On T-duality of $R^2$-corrections to DBI action at all orders of gauge field

0.15 MB·

by  Ahmad Ghodsi

On T-duality of R2-corrections to DBI action
at all orders of gauge field
Ahmad Ghodsi ∗, Mohammad R. Garousi †and Ghadir Jafari ‡
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Department of Physics, Ferdowsi University of Mashhad,
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n P.O.Box 1436, Mashhad, Iran
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January 28, 2016

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Abstract
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v Recently, it has been observed that in a T-duality invariant world-volume theory
0
in flat spacetime, all orders of gauge field strength and all orders of the D-brane
3
velocity appear in two specific matrices. Using these two matrices, we construct the
5
7 world-volume couplings of two massless NSNS states at order α′2 and all orders of
0
the velocity and the gauge field strength, by requiring them to be invariant under the
.
1 linear T-duality. The standard extension F → F +P[B], then produces all orders of
0
the pull-back of B-field into the action. We compare the resulting couplings for zero
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1 velocity andgaugefieldstrength,withtheα′2 termsofthedisk-levelS-matrixelement
:
v of two massless NSNS vertex operators in the presence of a constant background B-
i field. We have found an exact agreement.
X
r
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∗[email protected]
†[email protected]
‡[email protected]
1
1 Introduction
D-branes are non-perturbative objects in superstring theory which play the central role in
exploring different aspects of the theory, from statistical computation of black hole entropy
[1] to realization of the AdS/CFT correspondence [2] or appearance of noncommutative
geometry in string theory [3]. These objects should be described by the supersymmetric
extension of the cubic string field theory [4] which includes massless and all infinite tower
of the massive excitations of the open strings. In the Wilsonian effective action, however,
the effect of all massive states appears in the higher derivatives of the massless fields.
At long wavelength limit, the higher derivative terms can be ignored and D-branes
are completely described by Dirac-Born-Infeld (DBI) action [5, 6] which includes constant
metric, B-field and dilaton, as well as the first derivative of the massless NS fields. A non-
Abelian extension for DBI action has been proposed in [7]. The leading higher derivative
corrections to the Abelian DBI action should include acceleration which appears through
the second fundamental form Ω (see e.g., [8]) and the first derivative of the gauge field
strength in the NS part, as well as the first and the second derivatives of the metric, B-field
and dilaton in the NSNS part. The first derivative of the metric appears also through Ω
and the second derivatives of the metric appears through the curvature terms.
Theleadinghigherderivativecorrectionsofthemetrichavebeenfoundin[9]byrequiring
the consistency of the effective action with the O(α′2) terms of the corresponding disk-level
scattering amplitude [10, 11]. For totally-geodesic embeddings of the world-volume in the
ambient spacetime (Ω = 0), the corrections to the DBI action in string frame for vanishing
gauge field and B-field and for a constant dilaton is the following action [9]1:
π2α′2T
S = p dp+1xe−φ −G R Rabcd −2Rˆ Rˆab −R Rabij +2Rˆ Rˆij ,(1.1)
abcd ab abij ij
48 Z q
(cid:16) (cid:17)
e
where Rˆ = G˜cdR , Rˆ = G˜cdR and G˜ is the pull-back of the bulk metric, G˜ =
ab cadb ij cidj ab ab
∂ Xµ∂ Xνg . In above equation, the Riemann curvatures are the projections of the bulk
a b µν
Riemann tensors into the world-volume or the transverse space. For example:
R = ∂ Xα∂ Xβ∂ Xµ∂ XνR . (1.2)
abcd a b c d αβµν
1In our index notation the Greek letters (µ,ν,···) are the indices of the space-time coordinates, the
Latin letters (a,b,c,···) are the world-volume indices and the letters (i,j,k,···) are corresponding to the
normal coordinates of the D-brane.
1
The above action includes all orders of the velocity through the pull-back operator in the
static gauge, i.e., Xa = σa and Xi = λi . The acceleration terms (Ω 6= 0) have been also
found in [9] (see also [12]). The consistency of these couplings with T-duality may include
all orders of dilaton, B-field and F.
