D=10 Chiral Tensionless Super p-Branes

12 Pages·0.15 MB·

by  P. Bozhilov

D=10 CHIRAL TENSIONLESS SUPER p-BRANES 1
P. Bozhilov 2
Bogoliubov Laboratory of Theoretical Physics,
JINR, 141980 Dubna, Russia
We consider a model for tensionless (null) super p-branes with N chiral super-
symmetries in ten dimensional flat space-time. After establishing the symmetries
9
9 of the action, we give the general solution of the classical equations of motion in
9
a particular gauge. In the case of a null superstring (p=1) we find the general
1
solution in an arbitrary gauge. Then, using a harmonic superspace approach, the
n
a initial algebra of first and second class constraints is converted into an algebra of
J
Lorentz-covariant, BFV-irreducible, first class constraints only. The corresponding
9
BRST charge is as for a first rank dynamical system.
2

1
v
0
1 Introduction
6
1
1
0 The tensionless (null) p-branes correspond to usual p-branes with their ten-
9 sion T taken to be zero. This relationship between null p-branes and the tensionful
p
9
ones may be regarded as a generalization of the massless-massive particles corre-
/
h
spondence. On the other hand, the limit T 0 corresponds to the energetic scale
t p
- →
p E >> MPlank. In other words, the null p-brane is the high energy limit of the
e tensionful one. There exist also an interpretation of the null and free p-branes as
h
: theories, corresponding to different vacuum states of a p-brane, interacting with a
v
scalar field background [1]. So, one can consider the possibility of tension generation
i
X
for null p-branes (see [2] and references therein). Another viewpoint on the connec-
r
a tion between null and tensionful p-branes is that the null one may be interpreted as
a ”free” theory opposed to the tensionful ”interacting” theory [3]. All this explains
the interest in considering null p-branes and their supersymmetric extensions.
Models for tensionless p-branes with manifest supersymmetry are proposed in
[4]. In [1] a twistor-like action is suggested, for null super-p-branes with N-extended
global supersymmetry in four dimensional space-time. In the recent work [5], the
quantum constraint algebras of the usual and conformal tensionless spinning p-
branes are considered.
Inapreviouspaper[6],weannouncedforanullsuperp-branemodel, andherewe
are going to formulate it, and to consider its classical properties. After establishing
1Work supported in part by the National Science Foundation of Bulgaria under contract φ
−
620/1996
2E-mail: [email protected]; permanent address: Dept.of Theoretical Physics,”Konstantin
Preslavsky”Univ. of Shoumen, 9700 Shoumen, Bulgaria
1
the symmetries of the action, we give the general solution of the classical equations
of motion in a particular gauge. In the case of a null superstring, (p=1), we find
the general solution in an arbitrary gauge. Then, in the framework of a harmonic
superspace approach, the initial algebra of first and second class constraints is con-
verted into an algebra of Lorentz-covariant, BFV-irreducible, first class constraints
only. The corresponding BRST charge is as for a first rank dynamical system.
2 Lagrangian formulation
We define our model for D = 10 N-extended chiral tensionless super p-branes
by the action:
S = dp+1ξL , L = VJVKΠµΠν η , (1)
J K µν
Z
N
Πµ = ∂ xµ +i (θAσµ∂ θA) , ∂ = ∂/∂ξJ,
J J J J
A=1
X
ξJ = (ξ0,ξj) = (τ,σj), diag(η ) = ( ,+,...,+),
µν
−
J,K = 0,1,...,p , j,k = 1,...,p , µ,ν = 0,1,...,9.
Here (xν,θAα) are the superspace coordinates, θAα are N left Majorana-Weyl space-
time spinors (α = 1,...,16 , N being the number of the supersymmetries) and
σµ are the 10-dimensional Pauli matrices (our spinor conventions are given in the
Appendix). Actions of this type are first given in [7] for the case of tensionless
superstring (p = 1,N = 1) and in [8] for the bosonic case (N = 0).
