The one-instanton moduli spaces of the Deligne–Cvitanovicseries of simple Lie algebras have an economical description Their coordinate rings are generated by adjoint-valued moment maps μg subject to the Joseph relations [29]. In the rank two theories, the Higgs branch chiral ring has as generators the moment maps μsu(2) and μg, which transform in the (1, 1) and (0, Adj) representations of su(2) × g, respectively, along with an additional multiplet of generators ω with SU (2)R charge This collection of generators can, for example, be read off from the two-instanton Hilbert series as computed in [31,32,33,34]. The numerical constant of proportionality in (2.10) can be taken to define the normalization of ω
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