In [15, 16], it has been shown that the consistency of the couplings in (1.1) at zero
velocity with linear T-duality transformation, requires ∇H and ∇∇φ couplings to appear
as:
π2α′2T
S = p dp+1xe−φ −G˜ R Rabcd −2Rˆ Rˆab −R Rabij +2Rˆ Rˆij
abcd ab abij ij
48 Z
p (cid:16)
1 1 1
− ∇ H ∇aHijk − ∇ H ∇iHabc + ∇ H ∇aHbci , (1.3)
a ijk i abc a bci
6 3 2
(cid:17)
where Rˆ = Rˆ +∇ ∇ φ and H is the field strength of the B-field. The consistency of the
µν µν µ ν
action(1.1) at zero velocity andnon-zero Ωwith nonlinear T-duality should also include H4
couplings in which we are not interested in this paper. Such couplings have been found in
[13, 14] for O-planes. In this paper, we are interested to include all orders of the constant
Abelian gauge field strength F in the above action. This can be done by releasing the
assumption in [15] that the velocity is zero, because the world-volume transverse scalars
transform to the gauge fields under the T-duality transformations.
Recently, it has been observed in [24] that in a T-duality invariant world-volume theory
in flat spacetime, all orders of gauge field strength and all orders of the D-brane velocity
appear in the following two matrices:
Gµν = ∂ Xµ∂ Xν Gab,
a b
Θµν = ∂ Xµ∂ Xν Θab, (1.4)
a b
where Gab and Θab are the symmetric and antisymmetric parts of ( 1 )ab, respec-
η+∂λi∂λi+F
tively. They transform into each other under T-duality [24]. Using these matrices, in
the next section we construct world-volume R2,Rφ,φ2, R∇H,φH and (∇H)2 couplings
by requiring them to be invariant under the linear T-duality and to reduce to (1.1) when
F = 0. In section three, in order to confirm our result with the S-matrix element, we use
the standard extension F → F +P[B], to include all pull-back of B-field into the action.
We then compare the resulting couplings for zero velocity and gauge field strength with the
α′2 terms of the disk-level S-matrix element of two massless NSNS vertex operators in the
presence of constant background B-field [25, 26]. We will show an exact agreement.
2
2 T-duality constraint
In this section, we are going to impose the constraint that the effective action should be
invariant under linear T-duality transformations, to include constant gauge field strength
into the action (1.3). Let us first review the strategy presented in [15] and [17]. As we know
under T-dualitytheNeumannandDirichlet boundaryconditionsareexchanging [18], thisis
due to the fact that a compact direction that is lying in the world-volume of the D-brane in
original picture, becomes a transverse direction after T-duality. Symbolically if we consider
S as an action for D-brane and y as a compact direction then under T-duality we will have:
T-duality
S [(a,y),i] ←−−−−→ S [a,(i,y)], (2.1)
Dp Dp−1
where a represents the tangent coordinate to the D-brane and i stands for normal coor-
dinate. The Killing index y on the left-hand side is a tangent coordinate whereas on the
right-hand side it is a normal coordinate.
The T-duality transformationforgaugefieldis A˜ = λy which isalinear transformation.
y
However, the T-duality transformation for NSNS fields are nonlinear [19, 20, 21, 22, 23].
Assuming NSNS fields as perturbations around the flat space, one can find the relevant
linear T-duality transformations for the closed string fields, i.e.,
1
φ˜= φ− h , h˜ = −h , h˜ = h ,
yy yy yy µν µν
2
h˜ = B , B˜ = h , B˜ = B . (2.2)
µy µy µy µy µν µν
One can use the T-duality implied by (2.1) as a constraint to find some new couplings from
the known ones, in fact this is the method have been used in [15] and [17].
