# nLab Kalb-Ramond field

Contents

## 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.

Its dual incarnation in KK-compactifications of heterotic string theory to 4d is a candidate for the hypothetical axion field (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 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 $(B_i \in \Omega^2(U_i))$ 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 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

$H|_{U_i} = d B_i \,.$

And just as a degree 2 Deligne cocycle is equivalently encoded in a $U(1)$-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 $H \in \Omega^3(X)$ 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 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.

$hol(\Sigma) = \prod_{f} \exp(i \int_f \Sigma^* B_{\rho(f)}) \prod_{e \subset f} \exp(i \int_{e} \Sigma^* A_{\rho(f) \rho(e)}) \prod_{v \subset e \subset f} \exp(i \lambda_{\rho(f) \rho(e) \rho(v)}) \,.$

That this is well defined requires that

$\lambda_{i j k} - \lambda_{i j l} + \lambda_{i k l} - \lambda_{j k l} = 0 \;mod \, 2\pi$

which says that $(B_i, A_{i j}, \lambda_{i j k})$ 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
$(D = 2n)$type IIA$\,$$\,$
D(-2)-brane$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$D=7 super Yang-Mills theory
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection$\,$
string$\,$B2-field2d SCFT
NS5-brane$\,$B6-fieldlittle string theory
D-brane for topological string$\,$
A-brane$\,$
B-brane$\,$
M-brane11D SuGra/M-theorycircle n-connection$\,$
M2-brane$\,$C3-fieldABJM theory, BLG model
M5-brane$\,$C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
S-brane
SM2-brane,
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

## 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 interpretation as a 4d axion is discussed in

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

• Krzysztof Gawedzki, Topological Actions in two-dimensional Quantum Field Theories, Cargese 1987 proceedings, Nonperturbative quantum field theory (1986) (web)

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 $AUT(U(1))$-principal 2-bundles for the automorphism 2-group $AUT(U)(1))$ of $U(1)$:

for the bosonic string this is discussed in

and for the refinement to the superstring in

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

Last revised on May 15, 2019 at 04:11:14. See the history of this page for a list of all contributions to it.