Three-Graviton Amplitude in Berkovits-Vafa-Witten Variables

14 Pages·0.13 MB·

by  K. Bobkov

hep-th/0201027
Three-Graviton Amplitude in
Berkovits-Vafa-Witten Variables
2
0
0
2

n
a
K. Bobkov and L. Dolan
J

4
Department of Physics

1
University of North Carolina, Chapel Hill, NC 27599-3255
v
7
2
0 We compute the three-graviton tree amplitude in Type IIB superstring theory compact-
1
0 ified to six dimensions using the manifestly (6d) supersymmetric Berkovits-Vafa-Witten
2
0 worldsheet variables. We consider two cases of background geometry: the flat space exam-
/
h 6 3
ple R K3, and the curved example AdS S K3 with Ramond flux, and compute
t 3
- × × ×
p the correlation functions in the bulk.
e
h
:
v
i
X
r
a
1. Introduction
We compute string tree correlation functions using the manifestly supersymmetric co-
variant formulation of Berkovits-Vafa-Witten (BVW) for Type IIB superstrings compacti-
fied to six dimensions [1-6]. Unlike the ten-dimensional covariant pure spinor quantization
[7-13], the six-dimensional version[1-3] we use here incorporates an N = 4 topological
string formulation to compute tree level scattering amplitudes.
For IIB superstrings either on flat six-dimensional space times K3, or on AdS
3
×
3
S K3, the massless degrees of freedom correspond to a D = 6, N = (2,0) supergravity
×
multiplet and 21 tensor multiplets. In this paper, we consider the string theory tree level
scattering of three gravitons both in the flat case and the AdS S3 case, where the latter
3
×
hasbackground Ramondflux. Forthese amplitudes, therelevant massless compactification
independent vertex operator, in the BVW worldsheet formalism, contains the graviton,
dilaton and two-form field which contribute to the supergravity and one of the tensor
multiplets.
In this formalism [1-3], the Type IIB superstring compactified on K3, which has 16
supercharges corresponding to 16 unbroken supersymmetries, has 8 which are manifest
in that they act geometrically on the target space. These are given by F ,F¯ described
a a
below and are related to the presence of 8 theta world sheet variables θa,θ¯a. The other 8
supersymmetries E ,E¯ are not manifest, but can still be expressed in terms of ordinary
a a
world sheet fields, i.e. not spin operators. Therefore, in addition to making some of the
6 3
supersymmetry geometric (either in R or AdS S ), this formalism is also advantageous
3
×
to describing background fields belonging to the Ramond-Ramond (RR) sector, since the
worldsheet fields which couple to the RR background fields are not spin fields. Thus for
strings on AdS that require RR backgrounds, the purpose of using BVW variables is that
RR background fields can be added to the worldsheet action without adding spin fields to
the worldsheet action [3].
6
On flat space (R ), the BVW variables describe a free worldsheet conformal field
theory. Their operator products (OPE’s) are reviewed in sect. 2, together with the N = 4
topological method for computing string correlation functions. In sect. 3 we use these
OPE’s to evaluate the flat space three graviton tree amplitude in position space, and show
that it reduces to the conventional answer.
On curved space, the BVW variables do not satisfy free operator product relations.
3
Nonetheless, we proceed to evaluate the three graviton correlation function on AdS S
3
×
1
by assuming a form for the OPE’s. It is motivated by [4], where the vertex operator
3
constraint equations were derived for AdS S by requiring them to be invariant under
3
×
the AdS supersymmetry transformations. It can be shown that our assumed OPE’s result
in the same constraint equations. In sect. 4, we compute the curved space three gravition
tree amplitude using these OPE’s.
