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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.
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.
That this is well defined requires that
which says that $(B_i, A_{i j}, \lambda_{i j k})$ is indeed a degree 3 Deligne cocycle.
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).
The name goes back to:
The interpretation as a 4d axion:
The interpretatin 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
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:
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
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
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):
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]
Last revised on February 9, 2024 at 09:48:23. See the history of this page for a list of all contributions to it.