Ramond–Ramond field

In theoretical physics, Ramond–Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory. The ranks of the fields depend on which type II theory is considered. As Joseph Polchinski argued in 1995, D-branes are the charged objects that act as sources for these fields, according to the rules of p-form electrodynamics. It has been conjectured that quantum RR fields are not differential forms, but instead are classified by twisted K-theory.

The adjective "Ramond–Ramond" reflects the fact that in the RNS formalism, these fields appear in the Ramond–Ramond sector in which all vector fermions are periodic. Both uses of the word "Ramond" refer to Pierre Ramond, who studied such boundary conditions and the fields that satisfy them in 1971.

As in Maxwell's theory of electromagnetism and its generalization, p-form electrodynamics, Ramond–Ramond (RR) fields come in pairs consisting of a p-form potential C_p and a (p + 1)-form field strength G_p+1. The field strength is, as usual defined to be the exterior derivative of the potential G_p+1 = dC_p.

As is usual in such theories, if one allows topologically nontrivial configurations or charged matter (D-branes) then the connections are only defined on each coordinate patch of spacetime, and the values on various patches are glued using transition functions which are gauge transformations. Unlike the case of electromagnetism, in the presence of a nontrivial Neveu–Schwarz 3-form field strength the field strength defined above is no longer gauge invariant and so also needs to be defined patchwise with the Dirac string off of a given patch interpreted itself as a D-brane. This extra complication is responsible for some of the more interesting phenomena in string theory, such as the Hanany–Witten transition.

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