2. Review of Components
The N = 4 topological prescription [1-3] for calculating superstring tree-level ampli-
tudes is
n
< V (z )(G˜+V (z ))(G+V (z )) dz G− G+V(z ) > . (2.1)
1 1 0 2 2 0 3 3 Z r −1 0 r
Y
r=4
±
where G are elements of the topological N = 4 super Virasoro algebra, and the notation
n
O V(z) denotes the pole of order d + n in the OPE of O(ζ) with V(z), when O is an
n
operator of conformal dimension d.
2.1. N = 4 Superconformal Algebra in Flat Space
For the IIB superstring there is both a holomorphic N = 4 superconformal algebra
and another anti-holomorphic one. The holomorphic generators, specialized for the IIB
string compactified to six dimensions, and in terms of BVW worldsheet variables, are [3]
T˜(z) = 1∂xm∂x p ∂θa 1∂ρ∂ρ 1∂σ∂σ+ 3∂2(ρ+iσ)+T˜
−2 m − a − 2 − 2 2 C
G+(z) = e−2ρ−iσ(p)4 + ie−ρp p ∂xab
− 2 a b
+eiσ( 1∂xm∂x p ∂θa 1∂(ρ+iσ)∂(ρ+iσ)+ 1∂2(ρ+iσ)) +G+
−2 m − a − 2 2 C
G−(z) = e−iσ +G−
C
J(z) = ∂(ρ+iσ)+J
C
J+(z) = eρ+iσJ+ = eρ+iσ+iHC (2.2)
C −
J−(z) = e−ρ−iσJ− = e−ρ−iσ−iHC
− C −
G˜+(z) = eiHC+ρ +eρ+iσG˜+
C
G˜−(z) = e−iHC[ e−3ρ−2iσ(p)4 ie−2ρ−iσp p ∂xab
− − 2 a b
+e−ρ( 1∂xm∂x p ∂θa 1∂(ρ+iσ)∂(ρ+iσ)+ 1∂2(ρ+iσ))]
−2 m − a − 2 2
e−ρ−iσG˜−.
− C
2
These currents are given in terms of the left-moving bosons ∂xm,ρ,σ, and the left-moving
fermionic worldsheet fields p ,θa, where 0 m 5;1 a 4. The conformal weights
a
≤ ≤ ≤ ≤
of p ,θa are 1 and 0, respectively. The BVW variables no longer exhibit the matter
a
times ghost sector structure familiar from the conventional Ramond-Neveu-Schwarz for-
malism, with cancelling contributions of 15 to the central charge. In BVW, the residual
±
ghost fields are ρ,σ. (In the ten-dimensional version, these are promoted to complex
worldsheet boson fields λα with a spacetime Majorana spinor index α, which are param-
eterized by eleven complex fields [7-13].) We define p4 1 ǫabcdp p p p = p p p p ; and
≡ 24 a b c d 1 2 3 4
∂xab = ∂xmσab wherethesigmamatricesinflatspacesatisfyσabσ +σabσ = η δb.
m m nac n mac mn c
Here lowered indices mean σ 1ǫ σcd. Note that eρ and eiσ are worldsheet
mab ≡ 2 abcd m
fermions. Also eρ+iσ eρeiσ = eiσeρ. Here J i∂H ,J+ eiHC,J−
≡ − C ≡ C C ≡ − C ≡
e−iHC. Both T˜,G±,J,J±,G˜± and the generators describing the K3 compactification
T˜ ,G±,J ,J±,G˜± satisfy the twisted N = 4, c = 6, superconformal algebra, i.e. both
C C C C C
T˜ and T˜ have zero central charge. However, c still appears in the twisted N = 4 and
C
N = 2 algebras in the products involving the supercurrents and the SU(2) currents; and
the N=2 generators in (2.2) T˜,G±,J decompose into a c = 0 six-dimensional part and a
c = 6 compactification-dependent piece.
The other non-vanishing OPE’s are xm(z,z¯)xn(ζ,ζ¯) = ηmnln z ζ ; for the left-
− | − |
moving worldsheet fermion fields p (z)θb(ζ) = (z ζ)−1δb; and for the left-moving world-
a a
−
sheet bosons ρ(z)ρ(ζ) = ln(z ζ); σ(z)σ(ζ) = ln(z ζ). Right-movers are denoted
− − − −
by barred notation and have similar OPE’s.
