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 (the so-called Ramond boundary conditions) and the fields that satisfy them in 1971.
Defining the fields
The fields in each theory 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 Cp and a (p + 1)-form field strength Gp+1. The field strength is, as usual defined to be the exterior derivative of the potential Gp+1 = dCp. 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. 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. The choices of allowed values of p depend on the theory. In type IIA supergravity, fields exist for p = 1 and p = 3. In type IIB supergravity, on the other hand, there are fields for p = 0, p = 2 and p = 4, although the p = 4 field is constrained to satisfy the self-duality condition G5 = *G5 where * is the Hodge star. The self-duality condition cannot be imposed by a Lagrangian without either introducing extra fields or ruining the manifest super-Poincaré invariance of the theory, thus type IIB supergravity is considered to be a non-Lagrangian theory. A third theory, called massive or Romans IIA supergravity, includes a field strength G0, called the Romans mass. Being a zero-form, it has no corresponding connection. Furthermore, the equations of motion impose that the Romans mass is constant. In the quantum theory Joseph Polchinski has shown that G0 is an integer, which jumps by one as one crosses a D8-brane.
The democratic formulation It is often convenient to use the democratic formulation of type II string theories, which was introduced by Paul Townsend in p-Brane Democracy. In D-brane Wess-Zumino Actions, T-duality and the Cosmological Constant Michael Green, Chris Hull and Paul Townsend constructed the field strengths and found the gauge transformations that leave them invariant. Finally in New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls the authors completed the formulation, providing a Lagrangian and explaining the role of the fermions. In this formulation one includes all of the even field strengths in IIA and all of the odd field strengths in IIB. The additional field strengths are defined by the star condition Gp=*G10−p. As a consistency check, notice that the star condition is compatible with the self-duality of G5, thus the democratic formulation contains the same number of degrees of freedom as the original formulation. Similarly to attempts to simultaneously include both electric and magnetic potentials in electromagnetism, the dual gauge potentials may not be added to the democratically formulated Lagrangian in a way that maintains the manifest locality of the theory. This is because the dual potentials are obtained from the original potentials by integrating the star condition.
Ramond–Ramond gauge transformations The type II supergravity Langragians are invariant under a number of local symmetries, such as diffeomorphisms and local supersymmetry transformations. In addition the various form-fields transform under Neveu–Schwarz and Ramond–Ramond gauge transformations. In the democratic formulation the Ramond–Ramond gauge transformations of the gauge potentials that leave the action invariant are
C p → C p + d Λ p − 1 + H ∧ Λ p − 3 {\displaystyle C_{p}\rightarrow C_{p}+d\Lambda _{p-1}+H\wedge \Lambda _{p-3}}
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