Super spin chain coherent state actions and $AdS_5 \times S^5$ superstring

TL;DR

The paper extends the coherent-state approach used to match spin-chain and string actions from purely bosonic sectors to fermionic subsectors of the AdS/CFT correspondence, focusing on SU(2|3) and SU(1|1). It derives a Landau-Lifshitz-type action on CP^{2|2} from the SU(2|3) one-loop dilatation operator and analyzes its connection to the AdS_5 × S^5 superstring action in a light-cone gauge, highlighting the quartic fermionic structure and necessary field redefinitions. Through explicit fermionic string solutions and careful truncations, it demonstrates how parts of the string action reduce to a massive 2D fermion in the SU(1|1) sector and how leading-order fermionic terms can be reconciled with the spin-chain description in the fast-string limit, while noting the challenges in fully matching quartic terms. The work outlines a program to achieve a consistent, higher-order correspondence between string and spin-chain pictures in fermionic sectors, clarifying the role of supercoset geometry and truncation procedures in AdS/CFT.

Abstract

We consider a generalization of the leading-order matching of coherent state actions for semiclassical states on the super Yang-Mills and the superstring sides of the AdS/CFT duality to sectors with fermions. In particular, we discuss the $SU(1|1)$ and $SU(2|3)$ sectors containing states with angular momentum $J$ in $S^5$ and spin in $AdS_5$. On the SYM side, we start with the dilatation operator in the $SU(2|3)$ sector having super spin chain Hamiltonian interpretation and derive the corresponding coherent state action which is quartic in fermions. This action has essentially the same ``Landau-Lifshitz'' form as the action in the bosonic SU(3) sector with the target space $CP^2$ replaced by the projective superspace $CP^{2|2}$. We then attempt to relate it to the corresponding truncation of the full $AdS_5 \times S^5$ superstring action written in a light-cone gauge where it has simple quartic fermionic structure. In particular, we find that part of the superstring action describing $SU(1|1)$ sector reduces to an action of a massive two-dimensional relativistic fermion, with the expansion in the effective coupling $λ/J^2$ being equivalent to a non-relativistic expansion.