2.2. N = 4 Superconformal Algebra for AdS S3
3
×
We also recall [3] the expression for the twisted holomorphic N = 4 generators when
R6 is replaced by AdS S3 in the presence of Ramond flux:
3
×
T˜(z) = 1ǫabcdK K FaEa 1∂ρ∂ρ 1∂σ∂σ+ 3∂2(ρ+iσ)+T˜
−8 ab cd − − 2 − 2 2 C
G+(z) = 1e−2ρ−iσǫabcdU U +ie−ρKabU
−6 ab cd ab
+eiσ( 1ǫabcdK K FaEa 1∂(ρ+iσ)∂(ρ+iσ)+ 1∂2(ρ+iσ)) +G+
−8 ab cd − − 2 2 C
G−(z) = e−iσ +G−
C
J(z) = ∂(ρ+iσ)+J
C
G˜+(z) = eiHC+ρ +eρ+iσG˜+,
C
(2.3)
3
where
U = (1 1eϕ+ϕ¯)−2[1F F 1e2ϕ¯E E ieϕ¯(E E +F F )], (2.4)
ab − 4 2 a b − 8 a b − 4 a b a b
and the remaining generators J±,G˜− can be constructed from T˜,G±,J [3]. Here eϕ
≡
e−ρ−iσ , and F ,E ,K are the fermionic and bosonic z-components of the right invariant
a a ab
PSU(2 2) currents. There is a corresponding anti-holomorphic N = 4 algebra in terms of
|
barred worldsheet fields. Although the N = 4 generators have definite holomorphicity, the
worldsheet fields F ,E ,K ,F¯ ,E¯ ,K¯ do not. They are each functions of z and z¯ and
a a ab a a ab
have non-free operator products, which are not known in closed form. They reduce to the
free conformal fields p ,∂θ ,∂xab,p¯ ,∂¯θ¯ ,∂¯xab only in the flat limit.
a a a a
3. Three-graviton tree amplitude in flat space
We first compute the six-dimensional three-graviton tree level amplitude in (6d) flat
6
space, for Type IIB superstrings on R K3 in the BVW formalism. It is contained in
×
the closed string three-point function
< V(z ,z¯ )(G+G¯+V(z ,z¯ ))(G˜+G¯˜+V(z ,z¯ )) > (3.1)
1 1 0 0 2 2 0 0 3 3
where the N = 4 supercurrents are found in (2.2) , and the vertex operators are given by
V(z,z¯) = eiσ(z)+ρ(z)eiσ¯(z¯)+ρ¯(z¯)θa(z)θb(z)θ¯a¯(z¯)θ¯¯b(z¯)σmσn φ (X(z,z¯)), (3.2)
ab a¯¯b mn
when the field
φ = g +b +g¯ φ
mn mn mn mn
satisfies theconstraint equations[3,4]∂mφ = 0, and φ = 0. These constraints imply
mn mn
the gauge conditions ∂mb = 0 for the two-form, and ∂mg = ∂ φ for the traceless
mn mn n
−
