- Email: [email protected]
Super-Poincaré invariant Koba-Nielsen formulas for the superstring
. f!l!ii
ca
26 September
1996
mm
‘53
ELSEWER
PHYSICS
Physics Letters B 385 (1996) 109-l
LETTERS
B
16
Super-Poincark invariant Koba-Nielsen forrnulas for the superstring Nathan Berkovits ’ Dept. de Fisica Matemdtica, Univ. de &To Paula, CP 66318 SEo Paula, SP 05389-970, Brazil and IMECC, Univ. de Campinas, CP 1170 Campinas, SP 13100, Brazil Received 21 June 1996 Editor: M. Dine
Abstract The new spacetime-supersymmetric description of the superstring is used to compute tree-level scattering amplitudes for an arbitrary number of massless four-dimensional states. The resulting Koba-Nielsen formula is manifestly SO(3,l) super-Poincare invariant and is easily generalized to scattering in the presence of a D-brane.
1. Introduction Before the work of this author, all spacetime-supersymmetric calculations of superstring amplitudes used the light-cone Green-Schwarz formalism, which requires non-trivial operators at the light-cone interaction points. [l-3] Because of this complication, only the four-point massless tree and one-loop amplitudes had been explicitly computed in a spacetime-supersymmetric manner. * Over the last few years, this author has developed an alternative spacetime-supersymmetric description of the superstring which has the advantage of being conformally invariant, and therefore not requiring non-trivial interaction-point operators. [ 51. The new formalism is manifestly SO(3,l) super-Poincare invariant and is suitable for any compactification to four dimensions which preserves N=l 4D supersymmetry. Furthermore, the new formalism contains critical N=2 worldsheet superconformal invariance, and is related by a field redefinition to the N=2 embedding of the N=l RNS superstring. [6] Together with Vafa, a simple “topological” method was developed for calculating scattering amplitudes in this formalism. [7] (An easily accessible review of this method can be found in Ref. [LX]) In this letter, the 1E-mail: nberkovi@snfn&.if.usp.br. ’ Although Ref. [3] contains explicit expressions for four-point multiloop amplitudes, these expressions contain unphysical divergences when interaction points coincide. It has not yet been determined how these expressions are affected by removing the divergences with contact terms. Also, Mandelstam has constructed an N-point tree amplitude which has manifest SU(4) xU( 1) super-Poincare invariance. [4] However the structure of his amplitude was obtained indirectly by unitarity arguments, rather than by explicit computations. It would be interesting to compare his amplitude with the Koba-Nielsen formula in this letter. 0370-2693/96/$12.00 Copyright PZZ SO370-2693(96)00873-S
Q 1996 Elsevier Science B.V. All rights reserved.
110
N. Berkouits/Pkysics
Letters B 385 (1996) 109-116
topological method will be used to explicitly compute tree-level scattering amplitudes for an arbitrary number of four-dimensional massless states. The result can be expressed as a Koba-Nielsen formula, and is manifestly SO( 3,l) super-Poincare invariant. It is straightforward to generalize the formula for scattering in the presence of a D-brane. This result should be useful for D-brane analysis since, unlike computations in the RNS formalism, [ 91 it is manifestly spacetimesupersymmetric, and therefore includes both NS-NS and R-R couplings.
2. Review As shown in Ref. [7], the topological prescription superstring states in any c=6 N=2 string theory is
for computing
the tree-level
amplitude
for N open
(2.1) where V( z ) is a U( 1 )-neutral N=2 primary field of weight 0, [ T, Gf ,G- , .I] are the twisted c=6 N=2 generators (after twisting, Gf has weight 1 and G- has weight 2)) G*(Y) means the contour integral of G’ around Y, @ = G- ( eiH) , and J = dH. The Chan-Paton factors will be supressed throughout this letter. When (2.1) is applied to the N=2 embedding of the N=l RNS superstring, it reproduces the standard RNS prescription in the “large” Hilbert space:
d =
Hzdr/;RNs
(2.2)
