Chiral Supergravitons Interacting with a 0-Brane N-Extended NSR Super-Virasoro Group

16 Pages·0.19 MB·

by  A. Boveia

UniversityofMarylandElementaryParticlePhysicsUniversityofMarylandElementaryParticlePhysicsUniversityofMarylandElementaryParticlePhysics
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§§ Jnauary 2002 (cid:6)(cid:5) UMDEPP 02-028 §§
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§ UIOWA 02-001 §
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§ CALT-68-2367 §
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§ Chiral Supergravitons Interacting with a 0-Brane §
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§§ N-Extended NSR Super-Virasoro Group1 §§
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n §§ A. Boveia2, Bjørg A. Larson, and V. G. J. Rodgers3 §§
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J § Department of Physics and Astronomy §
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4 §§ The University of Iowa §§
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§ Iowa City, IA 52246-1479 USA §
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§ S. James Gates, Jr.4,5, W. D. Linch, III6, and J. A. Phillips7 §
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1 § Department of Physics, University of Maryland §
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0 §§ College Park, MD 20742-4111 USA §§
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h/ §§§ Dagny M. Kimberly §§§
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p §§ Department of Physics, Brown University §§
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h §§ Providence, RI 02912 USA §§
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i §§ ABSTRACT §§
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a § We continue the development of AFF by examining the cases where there §
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§ PACS: 04.65.+e, 11.15.-q, 11.25.-w, 12.60.J §
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§§ 1 Supportedin part by National Science Foundation Grants PHY-01-5-23911 §§
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§ 2 Present Address: University of California at Santa Barbara, USA §
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§ [email protected] §
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§§ 4On Sabbatical leave at the California Inst. of Technology, Sept. 2001 thru July 2002 §§
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§§TheUniversityofIowaParticleTheoryGroupTheUniversityofIowaParticleTheoryGroupTheUniversityofIowaParticleTheoryGroupTheUniversity§§
1 Introduction
In the literature there has been some focus on the role that diffeomorphisms play in the
development of both classical and quantum gravity through string theories. One particular
approach uses the Virasoro algebra and its dual (the coadjoint representation) as a guide
to these developments [1, 2]. This approach differs from theories of gravity that are based
purely on geometry since it is the Lie derivative, as opposed to the covariant derivative,
which is at the center stage. Since the Lie derivative exists in any dimension, this approach
allows one to get a foothold on gravitation on the two-dimensional string worldsheet. From
this viewpoint, the Virasoro algebra corresponds to the time independent Lie derivative
of contravariant vector fields, while the isotropy equations on the coadjoint orbits serve as
extensionsoftheGauss’LawconstraintsfromYang-Millstheories. Actionsthatadmitthese
constraints whenevaluated on two-dimensional surfaceswereconstructed in[1,2]. Recently
[3], these actions were extended to the super Virasoro algebra for one supersymmetry (N
= 1). Since the two genders of particles, fermions and bosons, are placed on equal footing
through smooth continuous transformations, theseactions were dubbedaffirmative actions.
Another way to conceptualize this work is to view our problem as an attempt to couple
a NSR 0-brane with N-extended supersymmetry to a background containing 2D, (N,0)
supergravity fields. This is an extension of the well-known conformal σ-model technique.
There one couples the massless compositions of a string theory via a world sheet action to
the string. Conformal invariance of the world sheet σ-model then imposes the equations of
motion on the massless composites.
In our study, we introduce an NSR 0-brane described by coordinates τ and ζI (I =
1, ..., N) such that vector fields constructed from these coordinates naturally carry an
arbitarary N and model-independent representation of the super-Virasoro algebra. These
vectorfieldsprovidearepresentationofthecenterlesssuper-Virasoroalgebra. Asabenefitof
such a geometrical approach, the generators of the super-Virasoro algebra naturally possess
a set of transformation laws under the action of the group. Consequently, the co-adjoint
elements to the generators also naturally obtain a set of transformation law. By demanding
that these co-adjoint elements be embedded within background fields and preserve the
symmetries of the co-adjoint elements, restrictions on the backgrounds are derived. Thus
a major problem for us is to find constructions that respect these symmetries within this
approach.
