RR field in nLab

Contents

Contents

Idea

The RR field or Ramond–Ramond field is a gauge field appearing in 10-dimensional type II supergravity.

Mathematically the RR field on a space XX is a cocycle in differential K-theory – or rather, in full generality, in 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 K-cocycle in deRham cohomology, is an inhomogeneous even or odd differential form

• in type IIA string theory

  F RR=R 0+R 2+⋯ F_{RR} = R_0 + R_2 + \cdots

• in type IIB string theory

  F RR=R 1+R 3+⋯ 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: a cocycle in the corresponding twisted K-theory.

Moreover, the RR field is constrained to be a self-dual differential K-cocycle in a suitable sense.

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.

• RR-field tadpole

• D-brane charge

• Hanany-Witten effect

• K-theory classification of D-brane charge

  • differential K-theory

  • KK-theory

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, doi:10.1103/PhysRevLett.75.4724]

Review in the context of the K-theory classification of D-brane charge:

• 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, doi:10.4310/SDG.2002.v7.n1.a6, spire:537392]

Discussion of the NSR string perturbation theory in RR-field background:

• David Berenstein, Robert Leigh, Superstring Perturbation Theory and Ramond-Ramond Backgrounds, Phys. Rev. D 60 (1999) 106002 [arXiv:hep-th/9904104]

Self-duality and quadratic form

self-duality for pregeometric RR-fields – references

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

• Gregory Moore, Edward Witten, Self-Duality, Ramond-Ramond Fields, and K-Theory, JHEP 0005:032,2000 (arXiv:hep-th/9912279)

and (Freed 00) for type I superstring theory, and for type II superstring theory in

• Edward Witten, Duality Relations Among Topological Effects In String Theory, JHEP 0005:031 (2000) [arXiv:hep-th/9912086, doi:10.1088/1126-6708/2000/05/032]

• Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 [arXiv:hep-th/0002027]

• Duiliu-Emanuel Diaconescu, Gregory Moore, Edward Witten, E 8E_8 Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6 (2003) 1031-1134 [arXiv:hep-th/0005090], summarised in A Derivation of K-Theory from M-Theory [arXiv:hep-th/0005091]

with more refined discussion in twisted differential KR-theory in

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2011) [arXiv:0906.0795, slides]

See at orientifold for more on this. The relation to 11d Chern-Simons theory is made manifest in

• Dmitriy Belov, Greg Moore, Type II Actions from 11-Dimensional Chern-Simons Theories [arXiv:hep-th/0611020]

Review is in

• Richard Szabo, section 3.6 and 4.6 of: Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology [arXiv:1209.2530]

Discussion of Lagrangian densities for type II supergravity making the nature of the pregeometric RR-fields and their self-duality manifest:

• Gianguido Dall'Agata, Kurt Lechner, Mario Tonin, D=10D=10, N=IIBN=IIB Supergravity: Lorentz-invariant actions and duality, JHEP 9807:017 (1998) [arXiv:hep-th/9806140, doi:10.1088/1126-6708/1998/07/017]

• Eric Bergshoeff, Renata Kallosh, Tomas Ortin, Diederik Roest, Antoine Van Proeyen, New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls, Class. Quant. Grav. 18 (2001) 3359-3382 [arXiv:hep-th/0103233, doi:10.1088/0264-9381/18/17/303]

• Karapet Mkrtchyan, Fridrich Valach, Democratic actions for type II supergravities, Phys.Rev.D 107 6 (2023) 066027 [arXiv:2207.00626, doi:10.1103/PhysRevD.107.066027]

Irrational RR-charge

An argument that RR-charge may occur in irrational ratios is due to

• Constantin Bachas, Michael Douglas, Christoph Schweigert, around (2.8) of Flux Stabilization of D-branes, JHEP 0005:048, 2000 (arXiv: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)

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)C^{(1)} is constant. (from Zhou 01)

• Peter Rajan, D2-brane RR-charge on SU(2)SU(2), Phys.Lett. B533 (2002) 307-312 (arXiv:hep-th/0111245)

• Jian-Ge Zhou, D-branes in B Fields, Nucl.Phys. B607 (2001) 237-246 (arXiv:hep-th/0102178)

Observation that this paradox is resolved under Hypothesis H, at least for fractional D-branes:

• Simon Burton, Hisham Sati, Urs Schreiber, Lift of fractional D-brane charge to equivariant Cohomotopy theory, J Geom Phys 161 (2021) 104034 (doi:10.1016/j.geomphys.2020.104034, arXiv:1812.09679)