# nLab Kalb-Ramond field

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

# Contents

## Idea

The Kalb-Ramond field or B-field is the higher U(1)-gauge field that generalizes the electromagnetic field from point particles to strings.

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 $U\left(1\right)$-connection on it.

When replacing particles with 1-dimensional trajectories by 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 Kalb-Ramond field .

Often it is also simply called the $B$-field , after the standard symbol used for the 2-forms $\left({B}_{i}\in {\Omega }^{2}\left({U}_{i}\right)\right)$ 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 $\left({A}_{i}\in {\Omega }^{1}\left({U}_{i}\right)\right)$ in a Cech cocycle presentation of a line bundle with connection encoding the electromagnetic field.

The field strength of the Kalb-Ramond field is a 3-form $H\in \Omega$. On each patch ${U}_{i}$ it is given by

$H{\mid }_{{U}_{i}}=d{B}_{i}\phantom{\rule{thinmathspace}{0ex}}.$H|_{U_i} = d B_i \,.

And just as a degree 2 Deligne cocycle is equivalently encoded in a $U\left(1\right)$-principal bundle with connection, the degree 3 differential cocycle is equivalently encoded in

The study of bundle gerbes 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.

## 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:

• classical background The field strength 3-form $H\in {\Omega }^{3}\left(X\right)$ is required to be closed, $d{H}_{3}=0$.

• quantum coupling The gauge interaction with the quantum string is required to yield a well-defined surface holomomy in $U\left(1\right)$ from locally integrating the 2-forms ${B}_{i}\in {\Omega }^{2}\left({U}_{2}\right)$ with $d{B}_{i}=H{\mid }_{{U}_{i}}$ over its 2-dimensional trajectory.

$\mathrm{hol}\left(\Sigma \right)=\prod _{f}\mathrm{exp}\left(i{\int }_{f}{\Sigma }^{*}{B}_{\rho \left(f\right)}\right)\prod _{e\subset f}\mathrm{exp}\left(i{\int }_{e}{\Sigma }^{*}{A}_{\rho \left(f\right)\rho \left(e\right)}\right)\prod _{v\subset e\subset f}\mathrm{exp}\left(i{\lambda }_{\rho \left(f\right)\rho \left(e\right)\rho \left(v\right)}\right)\phantom{\rule{thinmathspace}{0ex}}.$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)}) \,.

That this is well defined requires that

${\lambda }_{ijk}-{\lambda }_{ijl}+{\lambda }_{ikl}-{\lambda }_{jkl}=0\phantom{\rule{thickmathspace}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}2\pi$\lambda_{i j k} - \lambda_{i j l} + \lambda_{i k l} - \lambda_{j k l} = 0 \;mod \, 2\pi

which says that $\left({B}_{i},{A}_{ij},{\lambda }_{ijk}\right)$ is indeed a degree 3 Deligne cocycle.

## Over D-branes

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 (Moore).

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$\left(D=2n\right)$type IIA$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D0-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$BFSS matrix model
D2-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D4-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D8-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
$\left(D=2n+1\right)$type IIB$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D1-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$2d CFT with BH entropy
D3-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$N=4 D=4 super Yang-Mills theory
D5-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D7-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D9-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
NS-branetype I, II, heteroticcircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
string$\phantom{\rule{thinmathspace}{0ex}}$B2-field2d SCFT
NS5-brane$\phantom{\rule{thinmathspace}{0ex}}$B6-fieldlittle string theory
M-brane11D SuGra/M-theorycircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
M2-brane$\phantom{\rule{thinmathspace}{0ex}}$C3-fieldABJM theory, BLG model
M5-brane$\phantom{\rule{thinmathspace}{0ex}}$C6-field6d (2,0)-superconformal QFT
M9-braneheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\phantom{\rule{thinmathspace}{0ex}}$C6-field on G2-manifold

## References

The name goes back to the article

• M. Kalb and Pierre Ramond, Classical direct interstring action, Phys. Rev. D. 9 (1974), 2273–2284

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

The later article

argues that the string $B$-field is a cocycle in Čech cohomologyDeligne cohomology using quantum anomaly-cancellation arguments.

This is expanded on in

A more refined discussion of the differential cohomology of the Kalb-Ramond field and the fields that it interacts with is in

In fact, in full generality the Kalb-Ramond field on an orientifold background is not a plain gerbe, but a Jandl gerbe , a connection on a nonabelian $\mathrm{AUT}\left(U\left(1\right)\right)$-principal 2-bundles for the automorphism 2-group $\mathrm{AUT}\left(U\right)\left(1\right)\right)$ of $U\left(1\right)$:

for the bosonic string this is discussed in

and for the refinement to the superstring in

See at orientifold for more on this.

The role of the KR field in twisted K-theory is discussed a bit also in

Revised on December 22, 2012 00:03:10 by Urs Schreiber (86.189.5.118)