Ramond-Ramond Couplings on Brane-Antibrane Systems

Transcription

1 hep-th/ Ramond-Ramond Couplings on Brane-Antibrane Systems Conall Kennedy and Andy Wilkins Department of Mathematics rinity College Dublin 2 Ireland May 1999) Couplings between a closed string RR field and open strings are calculated in a system of coincident branes and antibranes of type II theory. he result can be written cleanly using the curvature of the superconnection. CDMAH conall@maths.tcd.ie andyw@maths.tcd.ie 0

2 1 Introduction Unstable D-branes can decay and produce branes of lower dimension. he most simple decay is that of a Dp-brane annihilating with a Dp-antibrane to yield a Dp 2)-brane. In this scenario the lightest open-string mode stretching between the brane-antibrane pair is tachyonic and it can condense into a configuration with non-trivial winding. hen, in order to have finite energy, the gauge field living on the branes must have non-zero first Chern class which implies the existence of a non-zero p 2)-brane charge. he happy state of affairs is that the negative energy density of the condensed tachyon field exactly cancels with the positive energy density of the brane-antibrane pair asymptotically, leaving a Dp 2)-brane with finite tension [1]. One can also start with non-supersymmetric Dp-branes in type II theory p is odd for IIA and even for IIB). Here too there is a tachyon in the spectrum of the open string connecting the brane to itself. It can condense into a kink and the unstable non-supersymmetric brane will decay to yield a stable Dp 1) brane. his type of brane production was studied in detail in [2] where it was argued that all supersymmetric D-branes in IIA can be constructed as bound states of a number of D9-branes. Further examples of such decay processes are studied in [3,8,4,5,2,6,7] and can be understood within the context of K-theory [9, 10]. he unstable and non-bps branes have also been used in testing various duality conjectures [11, 12, 1, 8], indeed it was in this context that they made their first major appearance. he configurations that are studied in this note are a straightforward generalisation of the scenario describedin the firstparagraph to N Dp-branes coincident with N Dp-antibranes. Without the antibranes the coupling of the UN) massless world-volume vectors denoted by A) to the closed-string RR fields denoted by C) is given by the Wess-Zumino action [13, 14] S = C r e F, 1) where is the world volume, r is over the Chan-Paton factors and F is the field strength for A. Numerical factors such as α have been suppressed, as have the contributions from the antisymmetric 2-form and the A-roof genus [15 17]. When the antibranes are included, the light degrees of freedom are two UN ) gauge fields one living on the branes A + ) and the other on the antibranes A ) anda tachyon ) and antitachyon ) living in the N, N) and N,N) representations respectively. We would like to know the generalisation of the Wess-Zumino action. o this end we perform a tree-level string calculation of the effective action to low orders in the tachyon field. We find a non-zero contact interaction of the form g C p 1) r d d. 2) In this note we will not calculate the overall normalisation of such terms except to say their coefficients are not zero) for they turn out to be unimportant. We also consider the brane-antibrane pairs to have 1

3 indistinguishable world volumes. 1 he result Eq. 2) is somewhat surprising since it contains the winding number, r d d,ofthe tachyon configuration. Recall that after the tachyon condenses with non-zero winding, the gauge field acquires a non-zero first Chern class, due to energy considerations. For example, in the case of a single brane-antibrane pair, the one-soliton configuration looks like r, θ) =fr)e iθ and D r 0, where r, θ) are polar coordinates on the plane perpendicular to the p 2)-brane, the function f assumes the tachyon s vacuum expectation value at infinity, and the covariant derivatives are D = d + A + A and D = d A + +A. However, Eqs. 1) and 2) imply that the resulting p 2)-brane will have charge proportional to the sum of the first Chern class with the winding number! µ p 2 F + F )+g Σ 2) d d, Σ 2) where the plane perpendicular to the p 2)-brane has been labeled Σ 2). o fix this charge problem there must be another interaction containing C p 1), namely, the full result must read C p 1) F + F 1 2 g { F +, } g{ F, } ) +gd D, 3) at this order in the tachyon field). he first two terms come from the usual WZ action 2 while the last term is Eq. 2) covariantised. In Sec. 2.3 we shall show the middle two terms also appear by considering the string amplitude involved and gauge invariance. For example, after tachyon condensation on the single brane-antibrane pair, this gives the desired result; [ C p 2) F + F gda + A ) + gd d + ga + A )d ) ] Σ p 1) Σ 2) [ = C p 2) F + F + gd d gd A + A )] ) Σ p 1) Σ 2) [ = C p 2) F + ] F, 4) Σ p 1) Σ 2) where the finite energy condition D =d+a + A ) 0 has been used in the second equality. From this analysis it is evident that the coupling, g, is unimportant for processes involving tachyon condensation. 1 further comments on this point are made in [18] in which the tachyon potential was shown to assume a Mexican hat shape for weak fields 2 he relative sign difference occurs because the antibranes carry negative RR charge. 2

