Spin chains for ADE quiver theories

Abstract

The spectral problem of four-dimensional superconformal quiver gauge theories can be mapped to one-dimensional spin chains with restricted Hilbert spaces, where the composition of neighbouring spins follows the path algebra of the quiver. To better understand such spin chains, we compute the one-loop planar dilatation operator for the 4d N=2 ADE quiver gauge theories obtained by orbifolding the N=4 Super-Yang-Mills theory and marginally deforming by independently varying the gauge couplings. This extends previous work which was mainly focused on the Z2 quiver. We characterise the general features of the resulting ADE spin-chain models and construct the 2-magnon Bethe ansatz for holomorphic states. We also evaluate, at large N, the N=2 superconformal index of these gauge theories and use it to study their protected spectrum in specific sectors.

Figures (22)

• Figure 1: The Dynkin Diagrams for the affine $\hat{A}, \hat{D}$ and $\hat{E}$ series. Each node has an associated Kac index $n_i$, which denotes the corresponding vector space dimension. When orbifolding by the corresponding subgroup of $\mathrm{SU}(2)$, these Dynkin diagrams become the quiver diagrams of the orbifolded theories, with each node associated to a $\mathrm{SU}(n_iN)$ gauge group.
• Figure 2: For $L=2$, the $1/N$ part of the adjoint propagators gives a leading contribution in the large $N$ limit. There are two more diagrams, for $\bar{Q}Q$ and $\bar{Q}\bar{Q}$.
• Figure 3: The $Z_3$ quiver. All the gauge groups are $\mathrm{SU}(N)$. For clarity, we only indicate the holomorphic fields. The arrows for the corresponding conjugate fields are reversed.
• Figure 4: The $L=2$ neutral/$L=3$ holomorphic $\mathbb{Z}_3$ spin chain spectrum for the case $\kappa_1=1-2k,\kappa_2=1-k,\kappa_3=1$. Notice that all orbifold-point degenerate twisted-sector states split under this deformation. We do not extend the plot beyond $k=0.5$ as our Hamiltonian is not applicable to the case of gauge groups becoming global.
• Figure 5: The $L=2$ neutral/$L=3$ holomorphic $\mathbb{Z}_3$ holomorphic spin chain spectrum for the deformation $\kappa_1=\kappa_2=1-k,\kappa_3=1$. The theory approaches SCQCD as $k\rightarrow 1$, where we see three initially non-protected states approaching $E=0$.