Finite size corrections and integrability of $\mathcal{N}=2$ SYM and DLCQ strings on a pp-wave

TL;DR

This work studies integrability in a planar N=2 SU(N)^M quiver theory realized as a Z_M orbifold of N=4, focusing on finite-size (1/M) corrections to two-impurity dilatation spectra and their DLCQ pp-wave string dual. It employs a twisted Beisert-Dippel-Staudacher approach and coordinate/perturbative Bethe ansatz techniques to compute corrections up to three loops, finding agreement with the string results at one and two loops but a known three-loop mismatch that is resolved by a universal dressing factor, mirroring the N=4 case. The results support the persistence of integrability in the MRV limit and establish a precise gauge/string correspondence for finite-size effects, including the role of worldsheet wrapping and the dressing phase in the orbifolded setting.

Abstract

We compute the planar finite size corrections to the spectrum of the dilatation operator acting on two-impurity states of a certain limit of conformal $\mathcal{N}=2$ quiver gauge field theory which is a $Z_M$-orbifold of $\mathcal{N}=4$ supersymmetric Yang-Mills theory. We match the result to the string dual, IIB superstrings propagating on a pp-wave background with a periodically identified null coordinate. Up to two loops, we show that the computation of operator dimensions, using an effective Hamiltonian technique derived from renormalized perturbation theory and a twisted Bethe ansatz which is a simple generalization of the Beisert-Dippel-Staudacher~\cite{Beisert:2004hm} long range spin chain, agree with each other and also agree with a computation of the analogous quantity in the string theory. We compute the spectrum at three loop order using the twisted Bethe ansatz and find a disagreement with the string spectrum very similar to the known one in the near BMN limit of $\mathcal{N}=4$ super-Yang-Mills theory. We show that, like in $\mathcal{N}=4$, this disagreement can be resolved by adding a conjectured ``dressing factor'' to the twisted Bethe ansatz. Our results are consistent with integrability of the $\mathcal{N}=2$ theory within the same framework as that of $\mathcal{N}=4$.