4 Interestingly, Eq. 3) can be written in a more compact form by employing the superconnection [19 21] ) A + A = A, which transforms under the UN) UN) symmetry as g 0 A GdG 1 +GAG 1 where G = 0 ḡ ). he curvature of this is F + F = ) D D F, and a supertrace is defined by ) a b Sr =ra r d. c d hen the terms in Eq. 3) are just those in an expansion of S = C Sr e F. 5) We propose that this generalises the usual Wess-Zumino action Eq. 1). Arguments similar to Eq. 4) show that our proposal gives the correct charges for all decay products. 2 he calculations he process of interest is one where a RR boson annihilates onto the brane-antibrane world volume to create some open strings. By string duality this is just an open string graph with insertions on the boundary the brane s world volume) and one RR insertion in the interior. Similar calculations have been performed in [22 27] and we will follow their notations and conventions: Map the disc to the upper-half plane so the boundary of the disc becomes the real axis. On this axis the worldsheet bosons X µ ) and fermions ψ µ ) obey the following boundary conditions Neumann: { X a z) = X a z) ψ a z) = ψ a z) and Dirichlet: { X i z) = X i z) ψ i z) = ψ i z), while the ghosts c) and superghosts φ) obey trivial boundary conditions. Indices µ = 0,...,9, while an index a lives on the world volume a =0,...,p)andifills the transverse space i =p+1),...,9). Because of these boundary conditions the correlators mix; Neumann: X a z)x b w) = η ab logz w) and X a z) X b w) = η ab logz w), Dirichlet: X i z)x j w) = η ij logz w) and X i z) X j w) = +η ij logz w). 6) 3

5 and similarly for the fermions we use the conventions α = 2). Now use the doubling trick in which the fields Xz) andψz) are extended to the lower-half plane by defining { Xz) Neumann Xz) = Xz) Dirichlet z LHP). hen if we think of w being in the LHP the correlation function of this extended holomorphic field X µ z)x ν w) = η µν logz w), correctly reproduces all of Eq. 6). hus, when considering scattering off p-branes, the rule is to replace X µ z) D ν µ Xν z), ψµ z) D ν µ ψν z), φ z) φ z) and c z) c z), where and then use the usual correlators 1p+1 0 D = p ), X µ z)x ν w) = η µν logz w), ψ µ z)ψ ν w) = η µν z w) 1, cz)cw) = z w), φz)φw) = logz w). he vertex operators for the tachyon are V 0) x) =k ψeik X x) and V 1) x) =e φ e ik X x), where the superscripts label the superghost number. he momentum k is constrained to lie in the world volume; k µ =k a,0) with k 2 =1/4 in our conventions α = 2). In the coincident brane-antibrane system the vertex operators for the tachyon and the antitachyon look the same in order to distinguish them a Chan-Paton factor must be understood. After doubling, the vertex operators become [28] he RR vertex operators are V 0) z) = k ψe2ik X z), V 1) z) = e φ e 2ik X z). V 1) RR w, w) = P H/ m) ) αβ : e φ/2 S α e ip X w) ::e φ/2 ip Sβ e X w): with the projector P = γ11 ) assuring that we are using the correct chirality and H/ m) = 1 m! H µ 1...µ m γ µ 1...γ µm, 4