Before going through the T-duality consideration we note that the curvature terms, and
those terms with ∇∇φ and ∇H in equation (1.3) are the projections of the corresponding
bulktensorsintotheworld-volumeorthetransversespace. Byusingtheprojectionoperator
onto the vector space normal to the D-brane defined as:
⊥µν = nµnµi = Gµν −G˜µν , (2.3)
i
where G˜µν = ∂ Xµ∂ XνG˜ab is the first fundamental form, one can transform the transverse
a b
indices in the action (1.3) into the bulk space-time coordinates. As an example:
Rˆ Rˆij = ⊥µν⊥αβRˆ Rˆ = ⊥µν⊥αβG˜ab∂ Xρ∂ XλR G˜cd∂ Xσ∂ XγR
ij αµ βν a b αρµλ c d βσνγ
= ⊥µν⊥αβG˜ρλG˜σγR R . (2.4)
αρµλ βσνγ
3
Apart from the overall constant factor, the Lagrangian density in the action (1.3) can be
rewritten as:
L = G˜αβG˜νµ 2R ρ κR −R ρκR −2G˜αβG˜νµG˜ρκ 2R λR −R λR
α β νρµκ αν βµρκ ανβ ρµκλ ανρ βµκλ
−1G˜αβ∇ Hγ(cid:0)δρ ∇ H +G˜αβG˜γδ∇ H ρ(cid:1)κ∇ H − 2G˜αβ(cid:0)G˜γδG˜ρκ∇ H ∇λH (cid:1)
3 α β γδρ γ α δ βρκ 3 λ βδκ αγρ
+4R G˜αβ∇µ∇λφ−8R G˜αβG˜λµ∇ν∇ φ+2∇ ∇ φ∇β∇αφ−4G˜αβ∇ ∇ φ∇µ∇ φ.
αλβµ λβµν α β α µ β α
(2.5)
Writing in this form, all the Riemann curvatures, ∇H and ∇∇φ will be the bulk tensors.
Moreover, in this form, the transverse scalar fields appear in the action just through the
first fundamental form G˜µν. To find the scalar couplings, it is sufficient to break the indices
of G˜µν to tangent and normal indices and to go to the static gauge i.e., Xa = σa and
Xi = λi as follows:
G˜ia = G˜ab∂ λi,
b
G˜ij = G˜ab∂ λi∂ λj, (2.6)
a b
where G˜ab is the inverse of the pull-back metric in the static gauge i.e., G˜ab = ( 1 )ab.
g+∂λi∂λi
The bulk metric in G˜µν is not flat, so the Lagrangian density (2.5) is covariant and
contains all orders of the D-brane velocity. It would be desirable to include all orders of
gauge field in a covariant expression. That expression may be found by expanding G˜µν in
(2.5) in terms of different orders of g and velocity, and then include appropriate B-field
and F to make each term to be invariant under the T-duality. To restrict the calculation
for finding only F-terms, we assume the bulk metric in G˜µν to be flat metric. Then the
extension G˜µν → Gµν where Gµν is given in (1.4), produces the following couplings:
L = GαβGνµ(2R ρ κR −R ρκR )−2GαβGνµGρκ 2R λR −R λR
α β νρµκ αν βµρκ ανβ ρµκλ ανρ βµκλ
(cid:0) (cid:1)
−1Gαβ∇ Hγδρ ∇ H +GαβGγδ∇ H ρκ∇ H − 2GαβGγδGρκ∇ H ∇λH
3 α β γδρ γ α δ βρκ 3 λ βδκ αγρ
+4R Gαβ∇µ∇λφ−8R GαβGλµ∇ν∇ φ+2∇ ∇ φ∇β∇αφ−4Gαβ∇ ∇ φ∇µ∇ φ.
αλβµ λβµν α β α µ β α
(2.7)
Obviously this action reduces to (2.5) when F → 0. When the indices of Gµν are not the
Killing index y, the above expression is invariant under linear T-duality. The matrix Gµν
has even number of F, so the above couplings are also invariant under the parity.