The action (1) has an obvious global Poincare´invariance. Under global infinites-
imal supersymmetry transformations, the fields θAα(ξ), xν(ξ) and VJ(ξ) transform
as follows:
δAα = ηAα , δ xµ = i (θAσµδ θA) , δ VJ = 0.
η η η η
A
X
As a consequence δ Πµ = 0 and hence δ L = δ S = 0 also.
η J η η
Toprovetheinvarianceoftheactionunderinfinitesimaldiffeomorphisms, wefirst
write down the corresponding transformation law for the (r,s)-type tensor density
of weight a
δ TJ1...Jr [a] = L TJ1...Jr [a] = εL∂ TJ1...Jr [a]
ε K1...Ks ε K1...Ks L K1...Ks
+ TJ1...Jr [a]∂ εK +...+TJ1...Jr [a]∂ εK (2)
KK2...Ks K1 K1...Ks−1K Ks
TJJ2...Jr[a]∂ εJ1 ... TJ1...Jr−1J[a]∂ εJr
− K1...Ks J − − K1...Ks J
+ aTJ1...Jr [a]∂ εL,
K1...Ks L
where L is the Lie derivative along the vector field ε. Using (2), one verifies that if
ε
xµ(ξ), θAα(ξ) are world-volume scalars (a = 0) and VJ(ξ) is a world-volume (1,0)-
type tensor density of weight a = 1/2, then Πν is a (0,1)-type tensor, ΠνΠ is a
J J Kν
(0,2)-type tensor and L is a scalar density of weight a = 1. Therefore,
δ S = dp+1ξ∂ (εJL)
ε J
Z
2
and this variation vanishes under suitable boundary conditions.
Let us now check the kappa-invariance of the action. We define the κ-variations
of θAα(ξ), xν(ξ) and VJ(ξ) as follows:
δ θAα = i(ΓκA)α = iVJ( Π κA)α, δ xν = i (θAσνδ θA), (3)
κ J κ κ
6 −
A
X
δ VK = 2VKVL (∂ θAκA).
κ L
A
X
Therefore, κAα(ξ) are left Majorana-Weyl space-time spinors and world-volume
scalar densities of weight a = 1/2.
−
From (3) we obtain:
δ (ΠνΠ ) = 2i [∂ θA Π +∂ θA Π ]δ θA
κ J Kν − J 6 K K 6 J κ
A
X
and
δ L = 2VJΠνΠ [δ VK 2VKVL (∂ θAκA)] = 0.
κ J Kν κ − L
A
X
The algebra of kappa-transformations closes only on the equations of motion,
which can be written in the form:
∂ (VJVKΠ ) = 0, VJVK(∂ θA Π ) = 0, VJΠνΠ = 0. (4)
J Kν J 6 K α J Kν
As usual, an additional local bosonic world-volume symmetry is needed for its clo-
sure. In our case, the Lagrangian, and therefore the action, are invariant under the
following transformations of the fields:
δ θA(ξ) = λVJ∂ θA, δ xν(ξ) = i (θAσνδ θA), δ VJ(ξ) = 0.
λ J λ λ λ
−
A
X
Now, checking the commutator of two kappa-transformations, we find:
[δ ,δ ]θAα(ξ) = δ θAα(ξ)+terms eqs. of motion,
κ1 κ2 κ ∝
[δ ,δ ]xν(ξ) = (δ +δ +δ )xν(ξ)+terms eqs. of motion,
κ1 κ2 κ ε λ ∝
[δ ,δ ]VJ(ξ) = δ VJ(ξ)+terms eqs. of motion.
κ1 κ2 ε ∝
Here κAα(ξ), λ(ξ) and ε(ξ) are given by the expressions:
κAα = 2VK [(∂ θBκB)κAα (∂ θBκB)κAα],
− K 1 2 − K 2 1
B
X
λ = 4iVK (κA Π κA) , εJ = VJλ.
