Instanton counting in Class $\mathcal{S}_k$

Abstract

We compute the instanton partition functions of $\mathcal{N}=1$ SCFTs in class $\mathcal{S}_k$. We obtain this result via orbifolding Dp/D(p-4) brane systems and calculating the partition function of the supersymmetric gauge theory on the worldvolume of $K$ D(p-4) branes. Starting with D5/D1 setups probing a $\mathbb{Z}_\ell\times \mathbb{Z}_k$ orbifold singularity we obtain the $K$ instanton partition functions of 6d $(1,0)$ theories on $\mathbb{R}^4 \times T^2$ in the presence of orbifold defects on $T^2$ via computing the 2d superconformal index of the worldvolume theory on $K$ D1 branes wrapping the $T^2$. We then reduce our results to the 5d and to the 4d instanton partition functions. For $k=1$ we check that we reproduce the known elliptic, trigonometric and rational Nekrasov partition functions. Finally, we show that the instanton partition functions of $SU(N)$ quivers in class $\mathcal{S}_k$ can be obtained from the class $\mathcal{S}$ mother theory partition functions with $SU(kN)$ gauge factors via imposing the `orbifold condition' $a_{\mathcal{A}} \rightarrow a_A e^{2πi j/k}$ with $\mathcal{A}=jA$ and $A=1,\dots, N$, $j=1,\dots, k$ on the Coulomb moduli and the mass parameters.

Figures (7)

• Figure 1: The quiver diagram in $\mathcal{N}=1$ notation with $\ell=5$, $k=3$. Circular nodes denote vector multiplets and coloured arrows denote chiral multiplets. Blue lines denote $\Phi^1_{(n,i)}$, green $\Phi^2_{(n,i)}$ and red $\Phi^3_{(n,i)}$. The quiver should be periodically identified in both '$\ell$' and '$k$' directions, with gluing indicated by the black arrowed lines, such that it has the topology of a torus.
• Figure 2: Left: Schematic overview for $k=1$ of two alternate ways to obtain the 4d $\mathcal{N}=2$$\tilde{A}_{\ell-1}$ circular quivers with $SU(N)^\ell$ gauge group in class $\mathcal{S}$ from compactifications of 6d SCFTs. Right: A schematic overview of the $k>1$ generalisations of compactifications of 6d SCFTs. The resulting 4d SCFTs are $\mathcal{N}=1$$\tilde{A}_{\ell-1}\times \tilde{A}_{k-1}$ torodial quivers in class $\mathcal{S}_k$ with gauge group $SU(N)^{\ell k}$.
• Figure 3: 5d circular (necklace) quiver $\mathcal{N}_{N,\ell}$ for $\ell=5$. Circular nodes denote $\mathcal{N}=1$ vector multiplets and solid lines connecting them denote bifundamental $\mathcal{N}=1$ hypermultiplets. Circle reduction of $\mathcal{N}_{N,\ell}$ results in 4d $\mathcal{N}=2$ circular $\tilde{A}_{\ell-1}$ quiver with the same structure.
• Figure 4: The $\mathcal{N}=(4,4)$ 2d quiver of the gauge theory on $K$ D1 branes in the presence of $N$ D5s. Using $\mathcal{N}=(4,4)$ notation, solid lines denote hypermultiplets, while the circular node denotes the $U(K)$ vector multiplet.
• Figure 5: The 5d $\mathcal{N}_{N,\ell}$ quiver for $\ell=5$ after taking the decoupling limit obtained by sending one of the couplings to zero. Similarly, circle reduction in 4d $\mathcal{N}=2$ linear $A_{\ell-1}$ quivers.