6 where m =2,4fortypeIIAandm=1,3,5 for type IIB. he spinorial indices are raised with the charge conjugation matrix, eg P H/ m) ) αβ = C αδ P H/ m) ) β δ further conventions and notations for spinors can be found in appendix B of [23]). he RR bosons are massless so p 2 = 0. he spin fields can also be extended to the entire complex plane. In calculations we replace S α w) M β α S β w), where M = 1 p! γa 0 γ a 1...γ ap ɛ a0...a p. Finally, a couple of correlators containing two spin fields are S α w)s β = w 5/4 Cαβ 1 ψ µ z)s α w)s β = z w) 1/2 z 1/2 w 3/4 γ µ αβ. where S β = S β 0) and C αβ is the charge conjugation matrix. 2.1 he two-point amplitude he amplitude containing one RR field and one tachyon or antitachyon) is A,RR = = dzdwd w V CKG V 1) z)v 1) RR w, w), dzdwd w e φz) e 2ik Xz) e φw)/2 S α w)e ip Xw) P H/ m) M) αβ V CKG e φ w)/2 S β w)e ip D X w). We have chosen the vertex operators according to the rule that the total superghost number must be 2. Including the ghost contribution, cz)cw)c w) =z w)z w)w w), and using the kinematic constraint, k a + p a = 0, the volume of the conformal Killing group can be canceled by fixing the three insertion points. he amplitude then reads ) A,RR =r P H/ m) M. Repeated use of {γ µ,γ ν }=2η µν yields r γ [µ 1...γ µm] γ [a 0...γ ap]) = p +1)!δ m,p+1η µma 0 η µ m 1a 1...η µ 1a p, implying that the amplitude vanishes since there is no H p+1) in the type II string. On the other hand, in the case of a non-bps brane as studied recently in [29]) this amplitude implies the effective action contains the term C d. 5

7 2.2 he three-point amplitude he three-point amplitude between one RR field, and two tachyonic particles is dzdz A,,RR dwd w = V 0) 1) z)v z )V 1) RR w, w), V CKG dzdz dwd w = k ψz)e 2ik Xz) e φz ) e 2ik Xz ) e φw)/2 S α w)e ip Xw) P H/ m) M) αβ V CKG e φ w)/2 S β w)e ip D X w). It is convenient to fix the points z,z,w, w)=x, x, i, i) to cancel the volume V CKG by inserting the ghost contribution cz )cw)c w) =z w)z w)w w). Introduce the Mandelstam variable t = k + k ) 2. hen using various kinematic constraints such as the amplitude reduces to A,,RR = k a + k a + p a =0 and p D p= 2t= 2p k+k ), 1 + x 2 ) 2 ) 1 2 +t 1 dx 16x 2 1+x 2r P H/ m) Mγ a )k a, = π Γ[ 2t] Γ[ 1 2 r P H/ m) Mγ a )k a. 7) t]2 Next the trace must be evaluated; r γ µ 1...γ µm γ a 0...γ ap γ a )H µ1...µ m ɛ a0...a p = p +1)!δ m,p 1) pp+1)/2 H a0...a p 1 ɛ a 1...a p 1 a. he trace containing the factor of γ 11 ensures the following results also hold for p>3withh m) H 10 m) for m 5. he prefactor of Eq. 7) has the interesting property that at non-negative integer values of t there is a pole corresponding to an open-string resonance with mass-squared m 2 = t = p 2 a. On the other hand, at positive half-integer values of t it vanishes, implying that strings with half-integer mass-squared do not propagate in this channel. In [24] it was shown that these are also properties of the amplitude for one NS-NS string to decay into two massless open strings stuck to a D-brane. he low-energy effective Lagrangian of massless and tachyonic particles will contain the terms L eff H p) F + L YM + D D, 8) as well as the terms we are looking for. Here L YM is the Yang-Mills Lagrangian and in keeping with the spirit of the rest of this note all constants have been omitted. At low energies t = p 2 a 0) the prefactor of Eq. 7) may be expanded π Γ[ 2t] 1 Γ[ 1 = 2 t]2 2t +2log2+Ot). 6