The purpose of this work is to show that we can extend these actions to N supersym-
metries for chiral two dimensional models using the results found in Ref.[4] as a guide.
However, in this work we will focus on a superfield formulation as opposed to the compo-
nent construction. Two superfields are used in the construction, one corresponds to the
N-extended prepotentials associated with the graviton while the other is the prepotential
associated with the gauge superfield that accompanies the internal SO(N) symmetry that
2
is now local.
N = 1
2 Affirmative Actions
For the sake of continuity we quickly review some of the results of Ref.[3]. Consider
the super Virasoro algebra which contains the bosonic Virasoro generators L ,mǫZ, and
m
fermionic generators G ,µ Z or Z+ 1, and a central charge cˆ. Its algebra is given by
µ ∈ 2
[L ,L ] = (m n)L + 1cˆ(m3 m)δ I ,
m n − m+n 8 − m+n,0
[L ,G ] = (1m µ)G ,
m µ 2 − m+µ
G ,G = i4L i1cˆ µ2 1 δ I . (1)
{ µ ν} − µ+ν − 2 (cid:16) − 4(cid:17) µ+ν,0
One can show [7] that this algebra can be represented with superfields by writing,
∞ ∞
1
A(z,ζ) = (Amzm+1)+2ζ (Aµzµ+2) , (2)
X X
m=−∞ µ=−∞
wherefornowζ issimplyaonedimensionalGrassmanncoordinate8 (laterweuseθtodenote
the space-time Grassmann supercoordinate variable). The generic element of the algebra
written in Eq.(2) has an equivalent representation as a doublet (A(Z),a) with Z = z,ζ .
{ }
Then, in terms of derivations on the superfield the commutation relations appear as [6, 7]
[[(A,a)(B,b)]] = ((∂A)B A∂B i1( A)( B), dZ(∂2 A)B) , (3)
− − 2 D D I D
where A and B are adjoint elements and
= ∂ + iζ ∂ , dZ = dz dζ . (4)
D ∂ζ 2 ∂z 2πi
implies 2 = i∂ = 1 , .
D D 2 z 2{D D}
Now we can take an adjoint superfield, say F, and act on a coadjoint element B⋆ as a
Lie derivative and supersymmetry transformation to find [5, 7]
δ B⋆ = F 2B⋆ 1 F B⋆ 3 2FB⋆+q 5F , (5)
F − D − 2D D − 2D D
whereF hasthedecompositionF = ξ+iζǫandB⋆ = (u+iζD,b⋆) = (u,D,b⋆).Theisotropy
equation (stability equation) for thecoadjoint element B⋆ is given by setting Eq.(5) to zero.
This determines the subalgebra that will leave the coadjoint vector B invariant. In terms
of component fields this becomes the two coupled equations
ξ∂D 1ǫ∂u 3∂ǫu+b⋆∂3ξ 2∂ξD =0 (6)
− − 2 − 2 −
ξ∂u 1ǫD 3∂ξu+q∂2ǫ =0 (7)
− − 2 − 2
8We can also refer to thisas theNSR0-brane Grassmann coordinate.
3
with∂ = ∂ . Theseequationsarepartofthefieldequationsoneextractsfromtheaffirmative
z
action. They have a direct analog to the Gauss’ Law constraints found in Yang-Mills.
Inordertoembedthisintotwodimensions,wereplacethe1DGrassmannvariableζ with
2-dimensional chiral Majorana spinors ζα. Then the supersymmetric covariant derivative
operator becomes
= ∂ iγNζν∂ . (8)
Dµ µ− 2 νµ N
With this
∂
, = iγN , (9)
{Dµ Dν} − µν∂zN
whereγM is the associated Gamma matrix. We will usecapital Latin indices for space-time
µν
indices and small Greek for spinor indices. With these γM’s we also introduce γaαβ such
αβ
that
γA γBβλ = 1δληAB + 1ΣABλ , (10)
αβ 2 α 2 α
where ΣABλ is anti-symmetric in its space-time indices.