graviton g and dilaton φ. The constraints follow from the physical state conditions
mn
which in this formalism are implemented by the N = 4 generators, as shown in [3]. ( Since
the N = 4 algebra istwisted, i.e. topological, the nilpotent generators G+,G˜+,G¯+,G˜¯+ are
dimension one, and their cohomology essentially determines the physical states.) There is
a residual gauge symmetry
g g +∂ ξ +∂ ξ , φ φ, b b (3.3)
mn mn m n n m mn mn
→ → →
4
with ξ = 0, ∂ ξ = 0. To evaluate (3.1), we first extract the simple poles
n
·
G+G¯+V(z,z¯) = eiσeiσ¯ ( 4) [φ (X)∂Xm∂¯Xn
0 0 − mn
p θbσmσpca∂¯Xn∂ φ (X)
a cb p mn
− (3.4)
p¯ θ¯¯bσn σpc¯a¯∂Xm∂ φ (X)
− a¯ c¯¯b p mn
+p θbp¯ θ¯¯bσmσpcaσn σqc¯a¯∂ ∂ φ (X)],
a a¯ cb c¯¯b p q mn
G˜+G¯˜+V(z,z¯) = eiHC+2ρ+iσeiH¯C+2ρ¯+iσ¯ θaθbθ¯a¯θ¯¯bσm σn φ (X). (3.5)
0 0 ab a¯¯b mn
Then, usingtheOPE’sfortheghostfieldsandH , wepartiallycompute(3.1)byexhibiting
C
their contribution as [14-17]
< V (z ,z¯ )(G+G¯+V (z ,z¯ ))(G˜+G¯˜+V (z ,z¯ )) >
1 1 1 0 0 2 2 2 0 0 3 3 3
= (z z )(z z )(z z )−1(z¯ z¯ )(z¯ z¯ )(z¯ z¯ )−1
1 2 2 3 1 3 1 2 2 3 1 3
− − − − − −
4 < eiHC(z3)eρ(z3)+2ρ(z3)e3iσ(z3)eiH¯C(z¯3)eρ¯(z¯3)+2ρ¯(z¯3)e3iσ¯(z¯3)
·
θa(z )θb(z )θ¯a¯(z¯ )θ¯¯b(z¯ )σmσn φ (X(z ,z¯ ))
· 1 1 1 1 ab a¯¯b mn 1 1
[φ (X(z ,z¯ ))∂Xj(z )∂¯Xk(z¯ )
jk 2 2 2 2
·
p (z )θf(z )σj σpue∂¯Xk(z¯ )∂ φ (X(z ,z¯ ))
− e 2 2 uf 2 p jk 2 2
p¯ (z¯ )θ¯f¯(z¯ )σk σpu¯e¯∂Xj(z )∂ φ (X(z ,z¯ ))
− e¯ 2 2 u¯f¯ 2 p jk 2 2
+p (z )θf(z )p¯ (z¯ )θ¯f¯(z¯ )σj σpueσk σqu¯e¯∂ ∂ φ (X(z ,z¯ ))]
e 2 2 e¯ 2 2 uf u¯f¯ p q jk 2 2
θc(z )θd(z )θ¯c¯(z¯ )θ¯d¯(z¯ )σg σh φ (X(z ,z¯ )) >
· 3 3 3 3 cd c¯d¯ gh 3 3
Computing the remaining z ,z operators products, and using the SL(2,C) invariance of
2 3
the amplitude to take the three points to constants z , z¯ , z 1,z¯ 1,
1 1 2 2
→ ∞ → ∞ → →
z 0,z¯ 0, we have
3 3
→ →
< V (z ,z¯ )(G+G¯+V (z ,z¯ ))(G˜+G¯˜+V (z ,z¯ )) >
1 1 1 0 0 2 2 2 0 0 3 3 3
= (z z )(z¯ z¯ )(z z )−1(z¯ z¯ )−1 4
2 3 2 3 2 3 2 3
− − − − ·