8 he first term corresponds to the RR particle decaying into a massless open string via the first term in Eq. 8)) which propagates resulting in the pole) and decays into two tachyons via the third term in Eq. 8)). Because the second term is non-zero, the effective action contains a coupling between the RR field, the tachyon and the antitachyon. In summary, in this subsection we have shown that the effective action contains the term S eff = H p) r d. which is Eq. 2) after integrating by parts and covariantising. As an aside, if there is no antitachyon in the case of the non-supersymmetric brane) the integration by parts gives zero. 2.3 he four-point amplitude It is left to prove that the effective action also contains the remaining terms of Eq. 3), which are λ H p) A + A ) = λ C p 1) A + A )d )+λ C p 1) F + F ). 9) he coupling, λ, is calculated from the finite part of the low energy limit of the associated string amplitude: ) λ = Finite part lim E 0 A,,A,RR = Finite part lim V 0) E 0 V 1 V 0) A V 1) RR. A direct calculation of this amplitude is difficult, even in the low-energy limit, because of the integrals involved. However, we know that λ = g in the units of Eq. 4)) because the first term on the RHS of Eq. 9) must appear by covariantising d d. he amplitude just written is the only one which can give this term, but because it contains the RR field strength it also gives the second term on the RHS of Eq. 9). hus, for the single brane-antibrane pair, the effective action at OC, 2 ) is indeed given by Eq. 4). 3 Summary and Discussion In coincident brane-antibrane systems we have shown, by calculating tree-level string amplitudes, that in addition to the usual Wess-Zumino terms the world-volume effective action contains [ C p 1) r D D 1 2 {F +, }+ 1 2 {F, } ], to O ). After tachyon condensation the correct charges for decay products are obtained. We propose that the full result to all orders in the tachyon) can be written in terms of the curvature of the superconnection: C Sr exp F. 7

9 It is amusing to note the connection of our proposal with noncommutative geometry. It would be very interesting to show that some sort of similar substitution F Fcould be made in the Dirac-Born-Infeld part of the effective action. Acknowledgments We thank Siddhartha Sen and Jim McCarthy for commenting on various drafts. his work was supported financially by the Enterprise Ireland grant F References [1] A. Sen achyon condensation on the brane antibrane system JHEP ) hep-th/ [2] P. Horava ype IIA D-branes, K-theory and Matrix theory CAL hep-th/ [3] O. Bergman and M. R. Gaberdiel Stable non-bps D-particles hep-th/ [4] A. Sen ype I D-particle and its interactions hep-th/ [5] A. Sen BPS D-branes on non-supersymmetric cycles JHEP ) hep-th/ [6] A. Sen Descent relations among bosonic D-branes hep-th/ [7] M. Frau, L. Gallot, A. Lerda and P. Strigazzi Stable non-bps D-branes in type I string theory DF hep-th/ [8] A. Sen SO32) spinors of type I and other solitons on brane-antibrane pair JHEP ) hep-th/ [9] E. Witten D-branes and K-theory hep-th/ [10] O. Bergman, E. G. Gimon and P. Horava Brane transfer operations and -duality of non-bps states hep-th/ [11] A. Sen Stable non-bps states in string theory JHEP ) hep-th/ [12] A. Sen Stable non-bps bound states of BPS D-branes JHEP ) hep-th/ [13] J. Polchinski Dirichlet-branes and Ramond-Ramond charges Phys.Rev.Lett ) hep-th/ [14] M. Li Boundary states of D-branes and dy-strings Nucl. Phys. B ) hep-th/

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