Higgs Branch and VOA of 4d $\mathcal{N}=2$ SCFTs from IIB

Abstract

We study the Higgs branch and associated vertex operator algebra (VOA) of 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) from the geometric engineering of IIB superstring on canonical threefold singularities. For terminal singularities, we explain how to derive the 4d Higgs branch from their small resolution. We also investigate singularities with compact 4-cycles in their crepant resolution, and discuss different ways to compute their Higgs branch. Using a symplectic duality argument, we propose the first examples of 4d $\mathcal{N}=2$ SCFTs with the E-type Kleinian singularities as their Higgs branches, and conjecture their associated VOA to be affine E-type W-algebra. Many new VOAs with no known W-algebra descriptions are found, with conjectured associated varieties. We investigate the singularities associated with lisse VOAs and propose predictions for the BPS quivers of $D_N^N[k]$ and $E_7^{14}[k]$ from the perspective of deformed singularities. We further analyze the structure of the Schur index using the Coulomb branch IR formula, derive the expressions for the Schur index corresponding to these two classes of singularities, and illustrate, in a general setting, how the Schur index is determined by the BPS quiver.

Figures (4)

• Figure 1: The inversion on the Coulomb branch Hasse diagram of an affine $E_8$ type unitary quiver EQ, leading to the Higgs branch Hasse diagram of EQ. One can conclude that the Higgs branch of EQ is a $\mathbb{C}^2/\Gamma_{E_8}$ Kleinian singularity.
• Figure 2: The inversion on the Coulomb branch Hasse diagram of a more complicated unitary quiver EQ, leading to the Higgs branch Hasse diagram of EQ.
• Figure 3: Consider two related singularities $\mathbf{X}=\mathrm{sing}(E_n)$ listed in Table \ref{['t:SingEn']} and $\mathbf{X}'=$ local $dP_n$ singularity, we can define 4d $\mathcal{N}=2$ SCFTs ${\mathcal{T}_{\mathbf{X}}^{\rm 4d}}$, $\mathcal{T}^{4\mathrm{d}}_{\mathbf{X}'}$ and 5d SCFTs ${\mathcal{T}_{\mathbf{X}}^{\rm 5d}}$, $\mathcal{T}^{5\mathrm{d}}_{\mathbf{X}'}$. Here we give the relation between their Higgs branches, via dimensional reduction, inversion and gauging.
• Figure :