4
However, the above action is not invariant under the linear T-duality when the indices
of Gµν are the Killing index. It has been shown in [24] that under T-duality Gµν transforms
to Θµν. So in the T-duality invariant theory there must be new couplings involving Θµν as
well. Using the fact that Gµν has even number of F and Θµν has odd number of F, the
invariance under parity requires the new terms contracted by even number of Θ in R2,Rφ
and H2 terms and odd number in RH and φH couplings.2 Moreover, the couplings R2
and H2 in (2.7) in which the indices of Gµν can be the Killing index y, contain two or
three Gµνs. As a result, T-duality requires the new terms with structure R2 and H2 to
have two or three Θ and/or G. The couplings Rφ in which the indices of Gµν can be the
Killing index, contain one or two Gµνs. As a result, T-duality requires the new terms with
structure Rφ to have two Θs, and the terms with structure Hφ to have one, two or three
Θ and/or G. Note that the indices of Gµν in φ2 term can not be the Killing index, so there
is no new coupling with structure φ2.
All possible independent R2 and Rφ terms are then the followings:
L = ΘαβΘκλ α R µνR +α R µ νR +α R µνR +ρR ∇µ∇ φ
1 1 ακ βλµν 2 α κ βµλν 3 αβ κλµν βκλµ α
(cid:0) (cid:1)
+ΘαβΘκλGµν α R ρR +α R ρR +α R ρR +α R ρR , (2.8)
4 ακµ βλνρ 5 αµκ βνλρ 6 αβµ κλνρ 7 ακβ λµνρ
(cid:0) (cid:1)
where α ,··· ,α and ρ are constants. Similarly for H2 terms we have:
1 7
L = ΘαβΘγδ β∇ H µν∇ H +β ∇ H µ∇νH +β ∇ H µν∇H +β ∇ H µ∇νH
2 1 β δ γ αµν 2 ν βδ αγµ 3 γ α δ βµν 4 ν γδ αβµ
(cid:0) (cid:1)
+ΘαβΘγδGµν β ∇ H λ∇ H +β ∇ H λ∇H +β ∇ H λ∇H +β ∇ H λ∇ H
5 β δν γ αµλ 6 γ αµ δ βνλ 7 β αµ δ γνλ 8 µ αγ ν βδλ
(cid:0)
+β ∇ H λ∇ H +β ∇ H λ∇ H +β ∇ H λ∇ H +β ∇ H ∇λH
9 µ αβ ν γδλ 10 β αγ ν δµλ 11 γ αβ ν δµλ 12 λ γδν αβµ
+β ∇ H ∇λH , (2.9)
13 λ βδν αγµ
(cid:1)
where β ,··· ,β are unknown coefficients. The new possible RH and φH couplings are:
1 13
L = ΘαβGγδ σ R ∇ H µν+σ R ∇ H µν +σ R ∇νH µ +σ R ∇νH µ
3 1 βδµν α γ 2 αβµν δ γ 3 γµδν αβ 4 βµδν αγ
(cid:0)
+σ R ∇ H µν +ΘαβGγδGµν σ R ∇ H ρ +σ R ∇ H ρ +σ R ∇ H ρ
5 βδµν γ α 6 βδνρ α γµ 7 δµνρ β αγ 8 δµνρ γ αβ
(cid:1) (cid:0)
+σ R ∇ H ρ +σ R ∇ H ρ +σ R ∇ H ρ +σ R ∇ H ρ
9 βµνρ δ αγ 10 αβνρ δ γµ 11 βδνρ µ αγ 12 βνδρ µ αγ
(cid:1)
+ΘαβΘγδΘµν σ R ∇ H ρ+σ R ∇ H ρ+σ R ∇ρH +σ R ∇ H ρ)
13 δµνρ β αγ 14 βδνρ µ αγ 15 βδνρ αγµ 16 δµνρ γ αβ
(cid:0)
+γ ΘαβΘγδΘθκ∇ ∇ φ∇ H +γ Θαβ∇ H ∇δ∇γφ+γ ΘαβGγδ∇ H θ∇ ∇ φ
1 γ α δ βθκ 2 δ αβγ 3 δ βγ θ α
+γ ΘαβGγδ∇ H θ∇ ∇ φ+γ ΘαβGγδ∇ H ∇θ∇ φ, (2.10)
4 δ αβ θ γ 5 θ αβδ γ
2In fact we have observedthat for example by considering a term such as RRΘ,the T-duality transfor-
mation leads to RHG couplings, which reduce to RH couplings in vanishing F limit, but such terms do
not exist in (2.5), so T-duality requires parity invariance.