< eiHC(0)+3ρ(0)+3iσ(0)eiH¯C(0)+3ρ¯(0)+3iσ¯(0)θaθbθcθdθ¯a¯θ¯¯bθ¯c¯θ¯d¯ >
· 0 0 0 0 0 0 0 0
[σmσg σn σh < φ (X( ))φ (X(1))∂j∂kφ (X(0)) >
· ab cd a¯¯b c¯d¯ mn ∞ jk gh
+2σm (σjσpσg) σn σh < φ (X( ))∂ φ (X(1))∂kφ (X(0)) >
ab cd a¯¯b c¯d¯ mn ∞ p jk gh
+2σm σg σn (σkσpσh) < φ (X( ))∂ φ (X(1))∂jφ (X(0)) >
ab cd a¯¯b c¯d¯ mn ∞ p jk gh
+4σm (σjσpσg) σn (σkσqσh) < φ (X( ))∂ ∂ φ (X(1))φ (X(0)) >]
ab cd a¯¯b c¯d¯ mn ∞ p q jk gh
5
= 4[ g¯mgg¯nh < φ (x )φ (x )∂j∂kφ (x ) >
mn 0 jk 0 gh 0
g¯nh(σmσjσpσg)d < φ (x )∂ φ (x )∂kφ (x ) >
d mn 0 p jk 0 gh 0
−
g¯mg(σnσkσpσh)d¯ < φ (x )∂ φ (x )∂jφ (x ) >
− d¯ mn 0 p jk 0 gh 0
+(σmσjσpσg)d (σnσkσqσh)d¯ < φ (x )∂ ∂ φ (x )φ (x ) >]
d d¯ mn 0 p q jk 0 gh 0
wherethesecondequalityfollowsfromthevacuumexpectationvalueoftheghostfields, H
C
andeightfermionzeromodes< eiHC(0)+3ρ(0)+3iσ(0)eiH¯C(0)+3ρ¯(0)+3iσ¯(0)θaθbθcθdθ¯a¯θ¯¯bθ¯c¯θ¯d¯ >=
0 0 0 0 0 0 0 0
1 ǫabcdǫa¯¯bc¯d¯, and various sigma matrix identities [3-4]. Further using (σmσnσpσq)d =
16 d
g¯mng¯pq +g¯mqg¯np g¯mpg¯nq where in flat space g¯ = η , we have
mn mn
−
< V (z ,z¯ )(G+G¯+V (z ,z¯ ))(G˜+G¯˜+V (z ,z¯ )) >
1 1 1 0 0 2 2 2 0 0 3 3 3
= 4[ g¯mgg¯nh < φ (x )φ (x )∂j∂kφ (x ) >
mn 0 jk 0 gh 0
g¯nh(g¯mjg¯pg +g¯mgg¯jp g¯mpg¯jg) < φ (x )∂ φ (x )∂kφ (x ) >
mn 0 p jk 0 gh 0
− −
g¯mg(g¯nkg¯ph +g¯nhg¯kp g¯npg¯kh) < φ (x )∂ φ (x )∂jφ (x ) >
mn 0 p jk 0 gh 0
− −
+(g¯mjg¯pg +g¯mgg¯jp g¯mpg¯jg)(g¯nkg¯qh +g¯nhg¯kq g¯nqg¯kh)
− −
< φ (x )∂ ∂ φ (x )φ (x ) >].
mn 0 p q jk 0 gh 0
·
Using the gauge condition ∂mφ = 0 again, then finally
mn
< V (z ,z¯ )(G+G¯+V (z ,z¯ ))(G˜+G¯˜+V (z ,z¯ )) >
1 1 1 0 0 2 2 2 0 0 3 3 3
= 4[ g¯mgg¯nh < φ (x )φ (x )∂j∂kφ (x ) >
mn 0 jk 0 gh 0
g¯nh(g¯mjg¯pg g¯mpg¯jg) < φ (x )∂ φ (x )∂kφ (x ) >
mn 0 p jk 0 gh 0
− −
g¯mg(g¯nkg¯ph g¯npg¯kh) < φ (x )∂ φ (x )∂jφ (x ) >
mn 0 p jk 0 gh 0
− −
+(g¯mjg¯pg g¯mpg¯jg)(g¯nkg¯qh g¯nqg¯kh))
− −
< φ (x )∂ ∂ φ (x )φ (x ) >]
mn 0 p q jk 0 gh 0
·
= 12[< φmn(x )φjk(x )∂ ∂ φ (x ) > +2 < φmn(x )∂ φjk(x )∂ φ (x ) >].
