Types of quantum field thories
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 -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 -field , after the standard symbol used for the 2-forms on patches of a cover of spacetime when the differential cocycle is expressed in a Cech cohomology realization of Deligne cohomology.
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 is required to be closed, .
quantum coupling The gauge interaction with the quantum string is required to yield a well-defined surface holonomy in from locally integrating the 2-forms with over its 2-dimensional trajectory.
That this is well defined requires that
which says that 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).
|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|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|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|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
The name goes back to the article
The later article
This is expanded on in
A more refined discussion of the differential cohomology of the Kalb-Ramond field and the fields that it interacts with is in
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 -principal 2-bundles for the automorphism 2-group of :
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 volume 83 AMS (2011) (arXiv:0906.0795)
See at orientifold for more on this.
The role of the KR field in twisted K-theory is discussed a bit also in