5
here again σ ,··· ,σ and γ ,··· ,γ are unknown coefficients. In writing the above terms
1 16 1 5
we have considered independency by taking care of the Bianchi identities (for more details
see appendix A). In this regard (2.8)-(2.10) are the most general Lagrangians. In our
calculation we are going to assume constant G and Θ and to work with the second order of
perturbations, so the terms with coefficients β ,β ,σ and γ are total derivatives. Hence,
3 6 15 1
we ignore them, i.e., β = β = σ = γ = 0. On the other hand, for some specific relations
3 6 15 1
between the coefficients of some of the above terms, there might be total derivatives which
should be dropped. We will find such terms after imposing the T-duality constraint.
In order to fix the unknown coefficients in above Lagrangians by the linear T-duality
(2.2), we need to expand the metric around the flat background as g = η +h and keep
µν µν µν
terms up to the second order of perturbation expansion. Moreover one needs to expand
Gµν and Θµν up to the third order of velocity and/or F in the static gauge. Performing
these steps for (2.7)-(2.10), we compute both sides of relation (2.1). Next we apply the
linear T-duality on left(right) hand side and identify it with right(left) hand side to fix the
unknown coefficients. After all we find the following relations:
α α
1 4
α = 0 , α = − , α = −4 , α = 2− , α = 8 , ρ = −8;
2 3 5 6 7
2 2
β
10
β = 0 , β = 0 , β = 1 , β = +β +β ,
1 2 4 7 11 5
2
β β β β
10 11 11 10
β = , β = , β = −2− , β = 2− ; (2.11)
8 9 12 13
4 4 4 4
σ
1
σ = , σ = −4 , σ = 0 , σ = 4 , σ = 16+2σ ,
2 3 4 5 6 10
2
σ
13
σ = σ , σ = −8 , σ = 16−σ , σ = −16 , σ = 8+ ;
7 11 8 9 11 12 14
2
γ = 2 , γ = 2γ , γ = −4−γ .
2 3 4 5 4
In obtaining the above results we have used the integration by part and assumed Gµν and
Θµν are constant. Substituting (2.11) into the Lagrangian (2.8) for R2 and Rφ terms, we
have found:
L = ΘαβΘγδGρκ −4R λR +2R λR +8R λR
1 αργ βκδλ αβρ γδκλ αγβ δρκλ
(cid:0) (cid:1)
−8ΘαβΘκλR ∇θ∇ φ. (2.12)
βκλθ α
We have also found that the unfixed coefficients, α and α appear in the following expres-
1 4
sions:
α ΘαβΘγδ(R ρκR − 1R ρκR )+α ΘαβΘγδGρκ(R λR − 1R λR ).
1 αγ βδρκ 2 αβ γδρκ 4 αγρ βδκλ 2 αβρ γδκλ
6
However, they are total derivatives up to the second order of the perturbations. Since
our calculations are valid up to the second order of the perturbations, we can ignore these
terms, i.e., α = α = 0. For H2 terms, we have found
1 4
L = ΘαβΘµν∇ H ρ∇σH +2ΘαβΘµνGρσ ∇ H ∇λH −∇ H ∇λH .(2.13)
2 σ µν αβρ λ βνσ αµρ λ µνσ αβρ
(cid:0) (cid:1)
Wehave alsoshowed thattheconstants β ,β andβ arethecoefficients oftotalderivative
5 10 11
terms, so we set them to zero. Finally for RH and φH couplings in (2.10) we get:
L = 8ΘαβGγδGµν 2R ∇ H ρ −R ∇ H ρ +2R ∇ H ρ −2R ∇ H ρ
3 βδνρ α γµ δµνρ γ αβ βµνρ δ αγ βνδρ µ αγ
(cid:0) (cid:1)
+4ΘαβGγδ R ∇ H µν−R ∇νH µ +8ΘαβΘγδΘµν R ∇ H ρ+R ∇ H ρ
βδµν γ α γµδν αβ δµνρ γ αβ βδνρ µ αγ
(cid:0) (cid:1) (cid:0) (cid:1)
+2Θαβ∇ H ∇δ∇γφ−4ΘαβGγδ∇ H ∇θ∇ φ. (2.14)
δ αβγ θ αβδ γ
We have also found that the constants σ ,σ ,σ ,σ and γ are again the coefficients of
1 10 11 13 4
total derivative terms, so we set them to zero too.
