Structure constants of $\beta$ deformed super Yang-Mills

TL;DR

This paper investigates the structure constants of the beta deformed super Yang-Mills theory at both perturbative one-loop and strong coupling levels, revealing protected quantities and consistency with R-symmetry predictions.

Contribution

It demonstrates that certain structure constants are protected from corrections and constructs dual supergravity modes in the deformed geometry, extending understanding of AdS/CFT correspondence.

Findings

One-loop corrections are determined by the anomalous dimension Hamiltonian.

Three point functions of chiral primaries are non-renormalized at one loop.

Strong coupling results match undeformed case for specific supergravity modes.

Abstract

We study the structure constants of the ${\cal N}=1$ beta deformed theory perturbatively and at strong coupling. We show that the planar one loop corrections to the structure constants of single trace gauge invariant operators in the scalar sector is determined by the anomalous dimension Hamiltonian. This result implies that 3 point functions of the chiral primaries of the theory do not receive corrections at one loop. We then study the structure constants at strong coupling using the Lunin-Maldacena geometry. We explicitly construct the supergravity mode dual to the chiral primary with three equal U(1) R-charges in the Lunin-Maldacena geometry. We show that the 3 point function of this supergravity mode with semi-classical states representing two other similar chiral primary states but with large U(1) charges to be independent of the beta deformation and identical to that found in the $AdS_5\times S^5$ geometry. This together with the one-loop result indicate that these structure constants are protected by a non-renormalization theorem. We also show that three point function of U(1) R-currents with classical massive strings is proportional to the R-charge carried by the string solution. This is in accordance with the prediction of the R-symmetry Ward identity.