since in the N=2 embedding, V = cVRNs, G+ = jBRST, G- = b, and @ = q. (Note that p = ate-6 and y = rle$ are the bosonic RNS ghosts, 2 = {Q, S} is the RNS picture-changing operator, Vms is in the -1 picture, and zvms = {Q, @ar=} is in the 0 picture). So G+(V) = Q(g%‘“““) = ZVms and G+(V) = VRNs. In the new spacetime-supersymmetric description of the superstring, a field redefinition is used to write the generators of the N=2 embedding in terms of spacetime-supersymmetric variables. [5] In terms of the fourvariables, the dimensional supersymmetric variables [x m, 8”) I?‘, pa, pk, p] and the compactification-dependent twisted c=6 N=2 generators are T=~dxn’dx,,+p,aea+p~.a~~+~(dpdp+id2p)+TC, G+ = e@(d)* + G;,
G- = e-@(d)*
+ GE,
J = -Jp
+ Jc,
(2.3)
where m takes values 0 to 3, (Y and & take values 1 or 2, pa and & are conjugates boson, d, =pe + ;i%xa,
- $(I?)%,
+ $$J(8)2,
& = j& + ;B”&&
to 8” and @‘, p is a chiral
- $(8)%&
JC] are the twisted c=9 N=2 generators and [Tc, Gs,G,, (d)2 means i@d,dp, theory used to describe the compactification manifold. The action in conformal gauge for these fields is
+ ~a&a(e)2,
of the superconformal
field
N. Ber!uwits/Physics
where (as in with 0 The
111
Letters B 385 (1996) 109-116
SC is the action for the compactification-dependent variables. When the R/L index has been suppressed (2.3) ) , we shall always mean left-moving. Note that z versus Z is correlated with L versus R, and not versus e. free-field OPE’s of the four-dimensional variables are
z),
PR(Y)PR(Z)
+ log(J - 9,
(2.5) Note that the chiral boson p can not be fermionized since &J’(Y) e-ip(z) + (y - z). It has the same behavior as the negative-energy field qb that appears when bosonizing the RNS ghosts. After twisting, the background charge is defined for open superstring tree amplitudes by (W2@)
2,-iP+‘Hc)
= 1
where Jc = dHc. As was shown by Siegel,
(2.6)
[lo]
d, and & satisfy the OPE that d,(y)
dp(z)
is regular,
(2.7) where II,& =
O-zkax, - $a&+ - $aO,.
The advantage ofworking with the variables d, and III, is that they commute with the spacetime supersymmetry generators, 4a =
3. Koba-Nielsen formula for open superstrings For massless states of the open superstring which are independent V only depends on the zero modes of x”‘, 0” and @. It can therefore which is the standard N=l 4D super-Yang-Mills prepotential (e.g. field A”$, and the 0(8)2 and (6r)28 components of V are the gluino The condition of being N=2 primary implies that
of the compactification, the vertex operator be represented by the superfield V( x, 8, e) , the &? component of V is the gluon gauge fields 5” and 4”).
where D, = dp + ~8”<&,, and D, = de& + $?“a:&?,,, are the covariant fermionic derivatives. Note that ( D)2V = (D)2V = 0 is the standard supersymmetric generalization of the Lorentz gauge condition d,Atn = 0. Using the free-field OPE’s of (2.5)) it is straightforward to compute that
J
dzG-(G+(V))
=
J
dz(d”
(D)2D,V+J’
(D)2~‘d!V+d0aDuV-ade’
~kV-iIIa’
[Da,Db]V).
N. Berkovits/Physics
112
Plugging
Letters B 385 (1996) 109-116
into (2.1) ,
d = (~(z,)ei~d”(z2)D~~(z~)e-2”p+Hc d-’ (z3)Dciv3(a) f+W(zr)
(D)*&K(zr)
r=4 + a+(&)
(D)*D&vr(z,)
+ ae”(z,)D,v,(z,)
- J@(z,)
D&K(Zr)
- irIa~(zz)
uLmVr(Zr)))
N
+
nn~(Z,)a~~])eif(Z~)e-*iP(Z3)+iH~(Z3)
wa=Du 2 29
@ = 2
+$ = (D )2Dh r r T)
a”
2
=
w: =
a& - @ 2-
= 0,
(D,.)*D,“,
(3.1)
f@ = Df,
w; = a!$ = @
a; = D,“, Z$!= -@,
for r > 3, and 0,” and Df are fermionic
derivatives
= a?“! = 0,
a:’ = -i[DF,@]
which act only on V, (z,) . For example,
N = K(z,)D%(z2)
D;;v,(i,)
Using the fact that (d”(z)...)= OPE’s of (2.7) to write
+
$c T,SJ
ErffsEt
fiv,(zr). r=3
C,($-$f),
where fS is the residue of the pole at z = zS, one can use the
d&(Zs)EfWraW; (zr - zs> (zt - zs> N
+C~,[d’(z~)Wf +~~~(z,)t$ r
+~~~(zr)a~LL])e’P(z2)e-2’P(Z3)+HC123)~~(zT)) ?-=I
where 0: is always ordered to the left of w,. and a,. The term proportional d” ( zr) with the residue of the pole of dfl (z,) at zs. One can similarly use the OPE’s for 2’ to write
(3.2)
to E,E,E~ comes from the pole of
(3.3) where @
is always ordered to the left of D,“, which is always ordered to the left of w, and a,..