0 0 m n jk 0 0 m 0 j nk 0
(3.6)
To make contact with the supergravity field theory expression, we can evaluate this dual
model amplitude in either momentum space or in position space. For this flat space case,
momentum space is often used, since momentum is conserved, i.e. k +k +k = 0. For
1 2 3
AdS, momentum is not conserved, so we will just work in the position space representation
in both cases.
6
The Einstein-Hilbert action is I = ddx g R . Expanding to third order in K
| |{−2K2}
R p
using g = η +2Kh , we find the three-point interaction I . In harmonic gauge, i.e.
µν µν µν 3
when ∂µh 1∂ hρ = 0, and on shell h = 0, it is given by
µν − 2 ν ρ µν
I = K ddx[hµνhρσ∂ ∂ h +2hµν∂ hρσ∂ h ]. (3.7)
3 − Z µ ν ρσ µ ρ νσ
The gauge transformations
h h +∂ ξ +∂ ξ (3.8)
µν µν µ ν ν µ
→
leave invariant the harmonic gauge condition and I , given in (3.7), when ξ = 0. Using
3 µ
this gauge symmetry, we could further choose hρ = 0, ∂µh = 0. Then I represents the
ρ µν 3
three-graviton amplitude, and is invariant under residual gauge transformations that have
∂ ξ = 0.
·
To extract the string theory three-graviton amplitude from (3.6), we set b to zero,
mn
and use the field identifications[4] that relate the string fields g ,φ to the supergravity
mn
field h via φ 1hρ and g h 1g¯ hρ, where h is in harmonic gauge. Then
mn ≡ −3 ρ mn ≡ mn−6 mn ρ mn
φ = h 1g¯ hρ, and from (3.6) the on shell string tree amplitude is
mn mn − 2 mn ρ
K < V (z ,z¯ )(G+G¯+V (z ,z¯ ))(G˜+G¯˜+V (z ,z¯ )) >
−12 1 1 1 0 0 2 2 2 0 0 3 3 3
= K ddx[φmn(x)φjk(x)∂ ∂ φ (x)+2φmn(x)∂ φjk(x)∂ φ (x)]
m n jk m j nk
− Z
(3.9)
= K ddx[hmnhjk∂ ∂ h +2hmn∂ hjk∂ h ] +K ddxhmn∂ hk ∂ hp
− Z m n jk m j nk Z m k n p
′
= I +I
3 3
′
where I is the one graviton - two dilaton amplitude, I is the three graviton interaction
3 3
′
in harmonic gauge, and d = 6. We note that I and I separately are invariant under the
3 3
gauge transformation (3.8) with ξ = 0 and ∂ ξ = 0, which corresponds to the gauge
n
·
symmetry of the string field φ φ +∂ ξ +∂ ξ . Furthermore, I is also invariant
mn mn m n n m 3
→
under gauge transformations for which ∂ ξ = 0, and these can be used to eliminate the
· 6
trace of h in I . In the string gauge, the trace of φ is related to the dilaton φm = 6φ,
mn 3 mn m
so even when b = 0, (3.6) contains both the three graviton amplitude and the one
mn
graviton - two dilaton interaction. From (3.9) we see that we could have extracted I from
3
(3.6) merely by setting both b = 0 and φ = 0, since then φ = g and ∂mg = 0.