1 6 K 2 −
A
X
WenotethatΓ = (VJ Π ) in(3)hasthefollowingpropertyontheequations
αβ 6 J αβ
of motion
Γ2 = 0.
3
This means, that the local kappa-invariance of the action indeed eliminates half of
the components of θA as is needed.
For transition to Hamiltonian picture, it is convenient to rewrite the Lagrangian
density (1) in the form (∂ = ∂/∂τ,∂ = ∂/∂σj):
τ j
1 2
L = (∂ µj∂ )x+i θAσ(∂ µj∂ )θA , (5)
4µ0 τ − j τ − j
h XA i
where
1 µj
VJ = (V0,Vj) = ,
−2√µ0 2√µ0!
The equations of motion for the Lagrange multipliers µ0 and µj which follow from
(5) give the constraints (p and pA are the momenta conjugated to xν and θAα):
ν θα
T = p2 , T = p ∂ xν + pA ∂ θAα. (6)
0 j ν j θα j
A
X
The remaining constraints follow from the definition of the momenta pA :
θα
DA = ipA ( pθA) . (7)
α − θα − 6 α
3 Hamiltonian formulation
The Hamiltonian which corresponds to the Lagrangian density (5) is a linear
combination of the constraint (6) and (7):
H = dpσ[µ0T +µjT + µAαDA] (8)
0 0 j α
Z A
X
It is a generalization of the Hamiltonians for the bosonic null p-brane and for the
N-extended Green-Schwarz superparticle.
The equations of motion which follow from the Hamiltonian (8) are:
(∂ µj∂ )xν = 2µ0pν (µAσνθA), (∂ µj∂ )p = (∂ µj)p ,
τ j τ j ν j ν
− − −
A
X
(∂ µj∂ )θAα = iµAα, (∂ µj∂ )pA = (∂ µj)pA +(µA p) . (9)
τ − j τ − j θα j θα 6 α
In (9), one can consider µ0, µj and µAα as depending only on σ = (σ1,...,σp), but
not on τ as a consequence from their equations of motion.
In the gauge when µ0, µj and µAα are constants, the general solution of (9) is
xν(τ,σ) = xν(z)+τ[2µ0pν(z) (µAσνθA(τ,σ))],
−
A
X
= xν(z)+τ[2µ0pν(z) (µAσνθA(z))]
−
A
X
p (τ,σ) = p (z), θAα(τ,σ) = θAα(z)+iτµAα, (10)
ν ν
pA (τ,σ) = pA (z)+τ(µAσν) p (z),
θα θα α ν
4
where xν(z), p (z), θAα(z) and pA (z) are arbitrary functions of their arguments
ν θα
zj = µjτ +σj.
In the case of null strings (p = 1), one can write explicitly the general solution
of the equations of motion in an arbitrary gauge: µ0 = µ0(σ), µ1 µ = µ(σ),
≡
µAα = µAα(σ). This solution is given by
σ σ
µ0(s) µAα(s)
xν(τ,σ) = gν(w) 2 dsfν(w)+ ds[σνζA(w)]
− µ2(s) µ(s) α
Z A Z
X
σ (µAσν) (s ) s1µAα(s)
α 1
i ds ds,
1
− µ(s ) µ(s)
A Z 1 Z
X
p (τ,σ) = µ−1(σ)f (w), (11)
ν ν
σ
µAα(s)
θAα(τ,σ) = ζAα(w) i ds,
− µ(s)
Z
σ
(µAσν) (s)
pA (τ,σ) = µ−1(σ) hA(w) α dsf (w) .
θα " α − µ(s) ν #
Z
Here gν(w), f (w), ζAα(w) and hA(w) are arbitrary functions of the variable
ν α
σ
ds
w = τ +
µ(s)
Z
The solution (10) at p = 1 differs from (11) by the choice of the particular solutions
of the inhomogenious equations. As for z and w, one can write for example (µ0, µ,
µAα are now constants)
p (τ,σ) = µ−1f (τ +σ/µ) = µ−1f [µ−1(µτ +σ)] = p (z)
ν ν ν ν
and analogously for the other arbitrary functions in the general solution of the
equations of motion.
