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

TL;DR

This paper extends the matching of coherent state actions in AdS/CFT to sectors with fermions, deriving a fermionic Landau-Lifshitz type action and relating it to the superstring action in a light-cone gauge.

Contribution

It introduces a fermionic coherent state action for the $SU(2|3)$ sector and connects it to the superstring action, including a reduction to a relativistic fermion in the $SU(1|1)$ sector.

Findings

Derived fermionic coherent state action in $SU(2|3)$ sector.

Connected the field theory action to the superstring action.

Reduced superstring action to a relativistic fermion in the $SU(1|1)$ sector.

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 $\lambda/J^2$ being equivalent to a non-relativistic expansion.