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

TL;DR

This paper investigates finite size corrections and integrability properties of a specific limit of $ ext{N}=2$ super Yang-Mills theory, comparing gauge theory calculations with string theory on a pp-wave background, and explores the role of a dressing factor in resolving spectrum discrepancies.

Contribution

It extends integrability and finite size correction analysis from $ ext{N}=4$ to $ ext{N}=2$ theories, introducing a twisted Bethe ansatz and examining three-loop spectrum agreement.

Findings

Two-loop results agree between gauge and string theory.

Three-loop spectrum shows a discrepancy similar to $ ext{N}=4$ case.

Adding a dressing factor may resolve the spectrum disagreement.

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 $\mathcal{N}=4$.