Let us now consider the properties of the constraints (6), (7). They satisfy the
following (equal τ) Poisson bracket algebra
T (σ ),T (σ ) = 0, T (σ ),DA(σ ) = 0,
{ 0 1 0 2 } { 0 1 α 2 }
T (σ ),T (σ ) = [T (σ )+T (σ )]∂ δp(σ σ ),
0 1 j 2 0 1 0 2 j 1 2
{ } −
T (σ ),T (σ ) = [δlT (σ )+δlT (σ )]∂ δp(σ σ ),
{ j 1 k 2 } j k 1 k j 2 l 1 − 2
T (σ ),DA(σ ) = DA(σ )∂ δp(σ σ ),
{ j 1 α 2 } α 1 j 1 − 2
DA(σ ),DB(σ ) = 2iδAB p δp(σ σ ).
{ α 1 β 2 } 6 αβ 1 − 2
From the condition that the constraints must be maintained in time, i.e. [9]
T ,H 0, T ,H 0, DA,H 0, (12)
{ 0 0} ≈ { j 0} ≈ { α 0} ≈
5
it follows that in the Hamiltonian H one has to include the constraints
0
TA = p DAβ
α 6 αβ
instead of DA. This is because the Hamiltonian has to be first class quantity, but
α
DA area mixture of first and second class constraints. TA has the following non-zero
α α
Poisson brackets
T (σ ),TA(σ ) = [TA(σ )+TA(σ )]∂ δp(σ σ ),
{ j 1 α 2 } α 1 α 2 j 1 − 2
TA(σ ),TB(σ ) = 2iδAB p T δp(σ σ ).
{ α 1 β 2 } 6 αβ 0 1 − 2
In this form, our constraints are first class and the Dirac consistency conditions (12)
(with DA replaced by TA) are satisfied identically. However, one now encounters a
α α
new problem. The constraints T , T and TA are not BFV-irreducible, i.e. they are
0 j α
functionally dependent:
( pTA)α DAαT = 0.
0
6 −
It is known, that in this case after BRST-BFV quantization an infinite number of
ghosts for ghosts appear, if one wants to preserve the manifest Lorentz invariance.
Thewayoutconsistsintheintroductionofauxiliaryvariables, sothatthemixtureof
first and second class constraints DAα can be appropriately covariantly decomposed
into first class constraints and second class ones. To this end, here we will use
the auxiliary harmonic variables introduced in [10] and [11]. These are ua and v±,
µ α
where superscripts a = 1,...,8 and transform under the ’internal’ groups SO(8)
±
and SO(1,1) respectively. The just introduced variables are constrained by the
following orthogonality conditions
uaubµ = Cab, u±uaµ = 0, u+u−µ = 1,
µ µ µ −
where
u± = v±σαβv±,
µ α µ β
Cab is the invariant metric tensor in the relevant representation space of SO(8) and
(u±)2 = 0 as a consequence of the Fierz identity for the 10-dimensional σ-matrices.
We note that ua and v± do not depend on σ.
ν α
Now we have to ensure that our dynamical system does not depend on arbitrary
rotations of the auxiliary variables (ua, u±). It can be done by introduction of first
µ µ
class constraints, which generate these transformations
1 1
Iab = (uapbν ubpaν + v+σabp+ + v−σabp−), σab = uaubσµν,
− ν u − ν u 2 v 2 v µ ν
1
I−+ = (v+p+α v−p−α), (13)
−2 α v − α v
1
I±a = (u±paµ + v∓σ±σap∓), σ± = u±σν, σa = uaσν.
− µ u 2 v ν ν
6
In the above equalities, paν and p±α are the momenta canonically conjugated to ua
u v ν
and v±.
