+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### String theory +-- {: .hide} [[!include string theory - contents]] =-- #### Differential cohomology +--{: .hide} [[!include differential cohomology - contents]] =-- =-- =-- \tableofcontents ## Idea The _RR field_ or _Ramond--Ramond field_ is a [[gauge theory|gauge field]] appearing in [[supergravity|10-dimensional type II supergravity]]. Mathematically the RR field on a space $X$ is a [[cocycle]] in [[differential K-theory]] -- or rather, in full generality, in [[twisted differential K-theory|twisted differential]] [[KR-theory]] subject to a [[self-dual higher gauge field]] constrained encoded by a quadratic form defining an [[11-dimensional Chern-Simons theory]] on twisted differential KR cocycles. Accordingly, the [[field strength]] of the RR field, i.e. the image of the [[differential cohomology|differential K-cocycle]] in deRham cohomology, is an inhomogeneous even or odd differential form * in [[type IIA string theory]] $$ F_{RR} = R_0 + R_2 + \cdots $$ * in [[type IIB string theory]] $$ F_{RR} = R_1 + R_3 + \cdots $$ The components of this are sometimes called the RR forms. In the presence of a nontrivial [[Kalb–Ramond field]] the RR field is [[twisted cohomology|twisted]]: a cocycle in the corresponding [[twisted K-theory]]. Moreover, the RR field is constrained to be a [[self-dual higher gauge theory|self-dual]] differential K-cocycle in a suitable sense. [[!include electric-magnetic duality -- table]] The RR field derives its name from the way it shows up when the [[supergravity]] theory in question is derived as an effective background theory in [[string theory]]. From the [[sigma-model]] perspective of the string the RR field is the condensate of fermionic 0-mode excitations of the type II superstring for a particular choice of boundary conditons called the _Ramond_ boundary condititions. Since these boundary conditions have to be chosen for _two_ spinor components, the name appears twice. ## Related concepts * [[RR-field tadpole]] * [[D-brane charge]] * [[Hanany-Witten effect]] * [[K-theory classification of D-brane charge]] * [[differential K-theory]] * [[KK-theory]] [[!include table of branes]] ## References ### General Early discussion in relation to [[D-branes]]: * [[Joseph Polchinski]], *Dirichlet-Branes and Ramond-Ramond Charges*, Phys. Rev. Lett. **75** (1995) 4724-4727 [[arXiv:hep-th/9510017](https://arxiv.org/abs/hep-th/9510017), [doi:10.1103/PhysRevLett.75.4724](https://doi.org/10.1103/PhysRevLett.75.4724)] Review in the context of the [[K-theory classification of D-brane charge]]: * {#Freed00} [[Daniel Freed]], example 2.10 in: *Dirac charge quantization and generalized differential cohomology_*, Surveys in Differential Geometry, Int. Press, Somerville (2000) 129-194 [[arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6), [spire:537392](https://inspirehep.net/literature/537392)] Discussion of the [[NSR-string|NSR]] [[string scattering amplitudes|string perturbation theory]] in RR-field [[background field|background]]: * {#BerensteinLeigh99} [[David Berenstein]], [[Robert Leigh]], _Superstring Perturbation Theory and Ramond-Ramond Backgrounds_, Phys. Rev. D **60** (1999) 106002 [[arXiv:hep-th/9904104](https://arxiv.org/abs/hep-th/9904104)] ### Self-duality and quadratic form {#ReferencesSelfDuality} The [[self-dual higher gauge field]] nature (see there for more) in terms of a [[quadratic form]] on [[differential K-theory]] is discussed originally around * {#MooreWitten99} [[Gregory Moore]], [[Edward Witten]], _Self-Duality, Ramond-Ramond Fields, and K-Theory_, JHEP 0005:032,2000 ([arXiv:hep-th/9912279](http://arxiv.org/abs/hep-th/9912279)) and ([Freed 00](#Freed00)) for [[type I superstring theory]], and for [[type II superstring theory]] in * {#Witten00} [[Edward Witten]], _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031 (2000) [[arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086), [doi:10.1088/1126-6708/2000/05/032](https://doi.org/10.1088/1126-6708/2000/05/032)] * {#FreedHopkins00} [[Daniel Freed]], [[Michael Hopkins]], *On Ramond-Ramond fields and K-theory*, JHEP 0005 (2000) 044 [[arXiv:hep-th/0002027](http://arxiv.org/abs/hep-th/0002027)] * {#FMW00} [[Duiliu-Emanuel Diaconescu]], [[Gregory Moore]], [[Edward Witten]], *$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory*, Adv. Theor. Math. Phys. **6** (2003) 1031-1134 [[arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)], summarised in _A Derivation of K-Theory from M-Theory_ [[arXiv:hep-th/0005091](http://arxiv.org/abs/hep-th/0005091)] with more refined discussion in [[twisted differential K-theory|twisted differential]] [[KR-theory]] in * {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2011) [[arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf)] See at _[[orientifold]]_ for more on this. The relation to [[11d Chern-Simons theory]] is made manifest in * {#BelovMooreII} Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ [[arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020)] [[!include self-duality for pregeometric RR-fields -- references]] ### Irrational RR-charge {#IrrationalRRCharge} An argument that RR-charge may occur in [[irrational number|irrational]] ratios is due to * {#BDS00} [[Constantin Bachas]], [[Michael Douglas]], [[Christoph Schweigert]], around (2.8) of _Flux Stabilization of D-branes_, JHEP 0005:048, 2000 ([arXiv:hep-th/0003037](https://arxiv.org/abs/hep-th/0003037)) In a sequence of followup articles, authors found this problematic and tried to make sense of it: * [[Washington Taylor]], _D2-branes in B fields_, JHEP 0007 (2000) 039 ([arXiv:hep-th/0004141](https://arxiv.org/abs/hep-th/0004141)) > In this article it was argued that the D0-brane charge arising from the integral over the D2-brane of the pullback of the B field is cancelled by the bulk contributions, but in this calculation it was implicitly assumed that the gauge field $C^{(1)}$ is constant. (from [Zhou 01](#Zhou01)) * Peter Rajan, _D2-brane RR-charge on $SU(2)$_, Phys.Lett. B533 (2002) 307-312 ([arXiv:hep-th/0111245](https://arxiv.org/abs/hep-th/0111245)) * {#Zhou01} Jian-Ge Zhou, _D-branes in B Fields_, Nucl.Phys. B607 (2001) 237-246 ([arXiv:hep-th/0102178](https://arxiv.org/abs/hep-th/0102178)) Observation that this paradox is resolved under [[Hypothesis H]], at least for [[fractional D-branes]]: * [[Simon Burton]], [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Lift of fractional D-brane charge to equivariant Cohomotopy theory|Lift of fractional D-brane charge to equivariant Cohomotopy theory]]*, J Geom Phys **161** (2021) 104034 ([doi:10.1016/j.geomphys.2020.104034](https://doi.org/10.1016/j.geomphys.2020.104034), [arXiv:1812.09679]([arXiv:1812.09679](https://arxiv.org/abs/1812.09679)) [[!redirects RR fields]] [[!redirects RR-field]] [[!redirects RR-fields]] [[!redirects Ramond-Ramond field]] [[!redirects Ramond–Ramond field]] [[!redirects Ramond--Ramond field]] [[!redirects Ramond-Ramond fields]] [[!redirects Ramond–Ramond fields]] [[!redirects Ramond--Ramond fields]] [[!redirects RR-charge]]