Quantum BRST charge and OSp(1|8) superalgebra of twistor-like p-branes with exotic supersymmetry and Weyl symmetry

0.22 MB·English

by  D. V. Uvarov

Quantum BRST charge and OSp(1|8) superalgebra of
twistor-like p-branes with exotic supersymmetry and
Weyl symmetry
D.V. Uvarova and A.A. Zheltukhina,b
a Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine
b Institute of Theoretical Physics, University of Stockholm, Albanova,
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0 SE-10691 Stockholm, Sweden
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2

n
a Abstract
J

5 Algebra of the constraints of twistor-like p-branes restoring 3 fraction of the spon-
4

taneously brokenD =4N = 1supersymmetryisstudiedusingtheconversion method.
2
v Classical and quantum realizations of the BRST charge, unified superalgebra of the
6 global generalized superconformal OSp(1|8) and Virasoro and Weyl symmetries are
3
constructed. It is shown that the quantum Hermitian BRST charge is nilpotent and
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6 the quantized OSp(1|8) superalgebra is closed.
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4
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1 Introduction
/
h
t
-
The physical interpretation of the central charges in supersymmetry algebra as topological
p
e charges carried by branes [1] advanced understanding of the phenomena of partial sponta-
h
neous breaking of supersymmetry [2]. Because branes are constituents of M-theory, sponta-
:
v
neously breaking supersymmetry, their global and local symmetries correlate with the sym-
i
X
metries of M-theory [3], [4]. Studying these symmetries resulted in the model independent
ar classical analysis of BPS states preserving 1, 1 or 3 fractions of the partially spontaneously
4 2 4
broken D = 4 N = 1 supersymmetry [5]. A special interest to construction of a physical
model with domain wall configurations preserving 3 fraction of the D = 4 N = 1 supersym-
4
metry against spontaneous breaking was subscribed there. That configurations were earlier
studied in superparticle dynamics [6] and algebraically realized as the brane intersections
in [7]. Then the tensionless string/brane model preserving 3 fraction of the D = 4 N = 1
4
supersymmetry and generating static solutions for these tensionless objects was proposed in
[8]. These results have sharpen the general question: whether quantum exotic BPS states
saturated by the p-brane states protect the same high M−1 fraction of N = 1 global super-
M
symmetry against spontaneous breaking as in the classical case?
We have started studying the question in [9] on the example of the p-branes preserving 3
4
fraction of the partially spontaneously broken D = 4 N = 1 supersymmetry and found some
obstacles for the quantization in the QˆPˆ-ordering previously studied in [10] (see also [11]).
Here we analyze the quantization problem applying the BFV approach [12] and construct
quantum Hermitian BRST operator and the generators of gauge Weyl, Virasoro and global
OSp(1|8) symmetries extended by the ghost contributions. We prove the nilpotency of the
1
quantum Hermitian BRST charge, its (anti)commutativity with the quantum Hermitian
generators of the OSp(1|8) superalgebra and the closure of this quantized superlgebra. At
the same time we show that the quantum QˆPˆ-ordered BRST operator and the OSp(1|8)
generators are nonHermitian and differ from the Hermitian ones by the presence of the
divergent terms. We discuss a possibility to overcome this obstacle by the choice of the
special regularization prescription for thep-brane world-volume delta function δp(~σ−~σ′)|~σ=~σ′
and its derivative ∂ δp(0). A possibility of the exact cancellation of the divergent terms in
M
the QˆPˆ-ordered quantum operators for other dimensions D = 2,3,4(mod8)is also discussed.
2 Conversion of tensionless super p-brane constraints
New models of tensionless string and p-branes evolving in the symplectic superspace Msusy
M
and preserving all but one fractions of N = 1 supersymmetry were recently studied in
[8], [13]. For M = 2[D2] (D = 2,3,4 mod 8) the space MsMusy extends the standard D-
dimensionalsuperspace-time(x ,θ ),(wherea,b = 1,2,...,2[D2])bythetensorcentralcharge
ab a
(TCC) coordinates z . The coordinates x = xm(γ C−1) and z = izmn(γ C−1) +
ab ab m ab ab mn ab
zmnl(γ C−1) + ... constitute components of the symmetric spin-tensor Y . In terms of
mnl ab ab
Y and the Majorana spinor θ the action [8], invariant under the spontaneously broken
