SO(d,d) Transformations of Ramond-Ramond Fields and Space-time Spinors

TL;DR

This work derives explicit $SO(d,d)$ transformations for Ramond-Ramond fields and potentials, along with space-time spinors and supersymmetry parameters, in type-II theories, including time-like dualities. By enforcing compatibility with space-time supersymmetry, Hassan constructs the spinor representation $oldsymbol{\Omega}$ of the induced twist between left- and right-moving worldsheet frames and shows that RR fields transform as $\widetilde{\boldsymbol{\cal F}}=\sqrt{\det Q_-}\,\boldsymbol{\Omega}\,\boldsymbol{\cal F}$, with RR potentials transforming in the same way. The paper provides detailed component formulas for $F^{(n)}$ and $C^{(n)}$ under general $SO(d,d)$, and presents an $SO(d,d)$-covariant form for RR kinetic terms, while clarifying the relation to the Majorana-Weyl spinor construction of RR fields. It also connects with alternative RR-spinor approaches and discusses checks against the $G_{ij}\to 0$ limit in Matrix theory/NC SYM, highlighting the framework’s utility for generating nontrivial D-brane bound states and for incorporating time-direction dualities.

Abstract

We explicitly construct the SO(d,d) transformations of Ramond-Ramond field strengths and potentials, along with those of the space-time supersymmetry parameters, the gravitinos and the dilatinos in type-II theories. The results include the case when the SO(d,d) transformation involves the time direction. The derivation is based on the compatibility of SO(d,d) transformations with space-time supersymmetry, which automatically guarantees compatibility with the equations of motion. It involves constructing the spinor representation of a twist that an SO(d,d) action induces between the local Lorentz frames associated with the left- and right-moving sectors of the worldsheet theory. The relation to the transformation of R-R potentials as SO(d,d) spinors is also clarified.