mn mn mn mn
7
4. Three-graviton tree amplitude in AdS S3
3
×
In this section, we compute the six-dimensional three-graviton amplitude in the Type
3
IIB superstring on AdS S K3 with background Ramond flux. Consider
3
× ×
< V (z ,z¯ )(G+G¯+V (z ,z¯ ))(G˜+G¯˜+V (z ,z¯ )) > (4.1)
1 1 1 0 0 2 2 2 0 0 3 3 3
where the N = 4 supercurrents are reviewed in (2.3),(2.4). They are expressed in terms of
the left action generators defined in [4] as
d d d
F = , K = θ +θ +t
a dθa ab − adθb bdθa Lab
(4.2)
d d
E = 1ǫ θb(tcd θc )+h ,
a 2 abcd L − dθ a¯bdθ¯
d ¯b
and corresponding right action generators F¯ ,E¯ ,K¯ . The vertex operators are
a a ab
V(z,z¯) = eiσ(z)+ρ(z)eiσ¯(z¯)+ρ¯(z¯)θa(z,z¯)θb(z,z¯)θ¯a¯(z,z¯)θ¯¯b(z,z¯)σmσn φ (X(z,z¯)).
ab a¯¯b mn
On AdS, in addition to the equation of motion, the string field φ = g +b +g¯ φ
mn mn mn mn
satisfies constraints given by tabhg hh σm σn φ = 0, ta¯¯bh g¯h h¯ σm σn φ = 0, which
L a¯ ¯b ab gh mn R a b g¯h¯ a¯¯b mn
werederivedin[4]wheretab,ta¯¯b describeinvariantderivativesontheSO(4)groupmanifold.
L R
These can be related to covariant derivatives cd σpcdD , c¯d¯ σpc¯d¯D , where
TL ≡ − p TR ≡ p
for example, acting on a function, = t and = t . But when acting acting on
L L R R
T T
fields that carry vector or spinor indices, they differ so that for example on spinor indices
tabV = abV + 1δaδbcV 1δbδacV . In terms of covariant derivatives, the constraints
L e TL e 2 e c − 2 e c
are Dmg = Dmg = D φ + H¯ brs, and Dmb = 0 = Dmb . These are the
mn nm n nrs mn nm
−
AdS S analog of the flat space constraints ∂mφ = 0. On AdS S3,
mn 3
× ×
G+G¯+V(z,z¯)
0 0
= eiσeiσ¯ ( 4)
−
[ 1Kab(z,z¯)K¯a¯¯b(z,z¯)σmσn φ (X)
· 4 ab a¯¯b mn (4.3)
1F (z,z¯)θb(z,z¯) K¯a¯¯b(z,z¯)(tac δac)σmσn φ (X)
− 2 a L − cb a¯¯b mn
1F¯ (z,z¯)θ¯¯b(z,z¯) Kab(z,z¯)(ta¯c¯ δa¯c¯)σmσn φ (X)
− 2 a¯ R − ab c¯¯b mn
+F (z,z¯)θb(z,z¯) F¯ (z,z¯)θ¯¯b(z,z¯) (ta¯c¯ δa¯c¯)(tac δac)σmσn φ (X)]
a a¯ R − L − cb c¯¯b mn
G˜+G¯˜+V(z,z¯) = eiHC+2ρ+iσeiH¯C+2ρ¯+iσ¯ θaθbθ¯a¯θ¯¯bσm σn φ (X). (4.4)
0 0 ab a¯¯b mn
8
In deriving (4.3), we keep only the contribution ie−ρKabU to G+ with U 1F F
ab ab ∼ 2 a b
since other terms in G+ do not survive the ghost measure in the vacuum expectation
value. We have also assumed that the OPE of F (z,z¯) with θe(ζ,ζ¯) can be replaced with
a
F (z,z¯)θe(ζ,ζ¯) (z ζ)−1δe, in accordance with (4.2). This is motivated by the obser-
a a
∼ −
vation that evaluating the OPE’s in this manner leads to the constraint equations found
in [4], where those equations were derived solely by requiring supersymmetric invariance
(and not from the action of the N = 4 generators).
