Bootstrapping the Finiteness of Leigh-Strassler Deformations and Uncovering Hidden Symmetries

Abstract

In this paper, we follow a Bootstrap-like approach to determine the most restricted form the finiteness constraint $\mathcal{F}(q,g,h,\kappa)$, which relates the four parameters of $\mathcal{N}=1$ Leigh-Strassler (LS) deformed models, by imposing mathematical and physical conditions. Focusing first on real parameters, we apply these conditions, together with a new symmetry of the superpotential we named ``q-symmetry'', to strongly constrains $\mathcal{F}$. Imposing only these mathematical conditions is enough, for example, to reproduces the \textit{structure} of the one-loop correction and the \textit{exact result} in the planar limit, which are known from the literature. Extending the analysis to complex parameters, we develop a similar method to obtain the more restricted form of $\mathcal{F}$, though the complex case obscures expansions in ``q-invariant'' variables. We also show how an additional pair $(q,h)$ of integrable deformations arises via q-invariance, and verify that the transformed R-matrix satisfies the Yang-Baxter equation. Moreover, we make two ansatz for the coefficients left in free in the finiteness $\mathcal{F}$ for the real parameters, and while it has some defects, it reveals interesting results when compared with literature: the first predicts the pair of integrable deformations derived in \cite{Mansson2010}, while the second ansatz gives the first correction only at fourth loop order $\kappa^8$ \cite{Mansson2010}, which is known to be true in the planar limit. Furthermore, we study the impact of this symmetry on the algebra of the deformed XXZ spin chain and the moduli-vacuum of LS, and find a gauge/gravity interpretation when $h=0$ for the q-symmetry, obtaining the simplest relation between $k$ (from TsT) and $\beta$ ($q = \exp(\pi i \beta)$) to be linear, in agreement with known results for the Lunin-Maldacena-Frolov deformation \cite{Frolov_2005,Lunin_2005}.