N. Berkovits/Physics
Finally, obtains
performing
+ i
C
GE,(
r&t
+
$c
the correlation
of (2.6),
one
+Et(Zr-Zt)(;r~~~~~zr-Z,))
(z,-z,>(z,-zr> _
Wsd.tT$Wr,WF ErEsErEu
(zr - zs) (zr - zu) (z, - z,) (zt - z,) ) :
(22 - 23)2rI(z,
- zd4Qsm
(3.4)
j-Jc(z,)
r
T,S where always and to It is implies
over the x’s and p, and using the normalization
wai+a
2w;iiifktcu‘+
r,u,u x
function
113
Letters B 385 (1996) 109-116
k? is the momentum of V,, : exp(> : is to remind that @ is always ordered to the left of 0,” which is ordered to the left of wI and a,, and s d28ad28a means to set P(z,) = ,9$ and @(z,.) = 8; for all I integrate over 13; and 8:. straightforward to check that A is gauge-invariant since under the transformation SV, = (B) 2@p (which a$ = D$, a; = kr, ad c$ = wp” =J$ = 0)
X
s
d28cd2&
: expF
: (22 - z~)'n(z,
where F is the object in the exponential
4. Koba-Nielsen
- z,)~~~*(~P)~@P
r,s
n
K(z,)
(3.5)
r+P
of (3.4).
formula for closed superstrings
For the closed superstring, the tree-level scattering amplitude of compactification independent massless states is obtained by multiplying d by its right-moving counterpart and by replacing the 4D N=l super-Yang-Mills prepotential V(x, 0, e) with the 4D N=2 supergravity plus tensor prepotential U(x, &, 8~; OR, 0~). In standard SU(2) notation for N=2 superspace, SE = e:, $2 = @&, 0;; = (?“_, and @jj= fi-&. As discussed in Ref. [ 111, U is a scalar superfield which describes an N=2 conformal supergravity multiplet and an -N=2 - tensor hypermultiplet. The NS-NS fields for the graviton, anti-symmetric tensor, and dilaton are in the ORB,&& component of U,
and the R-R field strengths for the U( 1) vector and complex (OR) 28R components of u, F,,, = a$
(DL)~D)L~(DR)~D~$J+c.c.,
scalar are in the ( 8L>28, (6R)‘8,
r&y = CJ$~D~,(DL)~(DR)~D~$I
and ~9~( 8L)2
114
N. Berkovits/Physics
For U to be an N=2 primary (D,)*U
= (&)*U
Letters B 385 (1996) 109-116
field, it must satisfy the constraints
= (DR)*U
= (&)*U
= &J”U
= 0.
The first four constraints are the N=2 supersymmetric generalization of the usual polarization the last constraint is the equation of motion in this gauge. The topological prescription for the N=2 closed superstring tree-level amplitude is
A=
(Ut(zl)
G;(GL+(UZ(ZZ)))
conditions,
and
G:(G;i(rr,(n)))~~~z~~irG~(Gf(G:(G~(U~(e)))))) r=4
where Gg and G: are the right and left-moving fermionic N=2 generators. Using the same methods as in the open superstring computation, one finds
X
d28Lod28Lo
s
d*&od*&o
: eXp(FL
+FR)
: IZ2 -
Z3/4nIZr
-
Z,12k’ku’zl--&(Zr)
(4.1)
r
I.S
where FL is the object in the exponential of (3.4) with k: replaced by ik;, and FR is its right-moving counterpart (i.e. FR is obtained from FL. by switching (ELM,DL~, WL~, aLr) to ( ERr, DRY, WRr, aRr ) and by switching zr to Z,).
5. Koba-Nielsen
formula
in the presence of a D-brane
For closed string scattering in the presence of a D-brane, there are boundary conditions on the fields which can be either Neumann or Dirichlet. For convenience, the boundary will be chosen to be the real line, and the closed strings to reside in the upper half-plane. The boundary conditions when z = Z for the four-dimensional fields are [ 121
where M> and M$ boundary
conditions
are the matrix
elements
in the mrh spacetime
of M = fli=,
direction.
~2,
and II,, = 0 or 1 for Neumann
(We are assuming
or Dirichlet
in this letter that cR=, n, is even. When
CL nnz is odd, one needs to switch t9;; with 8$ and dg with zg.) Note that because the four-dimensional variables are independent of the compactification, (5.1) is independent of the boundary conditions in the compactification directions. The first step in calculating
Comparing
the scattering
with the RNS prescription,
A = (~L(Z~~~~(ZI)
dZz{bL~ZRZL&wS .I
and using the N=l /N=2 prescription is
relationship
with a D-brane
is to write down the topological
prescription.