α
Thenewlyintroducedconstraints(13)obeythefollowingPoissonbracket algebra
Iab,Icd = CbcIad CacIbd +CadIbc CbdIac,
{ } − −
I−+,I±a = I±a,
{ } ±
Iab,I±c = CbcI±a CacI±b,
{ } −
I+a,I−b = CabI−+ +Iab.
{ }
This algebra is isomorphic to the SO(1,9) algebra: Iab generate SO(8) rotations,
I−+ is the generator of the subgroup SO(1,1) and I±a generate the transformations
from the coset SO(1,9)/SO(1,1) SO(8).
×
Now we are ready to separate DAα into first and second class constraints in a
Lorentz-covariant form. This separation is given by the equalities [12]:
1
DAα = [(σav+)αDA +( pσ+σav−)αGA], p+ = pνu+, (14)
p+ a 6 a ν
1
DAa = (v+σa p) DAβ, GAa = (v−σaσ+) DAβ.
β β
6 2
Here DAa are first class constraints and GAa are second class ones:
DAa(σ ),DBb(σ ) = 2iδABCabp+T δp(σ σ )
{ 1 2 } − 0 1 − 2
GAa(σ ),GBb(σ ) = iδABCabp+δp(σ σ ).
{ 1 2 } 1 − 2
It is convenient to pass from second class constraints GAa to first class constraints
GˆAa, without changing the actual degrees of freedom [12], [13] :
GAa GˆAa = GAa +(p+)1/2ΨAa GˆAa(σ ),GˆBb(σ ) = 0,
1 2
→ ⇒ { }
where ΨAa(σ) are fermionic ghosts which abelianize our second class constraints as
a consequence of the Poisson bracket relation
ΨAa(σ ),ΨBb(σ ) = iδABCabδp(σ σ ).
{ 1 2 } − 1 − 2
It turns out, that the constraint algebra is much more simple, if we work not
with DAa and GˆAa but with TˆAα given by
TˆAα = (p+)−1/2[(σav+)αDA +( pσ+σav−)αGˆA]
a 6 a
= (p+)1/2DAα +( pσ+σav−)αΨA.
6 a
After the introduction of the auxiliary fermionic variables ΨAa, we have to modify
some of the constraints, to preserve their first class property. Namely T , Iab and
j
I−a change as follows
i
Tˆ = T + Cab ΨA∂ ΨA,
j j 2 a j b
A
X
i
Iˆab = Iab +Jab, Jab = dpσjab(σ), jab = (v−σ σabσ+σ v−) ΨAcΨAd,
c d
4
Z A
X
Iˆ−a = I−a +J−a, J−a = dpσj−a(σ), j−a = (p+)−1jabp .
b
−
Z
7
As a consequence, we can write down the Hamiltonian for the considered model in
the form:
H = dpσ[λ0T (σ)+λjTˆ (σ)+ λAαTˆA(σ)]+
0 j α
Z A
X
λ Iˆab +λ I−+ +λ I+a +λ Iˆ−a.
ab −+ +a −a
The constraints entering H are all first class, irreducible and Lorentz- covariant.
Their algebra reads (only the non-zero Poisson brackets are written):
T (σ ),Tˆ (σ ) = (T (σ )+T (σ ))∂ δp(σ σ ),
{ 0 1 j 2 } 0 1 0 2 j 1 − 2
Tˆ (σ ),Tˆ (σ ) = (δlTˆ (σ )+δlTˆ (σ ))∂ δp(σ σ ),
{ j 1 k 2 } j k 1 k j 2 l 1 − 2
1
Tˆ (σ ),TˆA(σ ) = (TˆA(σ )+ TˆA(σ ))∂ δp(σ σ ),
{ j 1 α 2 } α 1 2 α 2 j 1 − 2
TˆA(σ ),TˆB(σ ) = iδABσ+ T δp(σ σ ),
{ α 1 β 2 } αβ 0 1 − 2
1
I−+,TˆA = TˆA, Iˆ−a,TˆA = (2p+)−1[paTˆA +(σ+σabv−) ΨAT ],
{ α} 2 α { α } α α b 0
Iˆab,Iˆcd = CbcIˆad CacIˆbd +CadIˆbc CbdIˆac,
{ } − −
I−+,I+a = I+a, I−+,Iˆ−a = Iˆ−a,
{ } { } −
Iˆab,I+c = CbcI+a CacI+b, Iˆab,Iˆ−c = CbcIˆ−a CacIˆ−b,
{ } − { } −
I+a,Iˆ−b = CabI−+ +Iˆab,
{ }
Iˆ−a,Iˆ−b = dpσ(p+)−2jabT .
