In this short note we review the construction and role of Wess-Zumino couplings of Dirichlet branes and Orientifold planes, and show how these combine to give the Green-Schwarz anomaly cancelling terms

Stefanski, B.

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Stefanski, B. (1999). WZ Couplings of D-branes and O-planes. Paper presented at the NATO Advanced Study Institute on Progress in String Theory and M-Theory, 24-05-1999 - 05-06-1999, Cargese, France. City Research OnlineOriginal citation: Stefanski, B. (1999). WZ Couplings of D-branes and O-planes. Paper presented at the NATO Advanced Study Institute on Progress in String Theory and M-Theory, 24-05-1999 - 05-06-1999, Cargese, France. Permanent City Research Online URL: http://openaccess.city.ac.uk/12562/ Copyright & reuseCity University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact with the author(s) of this paper, please email the team at publications@city.ac.uk.arXiv:hep-th/9909105v2  16 Sep 1999WZ COUPLINGS OF D-BRANES AND O-PLANESB. Stefan´ski, jr.Department of Applied Mathematics and Theoretical PhysicsCambridge University, Silver Street, Cambridge CB3 9EW, UKB.Stefanski@damtp.cam.ac.ukAbstract In this short note we review the construction and role of Wess-Zuminocouplings of Dirichlet branes and Orientifold planes, and show how thesecombine to give the Green-Schwarz anomaly cancelling terms.The low energy effective theory of a D-brane has two types of cou-plings, the Dirac-Born-Infeld (DBI) and Wess-Zumino (WZ) terms. Theformer describes the coupling of the vector potential and scalars couplingto the NS⊗NS fields, while the latter gives the coupling of the vectorpotential and the pullback of the curvature fields to the R⊗R potentials.The first D-brane WZ term [1] simply states that a Dp-brane is chargedunder C(p+1), the p+ 1-form R⊗R potentialSDWZ = µp∫BpC(p+1) , (1.1)where µp is the charge density of a D-brane and Bp its worldvolume. Itwas then shown that D-branes couple to lower dimensional R⊗R poten-tials [2] and that a ‘brane within brane’ picture emergedSDWZ = µp∫BpC ∧tr exp(iF/2π) . (1.2)In the above C represents a formal sum of R⊗R potentials, F = F −B,F is the worldvolume gauge field and B the NS⊗NS two form.1An R4 term found in [3] required, as a consequence of duality, thepresence of a gravitational WZ coupling [4]. Later the entire WZ action1F , rather than F , is present in the action as the latter quantity is not gauge invariant.12was found using the anomaly inflow mechanism [5, 6] and readsSDWZ = µp∫BpC ∧tr exp(iF/2π) ∧√√√√ Aˆ(RT )Aˆ(RN ), (1.3)where Aˆ is the Dirac or A-roof genus and RT , RN are the pull-backs ofthe tangent and normal bundle curvatures to the D-brane world-volume,respectively.This coupling has a natural interpretation within K-theory [7]. In K-theory there is a natural bilinear pairing of bundles given by the indexof the Dirac operator on the tensor product of the two bundles. For Ea bundle over a manifold X, with TX the tangent space of X, the mapE → ch(E)√Aˆ(TX) , (1.4)is an isometry with respect to this pairing and the DeRham pairing inH∗(X), the cohomology of X. In fact,following [13], it was shown [14]that D-brane charges are indeed classified by K-theory.The D-brane WZ couplings were confirmed by string amplitude cal-culations. In [9] the four-form couplings were computed at tree level.A one-loop amplitude in [10] and tree-level amplitude in [11] confirmedthe presence of all the couplings, while in [12] extra, non-anomalouscouplings, as well as the normal bundle contributions were determined.In [8], it was first observed that O-planes too carry gravitational WZcouplings. For consider Type I theory, in which 16 D9-branes, theirimages