α
Then, for example, an adjoint element is promoted to a vector superfield, F, and has a
ζ expansion FM = (ξM +ζαγMǫβ +σζµζαγN ) while a two dimensional coadjoint element
αβ µα
is promoted to the 3 spin superfield B = (Υ +ζαγN D +ζαζβγN A ). The
2 µM µM αβ MN µ[α β]MN
transformation of B with respect to FN is
µM
δ B = FN∂ B +∂ FNB + 1(∂ FN)B +i ( FNγλν) B . (11)
F µM N µM M µN 2 N µM Dλ N Dν µM
ThisisseenastheLiederivativewithrespecttoFM onthespace-timeindexinthefirstthree
summands followed by a supersymmetry transformation on B with ǫν ( FNγλν).
µM ≡ Dλ N
This combination of a Lie derivative and supersymmetry transformation is a natural exten-
sion of the Lie derivative. The supersymmetric isotropy equation in higher dimensions can
now be thought of as setting δ B to zero. Then those fields F that satisfy this condition
F µM
make up a subspace of vector fields that form the isotropy algebra for B . In order to be
µM
consistent with conformal field theory, every spinor index will carry a density weight of 1.
2
Thus the superfield B is a tensor density with weight 1.
µN 2
N
3 Prepotentials for Superdiffeomorphisms and SO( )
3.1 The Components of the Algebra and Its Dual
Therehave beenmany N-Extended SuperConformalalgebras posedin the literature [8].
However we will focus on the N-extended Super Virasoro algebra proposed in [11]. There
4
one has
1 1
G I iτA+2 ∂I i2ζI∂ + 2( + 1)τA−2ζIζK∂ ,
A ≡ h − τ i A 2 K
L τA+1∂ + 1( + 1)τAζI∂ ,
A ≡ − h τ 2 A I i
TAIJ ≡ τAh ζI∂J − ζJ∂I i , (1)
q (q−2)
UI1···Iq i(i)[2]τ(A− 2 )ζI1 ζIq−1∂Iq , q = 3, ... , N + 1 ,
A ≡ ···
p (p−2)
RI1···Ip (i)[2]τ(A− 2 )ζI1 ζIp∂ , p = 2, ... , N ,
A ≡ ··· τ
where N is the number of supersymmetries. Here we have used the notational conventions
of [11]. In [4] one uses the dual representation of the algebra in order to develop a field
theory. This idea is very different from that used in so called conformal field theories as in
this approach there are no fields external to the algebra (actually its dual) introduced, that
is no modelor Lagrangian is introduced that is exterior to thealgebra. Theimpetus behind
the construction of the Lagrangian is for its field equations to correspond to constraints
that arise on the coadjoint orbits.
The elements of the algebra can be realized as fields whose tensor properties can be
determined by they way they transform under one dimensional coordinate transformations.
HoweachfieldtransformsunderaLiederivativewithrespecttoξ aresummarizedbelow. In
conformal field theory these transformation laws characterize the fields by weight and spin.
Here we treat the algebraic elements as one dimensional tensors so that higher dimensional
realizations can be achieved.
Table 1: Tensors Associated with the Algebra
Elementofalgebra TransformationRule TensorStructure
L η η ξ′η+ξη′ ηa
A
→ → −
GI χI χI ξ(χI)′+ 1ξ′χI χI;α
A → → − 2
TRS tRS tRS ξ(tRS)′ tRS
→ → −
UV1···Vn → wV1···Vn wV1···Vn → −ξ(wV1···Vn)′− 21(n−2)ξ′wV1···Vn wV1···αV1n·;··aαn
RAT1···Tn → rT1···Tn rT1···Tn → −(rT1···Tn)′ξ− 12(n−2)ξ′rT1···Tn rT1···αT1n··;·aαn
In the above table we have used capital Latin letters, such as I,J,K to represent SO(N)
indices, small Latin letters to represent tensor indices, and small Greek letters for spinor
indices. Spinors with their indices up transform as scalar tensor densities of weight one
(1) while those with their indices down transform as scalar densities of weight minus one
(-1). For example the generator UV1V2V3 has a tensor density realization of contravariant
tensor with rank one and weight 3 living in the N N N representation of SO(N),
−2 × ×
i.e. ωV1V2V3;a. This is the first step at identifying a spectrum of physical fields that have
α1α2α3
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a natural connection to the algebra. The completition of this identification comes from
identifying the tensors related to the coadjoint representation. The transformation laws for
the coadjoint elements and tensor representation are tabulated below. We will raise and
lower the SO(N) indices using the 1 matrix, while the spinor indices are raised and lowered
with Cαβ and C .
