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\section*{Kalb-Ramond field}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\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{mathematical_model_from_formal_physical_input}{Mathematical model from (formal) physical input}\dotfill \pageref*{mathematical_model_from_formal_physical_input} \linebreak
\noindent\hyperlink{over_dbranes}{Over D-branes}\dotfill \pageref*{over_dbranes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

The \textbf{Kalb-Ramond field} or \textbf{B-field} is the [[higher U(1)-gauge field]] that generalizes the [[electromagnetic field]] from point [[particles]] to [[string theory|strings]].

Its dual incarnation in [[KK-compactifications]] of [[heterotic string theory]] to 4d is a candidate for the hypothetical [[axion]] field (\hyperlink{SvrcekWitten06}{Svrcek-Witten 06, p. 15}).

Recall that the [[electromagnetic field]] is modeled as a [[cocycle]] in degree 2 [[ordinary differential cohomology]] and that this mathematical model is fixed by the fact that charged particles that trace out 1-dimensional trajectories couple to the electromagnetic field by an [[action functional]] that sends each trajectory to the holonomy of a $U(1)$-[[connection on a bundle|connection]] on it.

When replacing particles with 1-dimensional trajectories by [[string theory|strings]] with 2-dimensional trajectories, one accordingly expects that they may couple to a higher degree background field given by a degree 3 [[ordinary differential cohomology]] cocycle.

In [[string theory]] this situation arises and the corresponding background field appears, where it is called the \emph{Kalb-Ramond field} .

Often it is also simply called the \textbf{$B$-field} , after the standard symbol used for the 2-forms $(B_i \in \Omega^2(U_i))$ on patches $U_i$ of a [[cover]] of spacetime when the differential cocycle is expressed in a [[Cech cohomology]] realization of [[Deligne cohomology]].

This is the analog of the local 1-forms $(A_i \in \Omega^1(U_i))$ in a Cech cocycle presentation of a [[line bundle]] with [[connection on a bundle|connection]] encoding the [[electromagnetic field]].

The [[field strength]] of the Kalb-Ramond field is a [[differential forms|3-form]] $H \in \Omega$. On each patch $U_i$ it is given by

\begin{displaymath}
H|_{U_i} = d B_i
  \,.
\end{displaymath}
And just as a degree 2 [[Deligne cohomology|Deligne cocycle]] is equivalently encoded in a $U(1)$-[[principal bundle]] [[connection on a bundle|with connection]], the degree 3 differential cocycle is equivalently encoded in

\begin{itemize}%
\item a degree 3 [[Deligne cohomology|Deligne cocycle]];


\item a $\mathbf{B}U(1)$-[[principal 2-bundle]] with connection;


\item a $U(1)$-[[bundle gerbe]] with connection.



\end{itemize}
The study of [[bundle gerbe]]s was largely motivated and driven by the desire to understand the Kalb-Ramond field.

The next higher degree analog of the electromagnetic field is the [[supergravity C-field]].

\hypertarget{mathematical_model_from_formal_physical_input}{}\subsection*{{Mathematical model from (formal) physical input}}\label{mathematical_model_from_formal_physical_input}

The derivation of the fact that the Kalb-Ramond field that is locally given by a 2-form is globally really a degree 3 cocycle in the [[Deligne cohomology]] model for [[ordinary differential cohomology]] proceeds in in entire analogy with the corresponding discussion of the [[electromagnetic field]]:

\begin{itemize}%
\item \textbf{classical background} The [[field strength]] 3-form $H \in \Omega^3(X)$ is required to be closed, $d H_3 = 0$.


\item \textbf{quantum coupling} The gauge interaction with the quantum string is required to yield a well-defined surface holonomy in $U(1)$ from locally integrating the 2-forms $B_i \in \Omega^2(U_2)$ with $d B_i = H|_{U_i}$ over its 2-dimensional trajectory.

\begin{displaymath}
hol(\Sigma)
  =
  \prod_{f} \exp(i \int_f \Sigma^* B_{\rho(f)})
  \prod_{e \subset f}
  \exp(i \int_{e} \Sigma^* A_{\rho(f) \rho(e)})
  \prod_{v \subset e \subset f}
  \exp(i \lambda_{\rho(f) \rho(e) \rho(v)})
  \,.
\end{displaymath}
That this is well defined requires that

\begin{displaymath}
\lambda_{i j k} - \lambda_{i j l} +
  \lambda_{i k l} - \lambda_{j k l}
  = 0 \;mod \, 2\pi
\end{displaymath}
which says that $(B_i, A_{i j}, \lambda_{i j k})$ is indeed a degree 3 [[Deligne cohomology|Deligne cocycle]].