< V (z ,z¯ )(G+G¯+V (z ,z¯ ))(G˜+G¯˜+V (z ,z¯ )) >
1 1 1 0 0 2 2 2 0 0 3 3 3
= (z z )(z z )(z z )−1(z¯ z¯ )(z¯ z¯ )(z¯ z¯ )−1
1 2 2 3 1 3 1 2 2 3 1 3
− − − − − −
4 < eiHC(z3)eρ(z3)+2ρ(z3)e3iσ(z3)eiH¯C(z¯3)eρ¯(z¯3)+2ρ¯(z¯3)e3iσ¯(z¯3)
·
θa(z )θb(z )θ¯a¯(z¯ )θ¯¯b(z¯ )σmσn φ (X(z ,z¯ ))
· 1 1 1 1 ab a¯¯b mn 1 1
[ 1 φ (X(z ,z¯ ))σj σk Kef(z ,z¯ )K¯e¯f¯(z ,z¯ )
· 4 jk 2 2 ef e¯f¯ 2 2 2 2
1F (z ,z¯ )θf(z ,z¯ ) K¯e¯f¯(z ,z¯ )(teℓ δeℓ)σj σk φ (X)
− 2 e 2 2 2 2 2 2 L − ℓf e¯f¯ jk
1F¯ (z ,z¯ )θ¯f¯(z ,z¯ ) Kef(z ,z¯ )(te¯ℓ¯ δe¯ℓ¯)σj σk φ (X)
− 2 e¯ 2 2 2 2 2 2 R − ef ℓ¯f¯ jk
+F (z ,z¯ )θf(z ,z¯ ) F¯ (z ,z¯ )θ¯f¯(z ,z¯ ) (te¯ℓ¯ δe¯ℓ¯)(teℓ δeℓ)σj σk φ (X)]
e 2 2 2 2 e¯ 2 2 2 2 R − L − ℓf ℓ¯f¯ jk
θc(z )θd(z )θ¯c¯(z¯ )θ¯d¯(z¯ )σg σh φ (X(z ,z¯ )) > .
· 3 3 3 3 cd c¯d¯ gh 3 3
(4.5)
Evaluating at z , z¯ , and restricting φ ,φ ,φ to be symmetric, we have
1 1 mn jk gh
→ ∞ → ∞
< V (z ,z¯ )(G+G¯+V (z ,z¯ ))(G˜+G¯˜+V (z ,z¯ )) >
1 1 1 0 0 2 2 2 0 0 3 3 3
= (z z )(z¯ z¯ )(z z )−1(z¯ z¯ )−1 4
2 3 2 3 2 3 2 3
− − − − ·
< eiHC(0)+3ρ(0)+3iσ(0)eiH¯C(0)+3ρ¯(0)+3iσ¯(0)θaθbθcθdθ¯a¯θ¯¯bθ¯c¯θ¯d¯ > 1 ǫabcdǫa¯¯bc¯d¯
· 0 0 0 0 0 0 0 0 ← 16
[ 1σm σn σj σk < φ (X( ))φ (X(1))[tef te¯f¯ σg σh φ (X(0))] > (4.6)
· 4 ab a¯¯b ef e¯f¯ mn ∞ jk L R cd c¯d¯ gh
+σm σn < φ (X( ))[teℓσj σk φ (X(1))][te¯f¯σg σh φ (X(0))] >
ab a¯¯b mn ∞ L ℓc e¯f¯ jk R ed c¯d¯ gh
+σm σn < φ (X( ))[te¯ℓ¯σj σk φ (X(1))][tef σg σh φ (X(0))] >
ab a¯¯b mn ∞ R ef ℓ¯c¯ jk L cd e¯d¯ gh
+4σmσn σg σh < φ (X( ))[te¯ℓ¯teℓσj σk φ (X(1))]φ (X(0)) >]
ab a¯¯b ed e¯d¯ mn ∞ R L ℓc ℓ¯c¯ jk gh
where the z ,z OPE’s have been evaluated in similar fashion, the SL(2,C) invariance sets
2 3
z 1, z 0, and cancellations occur among the contributions to the OPE’s from the
2 3
→ →
four terms in the sum in (4.5).
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