αβ
Table 2: Tensors Associated with the Dual of the Algebra
Dualelementof algebra TransformationRule TensorStructure
L⋆ D D 2ξ′D ξD′ D
A ab
→ → − −
G⋆I ψI ψI ξ(ψI)′ 3ξ′ψI ψI
A → → − − 2 aα
T⋆RS ARS ARS (ξ)′ARS ξ(ARS)′ ARS
a
→ → − −
U⋆V1···Vn ωV1···Vn ωV1···Vn ξ(ωV1···Vn)′ (2 n)ξ′ωV1···Vn ωV1···Vn;α1···αn
→ → − − − 2 ab
R⋆T1···Tr ρT1···Tr ρT1···Tr (ρT1···Tr)′ξ (2 r)ξ′ρT1···Tr ρT1···Tr;α1···αr
A → → − − − 2 ab
Thus for N supersymmetries there is one rank two tensor D , N spin-3 fields ψI, a spin-1
ab 2
covariant tensorARS thatserves as theN(N 1)/2 SO(N)gauge potentials associated with
−
the supersymmetries, and N (2N N 1) fields for both the ωV1···Vp fields and ρT1···Tp
− −
sectors.
We would like to capture these component fields into superfields and write an action
in terms of these superfields. One can see from the above table that at least two distinct
superfields will be needed to absorb the field content. These two superfields will constitute
a diffeomorphism sector and a gauge sector for the SO(N) gauge symmetry.
3.2 The Chiral Diffeomorphism Superfield
Instead of using component fields we would like a superfield formulation of the algebra
that can be used to construct an N-extended version of the affirmative action found in [3].
Just as in the above reference we will write the action using tensor notation so that future
extension to higher dimensions and non-chirality can follow easily. Our focus will be on the
two dimensional models, so we will assume in what follows that the Grassmann variables
are Majorana and chiral. Therefore in this section we will display a fermionic index, ′′α′′
say, with the understanding that it is a one dimensional index.
Recall that up to the central extension the Virasoro algebra (or Witt algebra) may be
realized as the one dimensional reduction of the Lie algebra of vector fields. Consider the
vector fields ξa and ηa. We know that the Lie derivative of ηa with respect to ξa is given by
ηa = ξb∂ ηa+ηb∂ ξa = (ξ η)a, (12)
ξ b b
L − ◦
and further that
[ , ]= . (13)
ξ η ξ◦η
L L L
6
Now consider the superfields = (ξn,χJ,β) and = (ηn,ψK;α). We are assuming that we
F G
have Majorana fermions. Define the derivative operator through
DI;ν
∂
, = iδ γn . (14)
{DI;µ DJ;ν} − IJ µν∂zn
This implies that
∂ ∂
= δ iζJ;νγm . (15)
DI;µ ζI;µ − IJ 2 νµ∂zn
The super Virasoro algebra contains both a diffeomorphism as well as a supersymmetry
transformation with . We can construct vector fields
I;µ
D
∂
F = ξn + 1χJ;β , (16)
∂zn 2 DJ;β
and
∂
G = ηn + 1ψJ;β . (17)
∂zn 2 DJ;β
The commutator of F and G is
∂ ∂
[F,G] = [ξn + 1χJ;β , ηm + 1ψI;α ]
∂zn 2 DJ;β ∂zm 2 DI;α
∂ ∂
= ξn ηm ηm ηn+ 1χJ;β( ηm) 1ψK;α( ξm)
{ ∂zn − ∂zm 2 DJ;β − 2 DK;α
∂
+i(1χJ;β)(1ψK;α)δJKγn
2 2 βα}∂zn
+ 1(χJ;β( ψK;α)+ψJ;β( χK;α)
{−4 DJ;β DJ;β
∂ ∂
+i(ξn ψK;α ηn χK;α) . (18)
2 ∂zn − ∂zn }DK;α
With this in place we can now naturally extend the vector fields ξn to N supersymmetries.
