Contents

Idea

The Kalb-Ramond field or B-field is the gauge field that generalizes the electromagnetic field from point particles to strings.

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 $BU(1)$-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 holomomy in $U(1)$ from locally integrating the 2-forms $B_i\in {\Omega }^2(U_2)$ with $d{B}_i=H{\mid }_{U_i}$ over its 2-dimensional trajectory.

  $$\mathrm{hol}(\Sigma )=\prod _f\exp (i{\int }_f{\Sigma }^{*}{B}_{\rho (f)})\prod _{e\subset f}\exp (i{\int }_e{\Sigma }^{*}{A}_{\rho (f)\rho (e)})\prod _{v\subset e\subset f}\exp (i{\lambda }_{\rho (f)\rho (e)\rho (v)}).$$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

  $${\lambda }_{ijk}-{\lambda }_{ijl}+{\lambda }_{ikl}-{\lambda }_{jkl}=0\mathrm{mod}2\pi$$\lambda_{i j k} - \lambda_{i j l} + \lambda_{i k l} - \lambda_{j k l} = 0 \;mod \, 2\pi

  which says that $(B_i,A_{ij},{\lambda }_{ijk})$ is indeed a degree 3 Deligne cocycle.

References

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 (1986)

The later article

• Dan Freed, Edward Witten, Anomalies in String Theory with D-Branes (arXiv)

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

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

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

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Precis (arXiv:0906.0795)