Leigh-Strassler deformation of N=4 SYM Theory

### Introduction

There is an enormous body of literature on the AdS-CFT correspondence. The first and the most well-studied example in this correspondence is the one that relates type IIB string theory on $AdS_5\times S^5$ and the four-dimensional superconformal field theory(SCFT), the $\mathcal{N}=4$ supersymmetric Yang-Mills theory. The SCFT has maximal supersymmetry in four-dimensions and thus is somewhat unrealistic. Theories with $\mathcal{N}=1$ supersymmetry have a richer behaviour and show phases that are confining, Coloumb-like, Higgs-like and so on. However, non-supersymmetric Yang-Mills theories such as pure QCD are not conformal — there is a dynamically generated mass in the quantum theory. The scale of the mass is $\Lambda_{\textrm{QCD}}$. Needless to say, people are pursuing more general non-conformal situations and this is usually called the gauge-gravity correspondence.

It is known from the work of Leigh and Strassler that the $\mathcal{N}=4$ supersymmetric Yang-Mills theory has two marginal deformations that preserve $\mathcal{N}=1$ supersymmetry — let us call this theory the LS theory. The marginality of the deformation implies that the LS theory remains conformal and thus provides the CFT side of the AdS-CFT correspondence.

### The gravity dual

The LS theory is expected to be dual to type IIB string theory compactified on $AdS_5\times X^{(5)}$, where $X^{(5)}$ is expected to be a ‘deformation’ of $S^5$ preserving a $U(1)$ isometry. The precise details of $X$ is not known beyond this. However, the answer when one of the deformations, the so-called beta-deformation, is turned on, the answer has been provided by Lunin and Maldacena. To third-order in perturbations, Aharony, Kol and Yankielowicz, have worked out the details of the fields that get non-zero background values due to the two marginal deformations in the CFT. However, what one is looking for is the finite, all-orders version.

### The SCFT

The LS theory has the same spectrum as that of $\mathcal{N}=4$ SYM theory — it contains one $\mathcal{N}=1$ vector multiplet and three chiral multiplets that we will denote by $\Phi_1,\Phi_2,\Phi_3$, each of which transform in the adjoint of $SU(N)$ (not $U(N)$). The superpotential for $\mathcal{N}=4$ SYM theory is

(1)
\begin{align} W_0=h\ \mathrm{Tr}\left(\Phi_1 \big[\Phi_2,\Phi_3]\big)\right)=h\ \mathrm{Tr}\Big(\Phi_1 \Phi_2\Phi_3-\Phi_1 \Phi_3\Phi_2\Big) \end{align}

The superpotential for the LS theory is of the form

(2)
\begin{align} W=W_0 + \frac1{3!}c^{ijk}\ \mathrm{Tr}\left(\Phi_i \Phi_j\Phi_k\right)\ , \end{align}

where $c^{ijk}$ is totally-symmetric in its indices. It is useful to think of the three chiral fields as complex coordinates on $\mathbb{C}^3$. This has 10 independent components — using simple linear redefinitions acting on the fields ($SL(3,\mathbb{C})$ acting on the three fields), we find only two non-trivial deformations. These are, effectively, the two marginal deformations of Leigh and Strassler. Note that we need $N>2$ as the term that we add to $W_0$ vanishes for $SU(2)$.

### Anomalous dimensions and the spin-chain

In the context of the $\mathcal{N}=4$ SYM, it was realised that the spectrum of (planar) anomalous dimensions of single-trace chiral operators can be obtained as the spectrum of a spin-chain — the length of the spin-chain being related to the number of fields appearing in chiral operator. This has been generalised in several ways.