0
{ } −
Z
Having in mind the above algebra, one can construct the corresponding BRST
charge Ω [14] ( =complex conjugation)
∗
Ω = Ωmin +π ¯M, Ω,Ω = 0, Ω∗ = Ω, (15)
M
P { }
where M = 0,j,Aα,ab, +,+a, a. Ωmin in (15) can be written as
− −
Ωmin = Ωbrane +Ωaux,
Ωbrane = dpσ T η0 +Tˆ ηj + TˆAηAα + [(∂ ηj)η0 +(∂ η0)ηj]+
{ 0 j α P0 j j
Z A
X
1 i
+ (∂ ηk)ηj + A[ηj∂ ηAα ηAα∂ ηj] ηAασ+ ηAβ ,
Pk j Pα j − 2 j − 2P0 αβ }
A A
X X
Ωaux = Iˆabη +I−+η +I+aη +Iˆ−aη
ab −+ +a −a
+ ( acηb. bcηa. +2 +aηb +2 −aηb)η
P .c −P .c P + P − ab
+ ( +aη −aη )η +( −+ηa + abη )η
P +a −P −a −+ P − P −b +a
1
+ dpσ AηAαη +(p+)−1 [pa A (σ+σabv−) ΨA ]ηAαη
2 { Pα −+ Pα − α b P0 −a
Z A A
X X
(p+)−2jab η η .
0 −b −a
− P }
These expressions for Ωbrane and Ωaux show that we have found a set of constraints
which ensure the first rank property of the model.
8
Ωmin can be represented also in the form
1 1 1
Ωmin = dpσ[(T + Tgh)η0 +(Tˆ + Tgh)ηj + (TˆA + TAgh)ηAα]
0 2 0 j 2 j α 2 α
Z A
X
1 1 1 1
+(Iˆab + Iab)η +(I−+ + I−+)η +(I+a + I+a)η +(Iˆ−a + I−a)η
2 gh ab 2 gh −+ 2 gh +a 2 gh −a
1 1
+ dpσ∂ ηkηj + AηAαηj .
j 2Pk 4 Pα
Z (cid:16) XA (cid:17)
Here a super(sub)script gh is used for the ghost part of the total gauge generators
Gtot = Ω, = Ωmin, = G+Ggh.
{ P} { P}
We recall that the Poisson bracket algebras of Gtot and G coincide for first rank
systems. The manifest expressions for Ggh are:
Tgh = 2 ∂ ηj +(∂ )ηj,
0 P0 j jP
Tgh = 2 ∂ η0 +(∂ )η0 + ∂ ηk + ∂ ηk +(∂ )ηk
j P0 j jP0 Pj k Pk j kPj
3 1
+ A∂ ηAα + (∂ A)ηAα,
2 Pα j 2 jPα
A A
X X
3
TAgh = A∂ ηj (∂ A)ηj i σ+ ηAβ +
α −2Pα j − jPα − P0 αβ
1
+ Aη +(2p+)−1 pa A (σ+σabv−) ΨA η ,
2Pα −+ Pα − α b P0 −a
h i
Iab = 2( acηb. bcηa.)+( +aηb +bηa)+( −aηb −bηa),
gh P .c −P .c P + −P + P − −P −
1
I−+ = +aη −aη + dpσ AηAα,
gh P +a −P −a 2 Pα
Z A
X
I+a = 2 +bηa. +aη + −+ηa + abη ,
gh P .b −P −+ P − P −b
I−a = 2 −bηa. + −aη −+ηa + abη +
gh P .b P −+ −P + P +b
+ dpσ (2p+)−1 pa A (σ+σabv−) ΨA ηAα (p+)−2jab η .