\end{itemize}
\hypertarget{over_dbranes}{}\subsection*{{Over D-branes}}\label{over_dbranes}

The restriction of the Kalb-Ramond field in the 10-dimensional [[spacetime]] to a [[D-brane]] is a twist (as in [[twisted cohomology]]) of the [[gauge field]] on the D-brane: its 3-class is [[magnetic charge]] for the [[electromagnetic field]]/[[Yang-Mills field]] on the D-brane. See also [[Freed-Witten anomaly cancellation]] or the discussion in (\hyperlink{Moore}{Moore}).

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

\begin{itemize}%
\item [[discrete torsion]]

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

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

The name goes back to the article

\begin{itemize}%
\item M. Kalb and [[Pierre Ramond]], \emph{Classical direct interstring action, Phys. Rev. D. 9 (1974), 2273--2284}

\end{itemize}
The interpretation as a 4d [[axion]] is discussed in

\begin{itemize}%
\item Peter Svrcek, [[Edward Witten]], \emph{Axions In String Theory}, JHEP 0606:051,2006 (\href{http://arxiv.org/abs/hep-th/0605206}{arXiv:hep-th/0605206})

\end{itemize}
The earliest reference where the gauge term in the standard string [[action functional]] is identified with the surface holonomy of a 3-cocycle in [[Deligne cohomology]] seems to be

\begin{itemize}%
\item [[Krzysztof Gawedzki]], \emph{Topological Actions in two-dimensional Quantum Field Theories}, Cargese 1987 proceedings, \emph{Nonperturbative quantum field theory} (1986) (\href{http://inspirehep.net/record/257658?ln=de}{web})

\end{itemize}
The later article

\begin{itemize}%
\item [[Dan Freed]], [[Edward Witten]], \emph{Anomalies in String Theory with D-Branes}, Asian J. Math.3:819,1999 (\href{http://arxiv.org/abs/hep-th/9907189}{arXiv:hep-th/9907189})

\end{itemize}
argues that the string $B$-field is a cocycle in [[Čech cohomology]]--[[Deligne cohomology|Deligne]] cohomology using [[quantum anomaly]]-cancellation arguments.

This is expanded on in

\begin{itemize}%
\item [[Alan Carey]], Stuart Johnson, [[Michael Murray]], \emph{Holonomy on D-Branes}, Journal of Geometry and Physics Volume 52, Issue 2, October 2004, Pages 186-216 (\href{http://arxiv.org/abs/hep-th/0204199}{arXiv:hep-th/0204199}, \href{https://doi.org/10.1016/j.geomphys.2004.02.008}{arXiv:10.1016/j.geomphys.2004.02.008})

\end{itemize}
A more refined discussion of the [[differential cohomology]] of the Kalb-Ramond field and the fields that it interacts with is in

\begin{itemize}%
\item [[Dan Freed]], \emph{Dirac Charge Quantization and Generalized Differential Cohomology} (\href{http://arxiv.org/abs/hep-th/0011220}{arXiv:hep-th/0011220})

\end{itemize}
In fact, in full generality the Kalb-Ramond field on an [[orientifold]] background is not a plain gerbe, but a \emph{Jandl gerbe} , a connection on a nonabelian $AUT(U(1))$-[[principal 2-bundle]]s for the [[automorphism 2-group]] $AUT(U)(1))$ of $U(1)$:

for the bosonic string this is discussed in

\begin{itemize}%
\item [[Urs Schreiber]], [[Christoph Schweigert]], [[Konrad Waldorf]], \emph{Unoriented WZW Models and Holonomy of Bundle Gerbes} (\href{http://arxiv.org/abs/hep-th/0512283}{arXiv:hep-th/0512283})

\end{itemize}
and for the refinement to the [[superstring]] in

\begin{itemize}%
\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Precis} in [[Hisham Sati]], [[Urs Schreiber]] (eds.), \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics volume 83 AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795})


\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Spin structures and superstrings}, Surveys in Differential Geometry, Volume 15 (2010) (\href{http://arxiv.org/abs/1007.4581}{arXiv:1007.4581}, \href{http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4}{doi:10.4310/SDG.2010.v15.n1.a4}) See at \emph{[[orientifold]]} for more on this.



\end{itemize}
The role of the KR field in [[twisted K-theory]] is discussed a bit also in

\begin{itemize}%
\item [[Greg Moore]], \emph{K-theory from a physical perspective} (\href{http://arxiv.org/abs/hep-th/0304018}{arXiv:hep-th/0304018})

\end{itemize}
[[!redirects Kalb--Ramond field]] [[!redirects Kalb–Ramond field]] [[!redirects B-field]] [[!redirects B-fields]]

[[!redirects B2-field]]



\end{document}