It is interesting to note that the consistency of the couplings (2.5) with the linear T-
dualitycoulduniquelyfixallordersofconstantF. Theyaregivenin(2.7), (2.12), (2.13)and
(2.14). Similar observation has been made in [24] in making the world-volume transverse
′
scalar couplings at order α in the bosonic string theory to be consistent with T-duality.
In the next section, we are going to compare the above couplings with the corresponding
disk-level S-matrix element.
3 Comparison with S-matrix
Intheprevioussection, wehavefoundthecouplingsoftwomasslessNSNSstatesatorderα′2
to all orders of gauge field which appear through G and Θ. It is known that the pull-back of
B-field canbe included into the D-braneeffective actionvia the replacement F → F+P[B].
This combination is invariant under B-field gauge transformation as P[B] → P[B]−P[dΛ],
F → F + P[dΛ]. After this replacement, we set the velocity and gauge field strength F
to zero. This produces then the couplings of two NSNS states at order α′2 to all orders of
constant B-field. Such couplings may be compared with the disk-level S-matrix element of
two NSNS vertex operators in the presence of background B-field.
The S-matrix element of two NSNS vertex operators in the presence of background
7
B-field has been calculated in [25, 26], i.e.,
T −det(η+B) Γ(−t/2)Γ(−2s) t
p
A = − −2sa + a , (3.1)
p 2 Γ(1−t/2−2s) (cid:18) 1 2 2(cid:19)
where t = −2p ·p is the square of the momentum transfer in the transverse directions
1 2
and s = −1p ·D·p is the momentum flow parallel to the world-volume of the D-brane.
2 1 1
D is defined such that for world-volume coordinates it is Dab = (η+B)ab and for transverse
η−B
directions it is Dij = −δij. The kinematic factors a and a are given by:
1 2
a = Tr(ε ·D)p ·ε ·p −p ·ε ·D·ε ·p −p ·ε ·εT·DT·p −p ·εT·ε ·D·p
1 1 1 2 1 1 2 1 2 1 2 1 1 1 2 1 1
1
− (p ·ε ·εT·p +p ·εT·ε ·p )−sTr(ε ·εT)+ 1 ←→ 2 ,
2 1 2 1 2 2 1 2 1 1 2
n o
a = Tr(ε ·D)(p ·ε ·D·p +p ·D·ε ·p +p ·D·ε ·D·p )+p ·D·ε ·D·ε ·D·p
2 1 1 2 2 2 2 1 2 2 2 1 1 2 2
1
− (p ·D·ε ·εT·DT·p +p ·DT·εT·ε ·D·p )−sTr(ε ·D·ε ·D)
2 2 2 1 1 1 1 2 2 1 2
+sTr(ε ·εT)+Tr(ε ·D)Tr(ε ·D)(s+t/4)+ 1 ←→ 2 ,
1 2 1 2
n o
where ε and ε are the polarizations of the NSNS states. In order to find the corresponding
1 2
effective action at order α′2, we need to expand it in powers of α′p2 which is given as
T −det(η+B) t 1 π2α′2
A = − p (−2sa + a ) + +O(α′4) . (3.2)
p 2 1 2 2 (cid:18)st 24 (cid:19)
The leading order of this amplitude contains a t-channel and s-channel in addition to some
contact terms. This order completely is described by supergravity action in the bulk plus
the DBI action on the D-brane [25]3. At all other orders, the amplitude just contains
contact terms which are the effective couplings in the momentum space. We are interested
in the α′2-contact terms.