[9]
dz,dz,[bR,{bL,ZRZLV,RNS(Z,)}l),
Cz*)Jfi/
(5.2)
r=3
which is discussed
after (2.2))
one sees that the correct
topological
N. Berkovits/Physics
A=
Leners B 385 (1996) 109-116
dz,d~,G,(G,(G~(Gr:(U,tzr)))))).
(G;uJl(zl>>
Plugging
115
in the vertex operator for the massless
(5.3)
fields,
(5.4)
where the only non-zero WC+= (D )2p r, r r
w’s and a’s are w; = (8,)2DF,
for Y > 1 when left-moving
a: = DF,
~2: = -@,
and for r > 2 when right-moving,
a?
= -i[D,“,@]
and
One can now use the techniques of Section 3 to compute the correlation function. The only difference is that the correlation function of left-moving fields must equal the correlation function of its right-moving partner when the field sits on the boundary. For example,
where f;s, is the pole residue of dz( z ) at z = zs and .f& is the pole residue of M’$d{ ( z > at Z = Zt. The resulting Koba-Nielsen formula in the presence of a D-bane is
DL~~w;~ x
+ DL~~*$,
+ k,,,a$ + ELsaLluG
+ a~,$$
I
:exp[[xELA waLriii"! Lsk tax+
+ icj$LrELs(
$,*fsaLtah
(zs - z,>(z, - zt> +ELt(Zs-zf)(zr-Zt)(Zr-Za))
r,s,t
+$lC
(z, - zs12
Zr - zs
r,s
@Ls&*&WLtaW;r ELrELsELtELu (zr - z,>(zt - z,,>(z,- z,)fzt - z,)
r,.T,t,u
+ {Lu --+ Ru}l x (21 - &J-J
where @ = (-f)Y7-ky,
+ {Lt
4
Rt}] + (Ls -+ Rs}] + {Lr -+ Rr}l : (5.5)
I& - Zs12~kSn11Zr - z_r12@S,‘IIUAZ,) r r,s Jd2&d2&
al] r and to integrate over
means to set @;j(z,) = M>@(%)
= 6; and @(Zr>
= M"p@(%)
8,” and fig, and {Lu + Ru} means to switch zu + .Z, and to switch
= 6; for
116
N. Berkovirs/Physics
Letters B 385 (1996) 109-116
Note that the switch affects everything inside the [] brackets, e.g. {Lu 4 Ru} affects only the E,E,E~E, term but {Lx -+ RY} affects all terms (including those coming from earlier switches) _ In summary, the manifestly super-Poincare invariant Koba-Nielsen formulas for massless tree amplitudes of the open superstring, closed superstring, and closed superstring in the presence of a D-brane can be found in Eqs. (3.4), (4.1), and (5.5).
Acknowledgements I would like to thank Vipul Periwal and @jvind Tafjord for suggesting sharing their results using light-cone Green-Schwarz calculations.
the D-brane
generalization
and for
References [ l] 123 [3] [4] [5] [6] [7]
[ 81 [9]
[lo] [ 111 [ 121
M.B. Green and J.H. Schwarz, Nucl. Phys. B 243 (1984) 475. S. Mandelstam, Prog. Theor. Phys. Suppl. 86 (1986) 163. A. Restuccia and J.G. Taylor, Phys. Rep. 174 (1989) 283. S. Mandelstam, Workshop on Unified String Theories, 29 July-16 August 1985, eds. M. Green and D. Gross (World Scientific, Singapore, 1986) p. 577. N. Berkovits, Nucl. Phys. B 431 (1994) 258. N. Berkovits, Nucl. Phys. B 420 ( 1994) 332; N. Berkovits and C. Vafa, Mod. Phys. Lett. A 9 (1994) 653. N. Berkovits and C. Vafa, Nucl Phys. B 433 (1995) 123. N. Berkovits, A New Description of the Superstring, preprint IFUSP-P-1212, April 1996, to appear on hep-th. J. Polchinski, Dirichlet-Branes and Ramond-Ramond Charges, hep-th 9510017; S.S. Gubser, A. Hashimoto, I.R. Klebanov and J.M. Maldacena, Gravitational Lensing by p-Branes, hep-th 9601057; M.R. Garousi and R.C. Meyers, Superstring Scattering from D-Branes, hep-th 9603194. W. Siegel, Nucl. Phys. B 263 (1986) 93. N. Berkovits and W. Siegel, Superspace effective actions for 4D compactifications of heterotic and type II superstrings, preprint IPUSP-P-1180, October 1995, to appear in Nucl. Phys. B, hep-th 9501016. V. Periwal and 0. Tafjord, private communication; M.B. Green, Phys. Len. B 329 (1994) 435.
