M5 branes on ADE singularities: BPS spectrum and partition functions

Abstract

The dynamics of a stack of M5 branes probing a transverse multi-centered Taub-NUT space are described by a class of 6d $\mathcal{N}=(1,0)$ superconformal field theories known as the M-string orbifold SCFTs. We determine the equivariant partition functions for this class of theories on a geometric background of type $T^2\times\mathbb{C}^2/Γ$, where $Γ\in\{\mathcal{C}_N,\mathcal{Q}_N, \mathcal{T},\mathcal{O},\mathcal{I}\}$ is an arbitrary finite subgroup of $SU(2)$. The partition functions are built out of contributions from BPS strings as well as BPS particles that arise upon putting the 6d theory on a circle. We find that BPS particle contributions can be expressed in terms of $Γ$-covariant Hilbert series which count holomorphic sections of vector bundles on the orbifold singularity with monodromy specified by an irreducible representation of $Γ$. The BPS string contributions, on the other hand, are given by the elliptic genera of 2d $\mathcal{N}=(0,4)$ $Γ$-dressed quiver gauge theories, obtained by stacking Kronheimer-Nakajima quivers of type $Γ$ between interfaces that support current algebras for the McKay dual affine Lie algebra $\widehat{\mathfrak{g}}$. We obtain explicit expressions for the elliptic genera of arbitrary BPS string configurations corresponding to fractional instanton strings on $\mathbb{C}^2/Γ$, and for the case of star-shaped quivers of type $Γ\in\{\mathcal{Q}_4,\mathcal{T},\mathcal{O},\mathcal{I}\}$ we give a prescription to compute the elliptic genera by gluing 2d analogues of Gaiotto and Witten's $T[SU(N)]$ theories.

Figures (25)

• Figure 1: The 5d $\mathcal{N}=1$ quiver gauge theory corresponding to the M-string orbifold SCFT $\mathcal{T}^{6d}_{r,W}$.
• Figure 2: Correspondence between discrete subgroups $\Gamma$ of $SU(2)$, simply-laced Lie algebras $\mathfrak{g}$, and affine Dynkin diagrams. The label $(\underline{r}_j,\boldsymbol{R}_j)$ of the $j$-th node in the Dynkin diagram indicates the corresponding irreducible representations of $\Gamma$ and $\mathfrak{g}$.
• Figure 3: $3d$$\mathcal{N}=4$ Kronheimer--Nakajima quiver $\mathcal{KN}^{\mathcal{I}}_{\kappa}$.
• Figure 4: The 2d $(0,4)$ M-string quiver for $\Gamma=\mathcal{I}$. Black solid lines are the twisted hypermultiplets $X_{ij}^{(a)}$ or $W^{(a)}$; purple solid lines are the hypermultiplets $Y^{(a)}_j$; green dashed lines are the Fermi multiplets $\Psi^{(a)}_{ij}$ and $\widetilde{\Psi}^{(a)}_{ij}$; black dashed lines are the Fermi multiplets $\Sigma^{(a)}$ and $\Theta^{(a)}$. Vertical blue lines represent the interfaces $\boldsymbol{NS}^\Gamma$ that support the $\widehat{\mathfrak{g}}_1$ current algebra.
• Figure 5: The $\mathcal{C}_N$-type Kronheimer--Nakajima quiver corresponding to theory $T_{\vb*{\rho}}(SU(N))$.