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

TL;DR

The paper investigates finiteness constraints for Leigh-Strassler deformations of N=4 SYM by introducing a bootstrap-inspired framework and a novel q-symmetry. It derives a structured, invariant representation of the finiteness function $\mathcal{F}(g,q,h,\kappa)$ in terms of symmetry-respecting variables $\gamma_i$, and extends the analysis to complex parameters with careful handling of Hermiticity and analyticity. The authors demonstrate that the q-symmetry robustly constrains $\mathcal{F}$, reproducing the planar one-loop structure and revealing a mechanism to generate new integrable deformations via $q$-invariance, with consistency checks against known results. They connect these field-theoretic constraints to integrable spin-chain structures (XXZ) and to gauge/gravity duality through TsT transformations, obtaining a simple relation between the gravity TsT parameter $k$ and the LS deformation parameter $q$ in the real-case limit, and laying groundwork for a broader AdS/CFT interpretation of LS-type models.

Abstract

In this paper, we follow a Bootstrap-like approach to determine the most restricted form the finiteness constraint $\mathcal{F}(q,g,h,κ)$, 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 $κ^8$ \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 $β$ ($q = \exp(πi β)$) to be linear, in agreement with known results for the Lunin-Maldacena-Frolov deformation \cite{Frolov_2005,Lunin_2005}.

Figures (2)

• Figure 1: A Venn diagram illustrating the aim of this paper. The regions representing mathematical and physical conditions (e.g., loop effects and consistency) can grow with higher-loop orders in $g$. The allowed region of $\mathcal{F}$ expands with these corrections, but remains far more constrained than generally expected, a "smaller infinity" in a sense.
• Figure 2: The black line represents the pairs in \ref{['intdef']}, while the marks in the graph represent the possible roots. The main marks is $\mathrm{X}$, where the function is nearly $0$.