Comments on T-dualities of Ramond-Ramond Potentials

TL;DR

This paper makes the Ramond-Ramond (RR) sector of type II supergravities on a torus manifestly invariant under the T-duality group $SO(d,d;Z)$ by introducing mixed RR-NSNS potentials $D_{p+1}$ that combine RR forms with the NS-NS 2-form. It shows that, when the KK-reduced RR forms are assembled into KK towers $D_\alpha$ and $D_{\mu\alpha}$ and organized as a Majorana-Weyl spinor of $O(d,d;Z)$, the RR+$CS$ action reduces to a duality-covariant form, with the RR field strength $F$ expressed as $F=e^{- ext{widehat }B_2}\wedge dD$. A fermionic-oscillator construction of the $O(d,d;R)$ spinor representation is developed, enabling a simple proof of invariance for arbitrary $d$ and KK degree, and the work highlights the simplifications gained in the Chern-Simons sector. The results provide a concrete, spinor-based mechanism to relate RR sectors under T-duality and suggest avenues for leveraging the full duality group in classical solutions and potential extensions to U-duality.

Abstract

The type IIA/IIB effective actions compactified on T^d are known to be invariant under the T-duality group SO(d, d; Z) although the invariance of the R-R sector is not so direct to see. Inspired by a work of Brace, Morariu and Zumino,we introduce new potentials which are mixture of R-R potentials and the NS-NS 2-form in order to make the invariant structure of R-R sector more transparent. We give a simple proof that if these new potentials transform as a Majorana-Weyl spinor of SO(d, d; Z), the effective actions are indeed invariant under the T-duality group. The argument is made in such a way that it can apply to Kaluza-Klein forms of arbitrary degree. We also demonstrate that these new fields simplify all the expressions including the Chern-Simons term.