ca
26 September
1996
mm
‘53
ELSEWER
PHYSICS
Physics Letters B 385 (1996) 109-l
LETTERS
B
16
Super-Poincark invariant Koba-Nielsen forrnulas for the superstring Nathan Berkovits ’ Dept. de Fisica Matemdtica, Univ. de &To Paula, CP 66318 SEo Paula, SP 05389-970, Brazil and IMECC, Univ. de Campinas, CP 1170 Campinas, SP 13100, Brazil Received 21 June 1996 Editor: M. Dine
Abstract The new spacetime-supersymmetric description of the superstring is used to compute tree-level scattering amplitudes for an arbitrary number of massless four-dimensional states. The resulting Koba-Nielsen formula is manifestly SO(3,l) super-Poincare invariant and is easily generalized to scattering in the presence of a D-brane.
1. Introduction Before the work of this author, all spacetime-supersymmetric calculations of superstring amplitudes used the light-cone Green-Schwarz formalism, which requires non-trivial operators at the light-cone interaction points. [l-3] Because of this complication, only the four-point massless tree and one-loop amplitudes had been explicitly computed in a spacetime-supersymmetric manner. * Over the last few years, this author has developed an alternative spacetime-supersymmetric description of the superstring which has the advantage of being conformally invariant, and therefore not requiring non-trivial interaction-point operators. [ 51. The new formalism is manifestly SO(3,l) super-Poincare invariant and is suitable for any compactification to four dimensions which preserves N=l 4D supersymmetry. Furthermore, the new formalism contains critical N=2 worldsheet superconformal invariance, and is related by a field redefinition to the N=2 embedding of the N=l RNS superstring. [6] Together with Vafa, a simple “topological” method was developed for calculating scattering amplitudes in this formalism. [7] (An easily accessible review of this method can be found in Ref. [LX]) In this letter, the 1E-mail: nberkovi@snfn&.if.usp.br. ’ Although Ref. [3] contains explicit expressions for four-point multiloop amplitudes, these expressions contain unphysical divergences when interaction points coincide. It has not yet been determined how these expressions are affected by removing the divergences with contact terms. Also, Mandelstam has constructed an N-point tree amplitude which has manifest SU(4) xU( 1) super-Poincare invariance. [4] However the structure of his amplitude was obtained indirectly by unitarity arguments, rather than by explicit computations. It would be interesting to compare his amplitude with the Koba-Nielsen formula in this letter. 0370-2693/96/$12.00 Copyright PZZ SO370-2693(96)00873-S
Q 1996 Elsevier Science B.V. All rights reserved.
110
N. Berkouits/Pkysics
Letters B 385 (1996) 109-116
topological method will be used to explicitly compute tree-level scattering amplitudes for an arbitrary number of four-dimensional massless states. The result can be expressed as a Koba-Nielsen formula, and is manifestly SO( 3,l) super-Poincare invariant. It is straightforward to generalize the formula for scattering in the presence of a D-brane. This result should be useful for D-brane analysis since, unlike computations in the RNS formalism, [ 91 it is manifestly spacetimesupersymmetric, and therefore includes both NS-NS and R-R couplings.
2. Review As shown in Ref. [7], the topological prescription superstring states in any c=6 N=2 string theory is
for computing
the tree-level
amplitude
for N open
(2.1) where V( z ) is a U( 1 )-neutral N=2 primary field of weight 0, [ T, Gf ,G- , .I] are the twisted c=6 N=2 generators (after twisting, Gf has weight 1 and G- has weight 2)) G*(Y) means the contour integral of G’ around Y, @ = G- ( eiH) , and J = dH. The Chan-Paton factors will be supressed throughout this letter. When (2.1) is applied to the N=2 embedding of the N=l RNS superstring, it reproduces the standard RNS prescription in the “large” Hilbert space:
d =
Hzdr/;RNs
(2.2)
