Landau-Lifshitz sigma-models, fermions and the AdS/CFT correspondence

TL;DR

This work establishes a unified framework linking Landau-Lifshitz sigma models on cosets to the AdS/CFT correspondence, by constructing the general LL action on cosets $G/H$, and then implementing it in the important case of $PSU(2,2|4)/PS(U(2|2)^2)$ to capture the thermodynamic limit of the ${ m N}=4$ SYM one-loop dilatation operator. It simultaneously analyzes fermionic sectors of the Type IIB Green-Schwarz action on $AdS_5\times S^5$ by identifying consistent truncations with 2 bosons and 4, 8, or 16 fermions, showing that κ-symmetry acts trivially in these sectors. The paper then demonstrates a precise mapping in the large-charge limit: the GS actions for these fermionic sub-sectors reduce to corresponding LL models on appropriate cosets, establishing a direct link between gauge-theory spin chains and non-relativistic worldsheet dynamics. This provides a natural κ-gauge choice aligned with the spin-chain picture and offers a concrete route to match higher-order fermionic terms between string and gauge theory in the thermodynamic (large-$J$) regime, deepening our understanding of fermionic contributions in AdS/CFT and integrable structures. Overall, the results clarify how to extend LL-type descriptions to fermionic sectors and illuminate the role of truncations and gauge choices in the gauge/gravity correspondence at large charges.

Abstract

We define Landau-Lifshitz sigma models on general coset space $G/H$, with $H$ a maximal stability sub-group of $G$. These are non-relativistic models that have $G$-valued Nöther charges, local $H$ invariance and are classically integrable. Using this definition, we construct the $PSU(2,2|4)/PS(U(2|2)^2)$ Landau-Lifshitz sigma-model. This sigma model describes the thermodynamic limit of the spin-chain Hamiltonian obtained from the complete one-loop dilatation operator of the N=4 super Yang-Mills (SYM) theory. In the second part of the paper, we identify a number of consistent truncations of the Type IIB Green-Schwarz action on $AdS_5\times S^5$ whose field content consists of two real bosons and 4,8 or 16 real fermions. We show that $κ$-symmetry acts trivially in these sub-sectors. In the context of the large spin limit of the AdS/CFT correspondence, we map the Lagrangians of these sub-sectors to corresponding truncations of the $PSU(2,2|4)/PS(U(2|2)^2)$ Landau-Lifshitz sigma-model.