Let
ζI1···Im;α1···αm ζ[I1;α1 ζIm;αm], (19)
≡ ···
then an N-extension of the vector field ξn is given by the vector superfield Fn, where
Fn = ξn+χn ζI1;α1 +rn ζI1I2;α1α2 + +rn ζI1···IN;α1···αN (20)
I1;α1 I1I2;α1α2 ··· I1···IN;α1···αN
Thissuperfieldcontains thesuperpartnerthatisusedtoperformthesupersymmetrictrans-
lation. With this we can write the superdiffeomorphism vector field that is analogous to
the one used in Eq.[18] as
∂
F = Fn +δAB( Fn)γαβ . (21)
∂zn DA;α n DB;β
The Fn superfield contains the field content of the L ,GI , and RT1···Tr generators seen in
A A A
Table 1. Note that χIα = δII1γα1αχn .
n I1;α1
7
However the centrally extended contributions demand that we also consider the prepo-
tential FnI1···IN;α1···αN given by
FnI1···IN;α1···αN ξnζI1···IN;α1···αN +χn[I1···IN−1;αN−1ζIN;αN]+
≡ 0
rn[I1IN−2;α1αN−2ζIN−1IN−2;αN−1αN−2]+ +rnI1···IN;α1···αN, (22)
0 ··· 0
where the “0” subscripted fields are the prepotentials defining the elements of Fn through
Fn = FnI1···IN;α1···αN. (23)
DI1;α1 ··· DIN;αN
Then one may write down the commutation relations for centrally extended elements as
[[(F,a),(G,b)]] = ([F,G],<< F,G >> ) (24)
where the two cocycle << F,G >> is defined as
<< F,G >> ic dzdζ ((Fn)′′′GmI1···IN;α1···αN (Gn)′′′FmI1···IN;α1···αN)).
≡ 24π Z I1···IN;α1···αN −
(25)
In one dimension the tensor indices m and n are not relevant but simply let us keep track
of the transformation properties of the fields which will beuseful in higher dimensions. The
integrand transforms as a scalar density in one dimension. This preserves the N = 0 form
of the two cocycle used for the central extension of the Virasoro algebra.
To continue we need the dual elements of these vector fields. This implies that there
exists a bilinear two form < > that is invariant under diffeomorphisms. Consider
∗|∗
the dual of Fn for N supersymmetries. In one dimension one has a pairing that can be
represented tensorially as the integral
< (F,a)(B,ˆb)>= dzpdζ dζ FnBI1···IN;α1···αN +aˆb. (26)
| Z α1;I1··· αN;IN np
InonedimensionFp isconsideredarankonecontravariantvector field,whileBI1···IN;α1···αN
np
is a quadratic differential with density N. In higher dimensions we will take advantage
−2
of the fact that √gFab transforms like Fa in one coordinate. The invariant integral in k
dimensions will be written as
< B F >= dzkdζ dζ √gFnpBI1···IN;α1···αN. (27)
| Z α1;I1··· αN;IN np
For now, we use the one dimensional commutators and pairing to write the transformation
law for the coadjoint elements as
δ BI1···IN;α1···αN = Fm∂ BI1···IN;α1···αN BI1···IN;α1···αN ∂ Fm
F np m np mp n
− −
BI1···IN;α1···αN ∂ Fm+ N (∂ Fm)BI1···IN;α1···αN
− nm p 2 m np
+ ˆbc FmI1···IN;α1···αN. (28)
n p m
∇ ∇ ∇
The transformation shows that BI1···IN;α1···αN transforms as a rank two tensor due to its n
np
and p indices and for each contravariant fermion index we have assigned a density of weight
1.
−2
8
In terms of the fields in Table 2, the field BI1···IN;α1···αN has a decomposition of
np
BI1···IN;α1···αN = D ζI1···IN;α1···αN +ψ[I1;α1ζI2···IN;α2···αN]
np np np
+ ρ[I1I2;α1α2ζI3···IN;α3···αN]+ +ρI1···IN;α1···αN, (29)
np np
···
with the 3 spin field ψI in Table 2 satisfying,
2 aα
ψI = ψI;α1γp . (30)
aα ap α1α
In the construction of the actions that follow we will need to define Fn in terms of the
superfield BI1···IN;α1···αN. For the moment let us ignore the SO(N) gauge symmetry that
np
has been induced by the TIJ generators. We will use a tilde as a reminder of this. Then
M
we may write
B˜ = BI1···IN;α1···αN. (31)
np DI1;α1 ··· DIN;αN np
Since Fn will generate time-independent coordinate transformations, we assign Fn = B˜n.
