The type II superstring to order theta^4

TL;DR

This work provides a complete construction of the Green-Schwarz type II superstring action in a general type II supergravity background up to quartic order in the fermionic coordinates $\theta$. By solving the superspace Bianchi identities via a normal-coordinate expansion, the authors obtain the supervielbeins and NSNS/RR fields to order $\theta^4$, and present explicit forms for the zeroth, quadratic, and quartic Lagrangians, including a Killing-spinor operator and related matrices that encode background supersymmetry. They show that for a broad class of backgrounds, fixing kappa-symmetry (e.g., Γ^+Θ=0) truncates the action at this order, making the quartic action a complete description in those gauges; this is particularly useful for semiclassical analyses in AdS/CFT contexts. The approach also provides a systematic framework to derive supergeometry to higher orders, enabling investigations into the integrable structure and dynamics of strings beyond supercoset models, with potential applications to backgrounds like AdS2×S2×T6.

Abstract

The Green-Schwarz superstring action in a general type IIA or IIB supergravity background is derived up to fourth order in the Grassmann-odd coordinates theta. This is done by solving the superspace Bianchi identities order by order in theta, to quadratic order for all superfields and to quartic order for the supervielbeins. For a large class of backgrounds it is possible to fix the kappa symmetry in such a way that the action actually terminates at the quartic order in theta.