In what follows we impose physical conditions for graviton (ε = ε ) and B-field
µν νµ
(ε = −ε ) to find two-graviton, two-B-field and one-graviton-one-B-field couplings in
µν νµ
the presences of background B-field. For two gravitons we can simplify O(α′)2 part as
3This is similar to bosonic string computations [27].
8
follows:
A ∼ s2Tr(ε ·ε )−2s Tr(ε ·V )p ·ε ·p + tTr(ε ·V )Tr(ε ·V )(4s+t)
1 2 1 S 1 2 1 4 1 S 2 S
+stTr(ε ·ε ·V )+2t p ·V ·ε ·ε ·V ·p +2s p ·ε ·V ·ε ·p −stTr(ε ·V ·ε ·V )
1 2 S 1 S 1 2 S 1 1 2 S 1 2 1 S 2 S
+4sp ·ε ·ε ·V ·p −2tTr(ε ·V )p ·ε ·V ·p +2tTr(ε ·V )p ·V ·ε ·V ·p
1 2 1 S 1 1 S 1 2 S 1 1 S 1 S 2 S 1
−2t p ·V ·ε ·V ·ε ·V ·p −stTr(ε ·V ·ε ·V )−4t p ·V ·ε ·V ·ε ·V ·p
1 S 2 S 1 S 1 1 A 2 A 1 A 2 A 1 S 1
+2tTr(ε ·V )p ·V ·ε ·V ·p −2t p ·V ·ε ·V ·ε ·V ·p + 1 ←→ 2 , (3.3)
1 S 1 A 2 A 1 1 A 2 S 1 A 1
n o
where V and V are the symmetric and antisymmetric parts of V = η+DT = 1 . The am-
S A 2 η+B
plitude is invariant under parity (V → V ,V → −V ). We observe that theB-dependence
S S A A
appears as 1 which is the same as the F-dependence in the T-duality invariant couplings
η+B
that we have found in the previous section. More precisely, the parts of amplitude (3.3)
that contain the symmetric matrix V , coincide with the R2 terms in (2.7). The other terms
S
containing the antisymmetric matrix V , also completely reproduce (2.12). This part of the
A
S-matrix calculations has been done already in [26]. Our results confirm the computations
of [26] after considering some identities for R2 structures (for more details see appendix A).
The contact terms for two antisymmetric B-fields can be found by imposing the corre-
sponding polarization tensors in (3.2), i.e.,
A ∼ 1s(4s+t)Tr(ε ·ε )−2s p ·ε ·ε ·p +stTr(ε ·V ·ε )−4s p ·ε ·ε ·V ·p
2 1 2 2 1 2 1 1 S 2 1 2 1 S 1
−st Tr(ε ·V ·ε ·V )+2s p ·ε ·V ·ε ·p −2t p ·V ·ε ·V ·ε ·V ·p
1 S 2 S 2 1 S 2 1 1 S 1 S 2 S 1
−st Tr(ε ·V ·ε ·V )+2t p ·V ·ε ·ε ·V ·p + 1t(2s+t)Tr(ε ·V )Tr(ε ·V )
1 A 2 A 1 A 2 1 A 1 4 1 A 1 A
−2t Tr(ε ·V )p ·ε ·V ·p −4t p ·V ·ε ·V ·ε ·V ·p −4t Tr(ε ·V )p ·V ·ε ·V ·p
1 A 1 2 A 1 1 A 2 A 1 S 1 1 A 1 A 2 S 1
−2t p ·V ·ε ·V ·ε ·V ·p + 1 ←→ 2 . (3.4)
1 A 2 S 1 A 1
n o
Again we see that the part without V is in agreement with H2 terms in (2.7) and the
A
remaining part is reproduced exactly by (2.13).
Theamplitudeforonegravitonandoneantisymmetric B-fieldhasnocounterpartatzero
background B-field limit because it has odd number of V . Imposing physical conditions
A
9