and an O9-plane, fill the spacetime. The WZ coupling of thetheory is the Green-Schwarz (GS) coupling [15]. This differs from theWZ couplings of 32 D9-branes, indicating that O9-planes too have WZcouplings. By studying one-loop [10], and tree-level [11] scattering am-plitudes it was found that the WZ coupling of Op-planes isSOWZ = −2p−4µp∫BpC ∧√L(RT /4)L(RN/4), (1.5)where L is the Hirzebruch polynomial. To see that the O9-plane andD9-brane WZ couplings match the GS term consider the following. Themassless chiral fields of Type I string theory are a neutral gravitino, aneutral fermion of opposite chirality and SO(32) fermions. The totalanomaly for this theory follows by descent from2I =122π(Aˆ(R)(treiR/2π − 2) + Aˆ(R)treiF/2π)∣∣∣∣12-form2Note that all of these are real fields.3= π1(4π)2(trR2 − trF 2) ∧ X8 , (1.6)whereX8 =1(4π)4(23trF 4 +112trR4 +148(trR2)2 −112trR2trF 2). (1.7)In the units of [16] the action extracted from the string theory ampli-tudes [10, 11] isS = −14κ210∫dC(2)∧ ∗ dC(2)+ µ9∫ (2(4π)2C(6)∧(trR2− trF 2) + C(2)∧X8). (1.8)Since H = dC(2) + . . . is gauge invariant and dC(6) = ∗dC(2) the gaugetransformation for C(2) isδC(2) = 4µ9κ210(ω12,Y − ω12,L) , (1.9)hence (1.8) has an anomalous variation which follows by descent fromIWZ = 4µ29κ210(trF2− trR2) ∧X8 . (1.10)The Type I charge density satisfies(µ9κ10)2 =π2, (1.11)and hence IWZ = I as required.O-planes cannot couple to gauge fields so their WZ couplings have tobe purely gravitational. Hence it is a consistency check for the D9-braneand GS gauge and mixed couplings to agree.AcknowledgmentsThe author would like to thank the conference organisers for financial support anda stimulating programme.References[1] J. Polchinski, Dirichlet-Branes and Ramond-Ramond Charges,Phys. Rev. Lett.75 (1995) 4724, hep-th/9510017;[2] M. Li, Boundary States of D-branes and Dy-Strings, Nucl. Phys.B460 (1996) 351, hep-th/9510161; C. Schmidhuber, D-Brane Ac-tions, Nucl. Phys.B467 (1996) 146, hep-th/9601003; M.R. Douglas,Branes within Branes, hep-th/9512077;4[3] C. Vafa and E. Witten, A One Loop Test of String Dualities, Nucl.Phys. B447 (1995) 261, hep-th/9505053;[4] M. Bershadsky, C. Vafa, V. Sadov, D-Branes and Topological FieldTheories, Nucl. Phys. B463 (1996) 420, hep-th/9511222;[5] M.B. Green, J.A. Harvey and G. Moore, I-Brane Inflow andAnomalous Couplings on D-branes, Class. Quant. Grav. 14 (1997)47, hep-th/9605033;[6] Y.-K.E. Cheung and Z. Yin, Anomalies, Branes, and Currents,Nucl. Phys. B517 (1998) 69, hep-th/9710206;[7] R. Minasian and G. Moore, K-theory and Ramond-Ramond Charge,JHEP 9711 (1997) 002, hep-th/9710230;[8] K. Dasgupta, D.P. Jatkar, S. Mukhi, Gravitational Couplings andZ2 Orientifolds, Nucl. Phys. B523 (1998) 465, hep-th/9707224;[9] B. Craps and F. Roose, Anomalous D-brane and Orientifold Cou-plings from the Boundary State, Phys. Lett. 445B (1998) 150, hep-th/9808074;[10] J.F. Morales, C.A. Scrucca, M. Serone, Anomalous Couplingsfor D-branes and O-planes, Nucl. Phys. B552 (1999) 291, hep-th/9812071;[11] B. Stefan´ski, jr., Gravitational Couplings of D-branes and O-planes,Nucl. Phys. B548 (1999) 275, hep-th/9812088;[12] B. Craps and F. Roose, (Non-)Anomalous D-brane and O-planecouplings: the Normal Bundle, Phys. Lett. 450B (1999) 358, hep-th/9812149;[13] For a review see A. Sen, Non-BPS States and Branes in StringTheory, hep-th/9904207;[14] E. Witten, D-Branes And K-Theory, JHEP 9812 (1998) 025, hep-th/9810188;[15] M.B. Green and J.H. Schwarz, Anomaly Cancellations in Supersym-metric D=10 Gauge Theory and Superstring Theory, Phys. Lett.149B (1984) 117;[16] J. Polchinski, String Theory, Volume 2, Superstring Theory andBeyond, CUP 1998;

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