Full Lagrangian and Hamiltonian for quantum strings on AdS_4 x CP^3 in a near plane wave limit

TL;DR

The paper derives the complete interacting Lagrangian and Hamiltonian for type IIA superstrings on AdS$_4\times\mathbb{CP}^3$ in a near plane wave limit, including curvature-induced cubic and quartic terms up to order $1/R^2$. It employs a superspace Green-Schwarz action with a carefully fixed κ-symmetry gauge in the light-cone framework, and uses fermionic field redefinitions together with a Dirac procedure to handle fermionic second-class constraints, reducing Dirac brackets to Poisson brackets. The authors present explicit bosonic and fermionic contributions to the Lagrangian and Hamiltonian, including a detailed pp-wave analysis, mode expansions, and the full light-cone Hamiltonian up to quartic order, providing a robust framework for computing finite-size and curvature corrections to the string spectrum in ABJM-like AdS$_4$/CFT$_3$ setups. This work delivers well-defined expressions for all interactions and clarifies the role of nontrivial κ-symmetry fixing and non-normal-ordered quartic terms in obtaining a consistent quantum string spectrum.

Abstract

We find the full interacting Lagrangian and Hamiltonian for quantum strings in a near plane wave limit of AdS_4 x CP^3. The leading curvature corrections give rise to cubic and quartic terms in the Lagrangian and Hamiltonian that we compute in full. The Lagrangian is found as the type IIA Green-Schwarz superstring in the light-cone gauge employing a superspace construction with 32 grassmann-odd coordinates. The light-cone gauge for the fermions is non-trivial since it should commute with the supersymmetry condition. We provide a prescription to properly fix the kappa-symmetry gauge condition to make it consistent with light-cone gauge. We use fermionic field redefinitions to find a simpler Lagrangian. To construct the Hamiltonian a Dirac procedure is needed in order to properly keep into account the fermionic second class constraints. We combine the field redefinition with a shift of the fermionic phase space variables that reduces Dirac brackets to Poisson brackets. This results in a completely well-defined and explicit expression for the full interacting Hamiltonian up to and including terms quartic in the number of fields.