0
Before we put these ingredients into an action we must deal with the SO(N) symmetry.
N
3.3 The SO( ) Gauge Superfield
In the previous section we ignored the SO(N) gauge field that has become manifest due
tothefactthattheTIJ generatorstransformasscalarfieldswhiletheirdualelements trans-
form as vector fields under diffeomorphisms. Again we would like to cast the algebra into
the tidy language of superfields. Consider the superfields ΛI;α that enjoy the ζ expansion
in terms of the fields found in Table 1,
ΛI;α = wI;α+tI ζJ;α+w ζI1I;α1α+ +w ζI1···IN−1,I;α1···αN−1,α. (32)
J I1;α1 ··· I1···IN−1;α1···αN−1
Here wI;α = wI;aγβα, while tIJ is the anti-symmetric field found in Table 1. A generic
β a
element is then
∂
Λ= ΛJ;α . (33)
∂ζJ;α
We introduce the algebraic prepotential ΛII1···IN;αα1···αN through
ΛJ;α = ΛI[I1···IN];α[α1···αN]. (34)
DI1;α1 ··· DIN;αN
Then the centrally extended commutation relations can be written as
[[(Λ,λ),(Ω,µ)]] = ([Λ,Ω],<< Λ,Ω >>). (35)
where the one dimensional two cocycle for the algebra is
<< Λ,Ω >> = k dzpdζ (( ΛI;α)′ ΩM[I1···IN];β[a1···αN]
Z I1···IN;α1···αN DJ;α DN;β −
( ΩI;α)′ ΛM[I1···IN];β[a1···αN])δJN. (36)
J;α N;β IM
D D
9
The δJN allows us to show that the SO(N) invariant trace and is defined as
IM
δJN δ δJN δN δJ . (37)
IM IM I M
≡ −
Again notice that the derivative in one dimension has the affect of changing a scalar into a
scalar density. Thus the integral is invariant under one dimensional diffeomorphisms.
Again wewouldliketoextractadimensionindependentdescriptionofthedualelements
of the SO(N) adjoint. Here we use the fact that in one dimension ΛI;α transforms the same
way as √gΛI;αp. This device allows us to find a dimension independent description of the
dual of the algebra. The pairing between a centrally extended element, say (ΛI;α, a), and
a coadjoint element is then
< (Λ, a) (A,β) >= dzkdζ √gΛI;αpA I1···IN;α1···αN + aβ. (38)
| Z I1···IN;α1···αN I;αp
The dual element has the ζ decomposition
A I1···IN;α1···αN = ω ζI1···IN;α1···αN +ω[I1;α1ζI2···IN;a2···aN]+ +ωI1···IN;α1···αN. (39)
I;αp I;αp I;αp ··· I;αp
From the algebra and two cocycle we find that the transformation of the dual element with
respect to ΩI;α is
δ AI1···IN;α1···αN = ΩJ;β AI1···IN;α1···αN +( ΩJ;β)AI1···IN;α1···αN
Ω I;αp − DJ;β I;αp DI;β J;αp
( ΩJ;β)AI1···IN;α1···αN
− DJ;β I;αp
+ ˆbk ΩM[I1···IN];β[α1···αN]δ JN , (40)
J;β N;α p IM
D D ∇
This transformation law is the the Lie derivative on the SO(N) group manifold where only
the “I” index is recognized, since the other SO(N) indices were contracted with the SO(N)
volume form. Notice that the coadjoint element transforms as a tensor density of weight 1
due to the penultimate term in the transformation law.
The gauge field can be extracted from the superpotential by writing
A I = A I[I2···IN];α[α2···αN]. (41)
Jp DI2;α2···DIN;αN J;αp
We can then build an SO(N) covariant derivative operator so that the covariant deriva-
tive on a vector living in the left regular representation, say BL, is
q
BM = ∂ BM Γ rBM +A J BL(δI δ M δ δIM) (42)
p q p q pq r Ip q L J JL
∇ − −
Then Eq.(31) may be written as
B = BI1···IN;α1···αN. (43)
np ∇I1;α1 ··· ∇IN;αN np
10