Pα − α b P0 − P0 −b
Z n XA h i o
Up to now, we introduced canonically conjugated ghosts (ηM, ), (η¯ , ¯M) and
M M
P P
momenta π for the Lagrange multipliers λM in the Hamiltonian. They have Pois-
M
son brackets and Grassmann parity as follows (ǫ is the Grassmann parity of the
M
corresponding constraint):
ηM, = δM, ǫ(ηM) = ǫ( ) = ǫ +1,
{ PN} N PM M
η¯ , ¯N = ( 1)ǫMǫNδN, ǫ(η¯ ) = ǫ(¯M) = ǫ +1,
{ M P } − − M M P M
λM,π = δM, ǫ(λM) = ǫ(π ) = ǫ .
{ N} N M M
The BRST-invariant Hamiltonian is
H = Hmin + χ˜,Ω = χ˜,Ω , (16)
χ˜
{ } { }
9
because from H = 0 it follows Hmin = 0. In this formula χ˜ stands for the
canonical
gauge fixing fermion (χ˜∗ = χ˜). We use the following representation for the latter
−
1
χ˜ = χmin +η¯ (χM + ρ πM), χmin = λM ,
M (M) M
2 P
where ρ are scalar parameters and we have separated the πM-dependence from
(M)
χM. If we adopt that χM does not depend on the ghosts (ηM, ) and (η¯ , ¯M),
M M
P P
the Hamiltonian H from (16) takes the form
χ˜
1
H = Hmin + ¯M π (χM + ρ πM)+ (17)
χ˜ χ PMP − M 2 (M)
1
+ η¯ χM,G ηN + ( 1)ǫN χM,UQ ηPηN ,
M { N} 2 − PQ{ NP}
h i
where
Hmin = χmin,Ωmin ,
χ { }
and generally χM,UQ = 0 as far as the structure coefficients of the constraint
{ NP} 6
algebra UM depend on the phase-space variables.
NP
One can use the representation (17) for H to obtain the corresponding BRST
χ˜
invariant Lagrangian
L = L+L +L .
χ˜ GH GF
Here L stands for the ghost part and L - for the gauge fixing part of the
GH GF
Lagrangian. If one does not intend to pass to the Lagrangian formalism, one may
restrict oneself to the minimal sector (Ωmin,χmin,Hmin). In particular, this means
χ
that Lagrange multipliers are not considered as dynamical variables anymore. With
this particular gauge choice, Hmin is a linear combination of the total constraints
χ
Hmin = Hmin +Hmin =
χ brane aux
= dpσ Λ0Ttot(σ)+ΛjTtot(σ)+ ΛAαTAtot(σ) +
0 j α
Z h XA i
+ Λ Iab +Λ I−+ +Λ I+a +Λ I−a,
ab tot −+ tot +a tot −a tot
and we can treat here the Lagrange multipliers Λ0,...,Λ as constants. Of course,
−a
this does not fix the gauge completely.
4 Comments and conclusions
To ensure that the harmonics and their conjugate momenta are pure gauge
degrees of freedom, we have to consider as physical observables only such functions
on the phase space which do not carry any SO(1,1) SO(8)indices. More precisely,
×
these functions are defined by the following expansion
F(y,u,v;p ,p ,p ) = [ua1...uakpak+1 ...pak+l ]
y u v ν1 νk uνk+1 uνk+l SO(8)singlet
X
v+ ...v+ v− ...v− p+β1...p+βrp−βr+1...p−βm−n+r
α1 αm αm+1 αm+n v v v v
Fα1...αm+n,ν1...νk+l(y,p ),
β1...βm−n+r y
10