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\begin{document}

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\section*{RR field}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesSelfDuality}{Self-duality and quadratic form}\dotfill \pageref*{ReferencesSelfDuality} \linebreak
\noindent\hyperlink{IrrationalRRCharge}{Irrational RR-charge}\dotfill \pageref*{IrrationalRRCharge} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{RR field} or \emph{Ramond--Ramond field} is a [[gauge theory|gauge field]] appearing in [[supergravity|10-dimensional type II supergravity]].

Mathematically the RR field on a space $X$ is a [[cocycle]] in [[differential K-theory]] -- or rather, in full generality, in [[twisted differential K-theory|twisted differential]] [[KR-theory]] subject to a [[self-dual higher gauge field]] constrained encoded by a quadratic form defining an [[11-dimensional Chern-Simons theory]] on twisted differential KR cocycles.

Accordingly, the [[field strength]] of the RR field, i.e. the image of the [[differential cohomology|differential K-cocycle]] in deRham cohomology, is an inhomogeneous even or odd differential form

\begin{itemize}%
\item in [[type IIA string theory]]

\begin{displaymath}
F_{RR} = R_0 + R_2 + \cdots
\end{displaymath}

\item in [[type IIB string theory]]

\begin{displaymath}
F_{RR} = R_1 + R_3 + \cdots
\end{displaymath}


\end{itemize}
The components of this are sometimes called the RR forms.

In the presence of a nontrivial [[Kalb–Ramond field]] the RR field is [[twisted cohomology|twisted]]: a cocycle in the corresponding [[twisted K-theory]].

Moreover, the RR field is constrained to be a [[self-dual higher gauge theory|self-dual]] differential K-cocycle in a suitable sense.

[[!include electric-magnetic duality -- table]]

The RR field derives its name from the way it shows up when the [[supergravity]] theory in question is derived as an effective background theory in [[string theory]]. From the [[sigma-model]] perspective of the string the RR field is the condensate of fermionic 0-mode excitations of the type II superstring for a particular choice of boundary conditons called the \emph{Ramond} boundary condititions. Since these boundary conditions have to be chosen for \emph{two} spinor components, the name appears twice.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[RR-field tadpole]]


\item [[D-brane charge]]


\item [[differential K-theory]]


\item [[KK-theory]]



\end{itemize}
[[!include table of branes]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

For a quick review see for instance

\begin{itemize}%
\item [[Daniel Freed]], example 2.10 \emph{Dirac charge quantization and generalized differential cohomology} (\href{http://arxiv.org/abs/hep-th/0011220}{arXiv})

\end{itemize}
Discussion of the [[NSR-string|NSR]] [[string perturbation theory]] in RR-field background is in

\begin{itemize}%
\item [[David Berenstein]], Robert Leigh, \emph{Superstring Perturbation Theory and Ramond-Ramond Backgrounds}, Phys. Rev. D 60, 106002 (1999) (\href{https://arxiv.org/abs/hep-th/9904104}{arXiv:hep-th/9904104})

\end{itemize}
\hypertarget{ReferencesSelfDuality}{}\subsubsection*{{Self-duality and quadratic form}}\label{ReferencesSelfDuality}

The [[self-dual higher gauge field]] nature (see there for more) in terms of a [[quadratic form]] on [[differential K-theory]] is discussed originally around

\begin{itemize}%
\item [[Gregory Moore]], [[Edward Witten]], \emph{Self-Duality, Ramond-Ramond Fields, and K-Theory}, JHEP 0005:032,2000 (\href{http://arxiv.org/abs/hep-th/9912279}{arXiv:hep-th/9912279})

\end{itemize}
and (\hyperlink{Freed00}{Freed 00}) for [[type I superstring theory]], and for [[type II superstring theory]] in

\begin{itemize}%
\item [[Edward Witten]], \emph{Duality Relations Among Topological Effects In String Theory}, JHEP 0005:031,2000 (\href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9912086})


\item [[Daniel Freed]], [[Michael Hopkins]], \emph{On Ramond-Ramond fields and K-theory}, JHEP 0005 (2000) 044 (\href{http://arxiv.org/abs/hep-th/0002027}{arXiv:hep-th/0002027})


\item D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], \emph{$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory}, Adv.Theor.Math.Phys.6:1031-1134,2003 (\href{http://arxiv.org/abs/hep-th/0005090}{arXiv:hep-th/0005090}), summarised in \emph{A Derivation of K-Theory from M-Theory} (\href{http://arxiv.org/abs/hep-th/0005091}{arXiv:hep-th/0005091})



\end{itemize}
with more refined discussion in [[twisted differential K-theory|twisted differential]] [[KR-theory]] in

\begin{itemize}%
\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides})

\end{itemize}
See at \emph{[[orientifold]]} for more on this. The relation to [[11d Chern-Simons theory]] is made manifest in

\begin{itemize}%
\item Dmitriy Belov, [[Greg Moore]], \emph{Type II Actions from 11-Dimensional Chern-Simons Theories} (\href{http://arxiv.org/abs/hep-th/0611020}{arXiv:hep-th/0611020})

\end{itemize}
Review is in

\begin{itemize}%
\item [[Richard Szabo]], section 3.6 and 4.6 of \emph{Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology} (\href{http://arxiv.org/abs/1209.2530}{arXiv:1209.2530})

\end{itemize}
\hypertarget{IrrationalRRCharge}{}\subsubsection*{{Irrational RR-charge}}\label{IrrationalRRCharge}

An argument that RR-charge may occur in [[irrational number|irrational]] ratios is due to

\begin{itemize}%
\item [[Constantin Bachas]], [[Michael Douglas]], [[Christoph Schweigert]], around (2.8) of \emph{Flux Stabilization of D-branes}, JHEP 0005:048,2000 (\href{https://arxiv.org/abs/hep-th/0003037}{arXiv:hep-th/0003037})

\end{itemize}
In a sequence of followup articles, authors found this problematic and tried to make sense of it:

\begin{itemize}%
\item [[Washington Taylor]], \emph{D2-branes in B fields}, JHEP 0007 (2000) 039 (\href{https://arxiv.org/abs/hep-th/0004141}{arXiv:hep-th/0004141})

\end{itemize}
\begin{quote}%
In this article it was argued that the D0-brane charge arising from the integral over the D2-brane of the pullback of the B field is cancelled by the bulk contributions, but in this calculation it was implicitly assumed that the gauge field $C^{(1)}$ is constant. (from \hyperlink{Zhou01}{Zhou 01})


\end{quote}
\begin{itemize}%
\item Peter Rajan, \emph{D2-brane RR-charge on $SU(2)$}, Phys.Lett. B533 (2002) 307-312 (\href{https://arxiv.org/abs/hep-th/0111245}{arXiv:hep-th/0111245})


\item Jian-Ge Zhou, \emph{D-branes in B Fields}, Nucl.Phys. B607 (2001) 237-246 (\href{https://arxiv.org/abs/hep-th/0102178}{arXiv:hep-th/0102178})



\end{itemize}
[[!redirects RR fields]]

[[!redirects RR-field]] [[!redirects RR-fields]]

[[!redirects Ramond-Ramond field]] [[!redirects Ramond–Ramond field]] [[!redirects Ramond--Ramond field]]

[[!redirects Ramond-Ramond fields]] [[!redirects Ramond–Ramond fields]] [[!redirects Ramond--Ramond fields]]

[[!redirects RR-charge]]



\end{document}