nLab
supergravity C-field

The field content of standard $11$-dimensional supergravity consists of —besides the gravitational field? and the various fermionic superpartner?s— a field that is locally modeled by a differential $3$-form.

Given that

• an ordinary connection form is often denoted by the symbol $A$ and
• the local $2$-form of the Kalb–Ramond field is usually denoted by the symbol $B$,

this $3$-form is usually denoted by the letter $C$ and just called “the $C$ field”.

In complete analogy to how

• quantum particles couple to the electromagnetic field and
• quantum strings couple to the Kalb–Ramond field,

so $(2+1)$-dimensional extended objects, quantum “membranes”, couple to the $C$-field. By the entirely same Dirac quantization argument described at electromagnetic field and Kalb–Ramond field, this alone implies already that the $C$ field is a degree-$3$ Deligne differential cocycle.

Moreover, a phenomenon closely related to the Green–Schwarz mechanism implies that the $C$-field is not an arbitrary degree-$3$ diffential cocycle, but the difference of two Chern–Simons differential cocycles, in that the curvature characteristic $4$-form is constrained to be in the de Rham cohomology class of the first Pontryagin characteristic class of the tangent bundle of the underlying space.