nLab
Kalb-Ramond field

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 $U(1)$-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

$$H{\mid }_{U_i}=d{B}_i.$$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

• a degree 3 Deligne cocycle;

• a $BU(1)$-principal 2-bundle with connection;

• a $U(1)$-bundle gerbe with connection.

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 holomomy in $U(1)$ from locally integrating the 2-forms $B_i\in {\Omega }^2(U_2)$ with $d{B}_i=H{\mid }_{U_i}$ over its 2-dimensional trajectory.

  $$\mathrm{hol}(\Sigma )=\prod _f\exp (i{\int }_f{\Sigma }^{*}{B}_{\rho (f)})\prod _{e\subset f}\exp (i{\int }_e{\Sigma }^{*}{A}_{\rho (f)\rho (e)})\prod _{v\subset e\subset f}\exp (i{\lambda }_{\rho (f)\rho (e)\rho (v)}).$$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

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

  which says that $(B_i,A_{ij},{\lambda }_{ijk})$ 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).

Related concepts

Table of branes appearing in supergravity/string theory

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D=2n)$	type IIA	$$	$$
D0-brane	$$	$$	BFSS matrix model
D2-brane	$$	$$	$$
D4-brane	$$	$$	Khovanov homology observables
$(D=2n+1)$	type IIB	$$	$$
D1-brane	$$	$$	2d CFT with BH entropy
D3-brane	$$	$$	N=4 D=4 super Yang-Mills theory
D5-brane	$$	$$	$$
D7-brane	$$	$$	$$
NS-brane	type I, II, heterotic	circle n-connection	$$
string	$$	B2-field	2d SCFT
NS5-brane	$$	B6-field	little string theory
M-brane	11D	circle n-connection	$$
M2-brane	$$	C3-field	ABJM theory, BLG model
M5-brane	$$	C6-field	6d (2,0)-superconformal QFT

References

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 (1986)

The later article

• Dan Freed, Edward Witten, Anomalies in String Theory with D-Branes (arXiv)

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

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

• Dan Freed, Dirac Charge Quantization and Generalized Differential Cohomology (arXiv:hep-th/0011220)

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

• Urs Schreiber, Christoph Schweigert, Konrad Waldorf, Unoriented WZW Models and Holonomy of Bundle Gerbes (arXiv:hep-th/0512283)

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Precis in Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics volume 83 AMS (2011) (arXiv:0906.0795)

• Jacques Distler, Dan Freed, Greg Moore, Spin structures and superstrings (arXiv:1007.4581)

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

• Greg Moore, K-theory from a physical perspective (arXiv:hep-th/0304018)