since in the N=2 embedding, V = cVRNs, G+ = jBRST, G- = b, and @ = q. (Note that p = ate-6 and y = rle$ are the bosonic RNS ghosts, 2 = {Q, S} is the RNS picture-changing operator, Vms is in the -1 picture, and zvms = {Q, @ar=} is in the 0 picture). So G+(V) = Q(g%‘“““) = ZVms and G+(V) = VRNs. In the new spacetime-supersymmetric description of the superstring, a field redefinition is used to write the generators of the N=2 embedding in terms of spacetime-supersymmetric variables. [5] In terms of the fourvariables, the dimensional supersymmetric variables [x m, 8”) I?‘, pa, pk, p] and the compactification-dependent twisted c=6 N=2 generators are T=~dxn’dx,,+p,aea+p~.a~~+~(dpdp+id2p)+TC, G+ = e@(d)* + G;,
G- = e-@(d)*
+ GE,
J = -Jp
+ Jc,
(2.3)
where m takes values 0 to 3, (Y and & take values 1 or 2, pa and & are conjugates boson, d, =pe + ;i%xa,
- $(I?)%,
+ $$J(8)2,
& = j& + ;B”&&
to 8” and @‘, p is a chiral
- $(8)%&
JC] are the twisted c=9 N=2 generators and [Tc, Gs,G,, (d)2 means i@d,dp, theory used to describe the compactification manifold. The action in conformal gauge for these fields is
+ ~a&a(e)2,
of the superconformal
field
N. Ber!uwits/Physics
where (as in with 0 The
111
Letters B 385 (1996) 109-116
SC is the action for the compactification-dependent variables. When the R/L index has been suppressed (2.3) ) , we shall always mean left-moving. Note that z versus Z is correlated with L versus R, and not versus e. free-field OPE’s of the four-dimensional variables are
z),
PR(Y)PR(Z)
+ log(J - 9,
(2.5) Note that the chiral boson p can not be fermionized since &J’(Y) e-ip(z) + (y - z). It has the same behavior as the negative-energy field qb that appears when bosonizing the RNS ghosts. After twisting, the background charge is defined for open superstring tree amplitudes by (W2@)
2,-iP+‘Hc)
= 1
where Jc = dHc. As was shown by Siegel,
(2.6)
[lo]
d, and & satisfy the OPE that d,(y)
dp(z)
is regular,
(2.7) where II,& =
O-zkax, - $a&+ - $aO,.
The advantage ofworking with the variables d, and III, is that they commute with the spacetime supersymmetry generators, 4a =
3. Koba-Nielsen formula for open superstrings For massless states of the open superstring which are independent V only depends on the zero modes of x”‘, 0” and @. It can therefore which is the standard N=l 4D super-Yang-Mills prepotential (e.g. field A”$, and the 0(8)2 and (6r)28 components of V are the gluino The condition of being N=2 primary implies that
of the compactification, the vertex operator be represented by the superfield V( x, 8, e) , the &? component of V is the gluon gauge fields 5” and 4”).
where D, = dp + ~8”<&,, and D, = de& + $?“a:&?,,, are the covariant fermionic derivatives. Note that ( D)2V = (D)2V = 0 is the standard supersymmetric generalization of the Lorentz gauge condition d,Atn = 0. Using the free-field OPE’s of (2.5)) it is straightforward to compute that
J
dzG-(G+(V))
=
J
dz(d”
(D)2D,V+J’
(D)2~‘d!V+d0aDuV-ade’
~kV-iIIa’
[Da,Db]V).
N. Berkovits/Physics
112
Plugging
Letters B 385 (1996) 109-116
into (2.1) ,
d = (~(z,)ei~d”(z2)D~~(z~)e-2”p+Hc d-’ (z3)Dciv3(a) f+W(zr)
(D)*&K(zr)
r=4 + a+(&)
(D)*D&vr(z,)
+ ae”(z,)D,v,(z,)
- J@(z,)
D&K(Zr)
- irIa~(zz)
uLmVr(Zr)))
N
+
nn~(Z,)a~~])eif(Z~)e-*iP(Z3)+iH~(Z3)
wa=Du 2 29
@ = 2
+$ = (D )2Dh r r T)
a”
2
=
w: =
a& - @ 2-
= 0,
(D,.)*D,“,
(3.1)
f@ = Df,
w; = a!$ = @
a; = D,“, Z$!= -@,
for r > 3, and 0,” and Df are fermionic
derivatives
= a?“! = 0,
a:’ = -i[DF,@]
which act only on V, (z,) . For example,
N = K(z,)D%(z2)
D;;v,(i,)
Using the fact that (d”(z)...)= OPE’s of (2.7) to write
+
$c T,SJ
ErffsEt
fiv,(zr). r=3
C,($-$f),
where fS is the residue of the pole at z = zS, one can use the
d&(Zs)EfWraW; (zr - zs> (zt - zs> N
+C~,[d’(z~)Wf +~~~(z,)t$ r
+~~~(zr)a~LL])e’P(z2)e-2’P(Z3)+HC123)~~(zT)) ?-=I
where 0: is always ordered to the left of w,. and a,. The term proportional d” ( zr) with the residue of the pole of dfl (z,) at zs. One can similarly use the OPE’s for 2’ to write
(3.2)
to E,E,E~ comes from the pole of
(3.3) where @
is always ordered to the left of D,“, which is always ordered to the left of w, and a,..
