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

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.

  $$hol(\Sigma) = \textstyle{\prod}_{f} \exp(i \textstyle{\int}_f \Sigma^* B_{\rho(f)}) \textstyle{\prod}_{e \subset f} \exp(i \textstyle{\int}_{e} \Sigma^* A_{\rho(f) \rho(e)}) \textstyle{\prod}_{v \subset e \subset f} \exp(i \lambda_{\rho(f) \rho(e) \rho(v)}) \,.$$

  That this is well defined requires that

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

  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

• WZW model, basic bundle gerbe

References

The name goes back to:

• M. Kalb, Pierre Ramond, Classical direct interstring action, Phys. Rev. D. 9 (1974) 2273-2284 [doi:10.1103/PhysRevD.9.2273]

The interpretation as a 4d axion:

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

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

• Krzysztof Gawędzki, Topological Actions in two-dimensional Quantum Field Theories, in: Nonperturbative quantum field theory, Nato Science Series B 185, Springer (1986) [spire:257658, doi:10.1007/978-1-4613-0729-7_5, pdf]

picked up in

• Daniel Freed, Edward Witten, Anomalies in String Theory with D-Branes, Asian J. Math. 3 4 (1999) 819-852 [arXiv:hep-th/9907189, InSpire:504509]

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

• Krzysztof Gawędzki, Nuno Reis, WZW branes and gerbes, Rev. Math. Phys. 14 (2002) 1281-1334 [arXiv:hep-th/0205233, doi:10.1142/S0129055X02001557]

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

See also:

• Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, Classifying A-field and B-field configurations in the presence of D-branes, JHEP 0812:078 (2008) [arXiv:0810.4291, doi:10.1088/1126-6708/2008/12/078]

• Fabio Ferrari Ruffino, Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes, Nuclear Physics, Section B 858 (2012) 377-404 [arXiv:1104.2798, doi:10.1016/j.nuclphysb.2012.01.013]

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

• Dan Freed, Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry 7 (2002) 129-194 [ arXiv:hep-th/0011220, doi:10.4310/SDG.2002.v7.n1.a6, spire:537392]

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 $AUT(U(1))$-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, Communications in Mathematical Physics 274 1 (2007) 31-64 [arXiv:hep-th/0512283, doi:10.1007/s00220-007-0271-x]

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 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; 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

• 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)

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:

• Peter D. Jarvis, Jean Thierry-Mieg, Antisymmetric tensor fields: actions, symmetries and first order Duffin-Kemmer-Petiau formulations [arXiv:2311.01675]

• Jean Thierry-Mieg, Peter D. Jarvis, Conformal invariance of antisymmetric tensor field theories in any even dimension [arXiv:2311.01701]