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.

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

• a degree 3 Deligne cocycle;

• a $\mathbf{B}U(1)$-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 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).

Related concepts

• discrete torsion

• h1-meson, b1-meson

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

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	$\,$	$\,$
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-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
M-brane	11D SuGra/M-theory	circle n-connection	$\,$
M2-brane	$\,$	C3-field	ABJM theory, BLG model
M5-brane	$\,$	C6-field	6d (2,0)-superconformal QFT
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-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-brane	6d (2,0)-superconformal QFT
self-dual string		self-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

• Peter Svrcek, Edward Witten, Axions In String Theory, JHEP 0606:051,2006 (arXiv:hep-th/0605206)

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, in: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds.) Cargese 1987 proceedings, Nonperturbative quantum field theory (1986) (spire:257658, doi:10.1007/978-1-4613-0729-7_5)

The later article

• Daniel Freed, Edward Witten, Anomalies in String Theory with D-Branes, Asian J. Math. 3:819, 1999 (arXiv:hep-th/9907189)

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

This is expanded on in

• Alan Carey, Stuart Johnson, Michael Murray, Holonomy on D-Branes, Journal of Geometry and Physics Volume 52, Issue 2, October 2004, Pages 186-216 (arXiv:hep-th/0204199, doi:10.1016/j.geomphys.2004.02.008)

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

for the bosonic string this is discussed in

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

and for the refinement to the superstring in

• 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, Surveys in Differential Geometry, Volume 15 (2010) (arXiv:1007.4581, doi:10.4310/SDG.2010.v15.n1.a4)

  See at orientifold for more on this.

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

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

In relation to Einstein-Cartan theory:

• Richa Kapoor, A review of Einstein Cartan Theory to describe superstrings with intrinsic torsion (arXiv:2009.07211)

• Tanmoy Paul, Antisymmetric tensor fields in modified gravity: a summary (arXiv:2009.07732)