N. Berkovits/Physics
Finally, obtains
performing
+ i
C
GE,(
r&t
+
$c
the correlation
of (2.6),
one
+Et(Zr-Zt)(;r~~~~~zr-Z,))
(z,-z,>(z,-zr> _
Wsd.tT$Wr,WF ErEsErEu
(zr - zs) (zr - zu) (z, - z,) (zt - z,) ) :
(22 - 23)2rI(z,
- zd4Qsm
(3.4)
j-Jc(z,)
r
T,S where always and to It is implies
over the x’s and p, and using the normalization
wai+a
2w;iiifktcu‘+
r,u,u x
function
113
Letters B 385 (1996) 109-116
k? is the momentum of V,, : exp(> : is to remind that @ is always ordered to the left of 0,” which is ordered to the left of wI and a,, and s d28ad28a means to set P(z,) = ,9$ and @(z,.) = 8; for all I integrate over 13; and 8:. straightforward to check that A is gauge-invariant since under the transformation SV, = (B) 2@p (which a$ = D$, a; = kr, ad c$ = wp” =J$ = 0)
X
s
d28cd2&
: expF
: (22 - z~)'n(z,
where F is the object in the exponential
4. Koba-Nielsen
- z,)~~~*(~P)~@P
r,s
n
K(z,)
(3.5)
r+P
of (3.4).
formula for closed superstrings
For the closed superstring, the tree-level scattering amplitude of compactification independent massless states is obtained by multiplying d by its right-moving counterpart and by replacing the 4D N=l super-Yang-Mills prepotential V(x, 0, e) with the 4D N=2 supergravity plus tensor prepotential U(x, &, 8~; OR, 0~). In standard SU(2) notation for N=2 superspace, SE = e:, $2 = @&, 0;; = (?“_, and @jj= fi-&. As discussed in Ref. [ 111, U is a scalar superfield which describes an N=2 conformal supergravity multiplet and an -N=2 - tensor hypermultiplet. The NS-NS fields for the graviton, anti-symmetric tensor, and dilaton are in the ORB,&& component of U,
and the R-R field strengths for the U( 1) vector and complex (OR) 28R components of u, F,,, = a$
(DL)~D)L~(DR)~D~$J+c.c.,
scalar are in the ( 8L>28, (6R)‘8,
r&y = CJ$~D~,(DL)~(DR)~D~$I
and ~9~( 8L)2
114
N. Berkovits/Physics
For U to be an N=2 primary (D,)*U
= (&)*U
Letters B 385 (1996) 109-116
field, it must satisfy the constraints
= (DR)*U
= (&)*U
= &J”U
= 0.
The first four constraints are the N=2 supersymmetric generalization of the usual polarization the last constraint is the equation of motion in this gauge. The topological prescription for the N=2 closed superstring tree-level amplitude is
A=
(Ut(zl)
G;(GL+(UZ(ZZ)))
conditions,
and
G:(G;i(rr,(n)))~~~z~~irG~(Gf(G:(G~(U~(e)))))) r=4
where Gg and G: are the right and left-moving fermionic N=2 generators. Using the same methods as in the open superstring computation, one finds
X
d28Lod28Lo
s
d*&od*&o
: eXp(FL
+FR)
: IZ2 -
Z3/4nIZr
-
Z,12k’ku’zl--&(Zr)
(4.1)
r
I.S
where FL is the object in the exponential of (3.4) with k: replaced by ik;, and FR is its right-moving counterpart (i.e. FR is obtained from FL. by switching (ELM,DL~, WL~, aLr) to ( ERr, DRY, WRr, aRr ) and by switching zr to Z,).
5. Koba-Nielsen
formula
in the presence of a D-brane
For closed string scattering in the presence of a D-brane, there are boundary conditions on the fields which can be either Neumann or Dirichlet. For convenience, the boundary will be chosen to be the real line, and the closed strings to reside in the upper half-plane. The boundary conditions when z = Z for the four-dimensional fields are [ 121
where M> and M$ boundary
conditions
are the matrix
elements
in the mrh spacetime
of M = fli=,
direction.
~2,
and II,, = 0 or 1 for Neumann
(We are assuming
or Dirichlet
in this letter that cR=, n, is even. When
CL nnz is odd, one needs to switch t9;; with 8$ and dg with zg.) Note that because the four-dimensional variables are independent of the compactification, (5.1) is independent of the boundary conditions in the compactification directions. The first step in calculating
Comparing
the scattering
with the RNS prescription,
A = (~L(Z~~~~(ZI)
dZz{bL~ZRZL&wS .I
and using the N=l /N=2 prescription is
relationship
with a D-brane
is to write down the topological
prescription.
