nLab Kalb-Ramond field




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Differential cohomology



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)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 BB-field , after the standard symbol used for the 2-forms (B iΩ 2(U i))(B_i \in \Omega^2(U_i)) on patches U iU_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 iΩ 1(U i))(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 HΩH \in \Omega. On each patch U iU_i it is given by

H| U i=dB i. H|_{U_i} = d B_i \,.

And just as a degree 2 Deligne cocycle is equivalently encoded in a U(1)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 HΩ 3(X)H \in \Omega^3(X) is required to be closed, dH 3=0d H_3 = 0.

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

    hol(Σ)= fexp(i fΣ *B ρ(f)) efexp(i eΣ *A ρ(f)ρ(e)) vefexp(iλ ρ(f)ρ(e)ρ(v)). hol(\Sigma) = \textstyle{\prod}_{f} \exp(i \textstyle{\int}_f \Sigma^* B_{\rho(f)}) \textstyle{\prod}_{e \subset f} \exp(i \textstyle{\int}_{e} \Sigma^* A_{\rho(f) \rho(e)}) \textstyle{\prod}_{v \subset e \subset f} \exp(i \lambda_{\rho(f) \rho(e) \rho(v)}) \,.

    That this is well defined requires that

    λ ijkλ ijl+λ iklλ jkl=0mod2π, \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,λ ijk)(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)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(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\,
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
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
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


The name goes back to:

The interpretation as a 4d axion:

The interpretation of the B-field as a 3-cocycle in Deligne cohomology is due to

picked up in

The equivalent formulation in terms of connections on bundle gerbes originates with

See also:

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

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

for the bosonic string this is discussed in

and for the refinement to the superstring in

See at orientifold for more on this; also at discrete torsion.

The role of the KR field in twisted K-theory (see K-theory classification of D-brane charge) is discussed a bit also in

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)

In the context of cosmology with the Kalb-Ramond field as a dark matter-candidate (cf, axion and fuzzy dark matter):

  • Christian Capanelli, Leah Jenks, Edward W. Kolb, Evan McDonough, Cosmological Implications of Kalb-Ramond-Like-Particles [arXiv:2309.02485]

See also:

Last revised on April 2, 2024 at 06:38:40. See the history of this page for a list of all contributions to it.