[9]
dz,dz,[bR,{bL,ZRZLV,RNS(Z,)}l),
Cz*)Jfi/
(5.2)
r=3
which is discussed
after (2.2))
one sees that the correct
topological
N. Berkovits/Physics
A=
Leners B 385 (1996) 109-116
dz,d~,G,(G,(G~(Gr:(U,tzr)))))).
(G;uJl(zl>>
Plugging
115
in the vertex operator for the massless
(5.3)
fields,
(5.4)
where the only non-zero WC+= (D )2p r, r r
w’s and a’s are w; = (8,)2DF,
for Y > 1 when left-moving
a: = DF,
~2: = -@,
and for r > 2 when right-moving,
a?
= -i[D,“,@]
and
One can now use the techniques of Section 3 to compute the correlation function. The only difference is that the correlation function of left-moving fields must equal the correlation function of its right-moving partner when the field sits on the boundary. For example,
where f;s, is the pole residue of dz( z ) at z = zs and .f& is the pole residue of M’$d{ ( z > at Z = Zt. The resulting Koba-Nielsen formula in the presence of a D-bane is
DL~~w;~ x
+ DL~~*$,
+ k,,,a$ + ELsaLluG
+ a~,$$
I
:exp[[xELA waLriii"! Lsk tax+
+ icj$LrELs(
$,*fsaLtah
(zs - z,>(z, - zt> +ELt(Zs-zf)(zr-Zt)(Zr-Za))
r,s,t
+$lC
(z, - zs12
Zr - zs
r,s
@Ls&*&WLtaW;r ELrELsELtELu (zr - z,>(zt - z,,>(z,- z,)fzt - z,)
r,.T,t,u
+ {Lu --+ Ru}l x (21 - &J-J
where @ = (-f)Y7-ky,
+ {Lt
4
Rt}] + (Ls -+ Rs}] + {Lr -+ Rr}l : (5.5)
I& - Zs12~kSn11Zr - z_r12@S,‘IIUAZ,) r r,s Jd2&d2&
al] r and to integrate over
means to set @;j(z,) = M>@(%)
= 6; and @(Zr>
= M"p@(%)
8,” and fig, and {Lu + Ru} means to switch zu + .Z, and to switch
= 6; for
116
N. Berkovirs/Physics
Letters B 385 (1996) 109-116
Note that the switch affects everything inside the [] brackets, e.g. {Lu 4 Ru} affects only the E,E,E~E, term but {Lx -+ RY} affects all terms (including those coming from earlier switches) _ In summary, the manifestly super-Poincare invariant Koba-Nielsen formulas for massless tree amplitudes of the open superstring, closed superstring, and closed superstring in the presence of a D-brane can be found in Eqs. (3.4), (4.1), and (5.5).
Acknowledgements I would like to thank Vipul Periwal and @jvind Tafjord for suggesting sharing their results using light-cone Green-Schwarz calculations.
the D-brane
generalization
and for
References [ l] 123 [3] [4] [5] [6] [7]
[ 81 [9]
[lo] [ 111 [ 121
M.B. Green and J.H. Schwarz, Nucl. Phys. B 243 (1984) 475. S. Mandelstam, Prog. Theor. Phys. Suppl. 86 (1986) 163. A. Restuccia and J.G. Taylor, Phys. Rep. 174 (1989) 283. S. Mandelstam, Workshop on Unified String Theories, 29 July-16 August 1985, eds. M. Green and D. Gross (World Scientific, Singapore, 1986) p. 577. N. Berkovits, Nucl. Phys. B 431 (1994) 258. N. Berkovits, Nucl. Phys. B 420 ( 1994) 332; N. Berkovits and C. Vafa, Mod. Phys. Lett. A 9 (1994) 653. N. Berkovits and C. Vafa, Nucl Phys. B 433 (1995) 123. N. Berkovits, A New Description of the Superstring, preprint IFUSP-P-1212, April 1996, to appear on hep-th. J. Polchinski, Dirichlet-Branes and Ramond-Ramond Charges, hep-th 9510017; S.S. Gubser, A. Hashimoto, I.R. Klebanov and J.M. Maldacena, Gravitational Lensing by p-Branes, hep-th 9601057; M.R. Garousi and R.C. Meyers, Superstring Scattering from D-Branes, hep-th 9603194. W. Siegel, Nucl. Phys. B 263 (1986) 93. N. Berkovits and W. Siegel, Superspace effective actions for 4D compactifications of heterotic and type II superstrings, preprint IPUSP-P-1180, October 1995, to appear in Nucl. Phys. B, hep-th 9501016. V. Periwal and 0. Tafjord, private communication; M.B. Green, Phys. Len. B 329 (1994) 435.