ab a
N = 1 supersymmetry and world-volume reparametrizations, is given by
1
S = dτdpσ ρµUaW Ub, (1)
p µab
2 Z
where W = W dξµ is the supersymmetric Cartan differential one-form
ab µab
W = ∂ Y −2i(∂ θ θ +∂ θ θ ), (2)
µab µ ab µ a b µ b a
and ∂ ≡ ∂ with ξµ = (τ,σM), (M = 1,2,...,p) parametrizing the p-brane world volume.
µ ∂ξµ
The local auxiliary Majorana spinor Ua(τ,σM) parametrizes the generalized momentum
Pab = 1ρτUaUb of tensionless p-brane and ρµ(τ,σM) is the world-volume vector density
2
providing the reparametrization invariance of S similarly to the null branes [14].
p
The action (1) has (M−1) κ−symmetries
δ θ = κ , δ Y = −2i(θ κ +θ κ ), δ Ua = 0, (3)
κ a a κ ab a b b a κ
which protect M−1 fraction of the N = 1 global supersymmetry to be spontaneously broken,
M
because of the one real condition Uaκ = 0 for the transformation parameters κ (τ,~σ)1.
a a
For the four-dimensional space-time the action (1) takes the form
1
S = dτdpσ ρµ 2uαω u¯α˙ +uαω uβ +u¯α˙ω¯ u¯β˙ . (4)
p 2 Z (cid:16) µαα˙ µαβ µα˙β˙ (cid:17)
This action is invariant under the OSp(1|8) symmetry, which is global supersymmetry of the
massless fields of all spins in D = 4 space-time extended by TCC coordinates [15], [16].
1To remove a possible misunderstanding in this terminology let us remind that from the world-volume
perspective the last fraction of the N = 1 supersymmetry is also the symmetry of the action (1), but it
is spontaneously broken, because of the presence of Goldstone fermion η˜ = −2i(Uaθ ) encoding the single
a
physical fermionic degree of freedom associated with θ (see [13] for details).
a
2
The Hamiltonian structure of the action (4), described in [17], is characterized by 3
fermionic and 2p+7 bosonic first-class constraints that generate its local symmetries, as well
as, 1 fermionic and 8 bosonic second-class constraints taken into account by the construction
of the Dirac bracket. We found that the D.B. algebra of the first-class constraints has the
rank equal two and it gives rise to the higher powers of the ghosts in the BRST generator.
To simplify transition to the quantum theory the conversion method [18]-[23], transform-
ing all the primary and secondary constraints to the first class, has been applied in [9]. To
this end the additional canonically conjugate pairs (Pα,qα), (P¯α˙,q¯α˙), (P(ϕ),ϕτ) and the self-
q q τ
conjugate Grassmannian variable f have been introduced. As a result, all the constraints
have been converted to the effective first-class constraints in the extended phase space.
The converted constraints for the auxiliary fields are the following
¯
Pα = Pα +Pα ≈ 0, Pα˙ = P¯α˙ +P¯α˙ ≈ 0, (5)
u u q u u q
e P(ρ) = P(ρ) +P(ϕe) ≈ 0, P(ρ) ≈ 0. (6)
τ τ τ M
¯
The converted bosonic consteraints Φ ≡ (Φα˙α,Φαβ,Φα˙β˙) originating from the Φ-constraints
[9] are given by
e e e e
Φα˙α = Pα˙α −ρ˜τu˜αu¯˜α˙ ≈ 0, Φαβ = παβ + 1ρ˜τu˜αu˜β ≈ 0, Φ¯α˙β˙ = π¯α˙β˙ + 1ρ˜τu¯˜α˙u¯˜β˙ ≈ 0 (7)
2 2
e e e
and have zero Poisson brackets (P.B.) with the constraints (5), (6) and among them-
¯
selves. The converted fermionic constraints Ψ = (Ψα,Ψα˙) originating from the primary
Ψ-constraints and generating four κ−symmetries take the form
e e e
Ψα = πα −2iθ¯ Pα˙α −4iπαβθ +2(ρ˜τ)1/2u˜αf ≈ 0,
α˙ β
(8)
¯
Ψα˙ =e−(Ψα)∗ = π¯α˙ −2iPα˙αθ −4iπ¯α˙β˙θ¯ −2(ρ˜τ)1/2u¯˜α˙f ≈ 0,
α β˙
where f∗ = f isean auxilieary Grassmannian variable characterized by the P.B.
{f(~σ),f(~σ′)} = −iδp(~σ −~σ′). (9)
P.B.
The addition of the field f(τ,~σ) restores the forth κ-symmetry and transforms all Ψ-
constraints to the first class. The Weyl symmetry constraint ∆ in the extended phase
W e
space is
∆ = (P˜˜αu˜ +P¯˜˜α˙u¯˜ )−2ρ˜τP˜˜(ρ) −2ρMP(ρ)e≈ 0, (10)
W u α u α˙ τ M
wherethevariables(u˜αe= uα−qα, P˜˜α = 1(Pα−Pα))and(ρ˜τ = ρτ−ϕτ, P˜˜(ρ) = 1(P(ρ)−P(ϕ)))
u 2 u q τ 2 τ τ
form canonically conjugate pairs [9]. Finally, the converted constraints L of the world-
M
volume ~σ−reparametrizations are
e
L = ∂ x Pα˙α +∂ z παβ +∂ z¯ π¯α˙β˙ +∂ θ πα +∂ θ¯ π¯α˙
M M αα˙ M αβ M α˙β˙ M α M α˙
(11)
e + ∂ u˜ P˜˜α +∂ u¯˜ P¯˜˜α˙ −ρ˜τ∂ P˜˜(ρ) −ρN∂ P(ρ) − if∂ f ≈ 0.
M α u M α˙ u M τ M N 2 M
The P.B. superalgebra of the converted first-class constraints (5)-(8), (10), (11) is described
by the following non zero relations
{Ψα(~σ),Ψβ(~σ′)} = −8iΦαβδp(~σ −~σ′), (12)
P.B.
e e e
3
{Ψα(~σ),Ψ¯β˙(~σ′)} = −4iΦβ˙αδp(~σ −~σ′), (13)
P.B.
{∆e (~σ),eP(ρ)(~σ′)} = 2Pe(ρ)δp(~σ −~σ′), (14)
W M P.B. M
{Le (~σ),P(ρ)(~σ′)} = ∂ P(ρ)δp(~σ −~σ′), (15)
M N P.B. M N
{LM(~σ),LeN(~σ′)}P.B. = (LM(~σ′)∂N′ −LN(~σ)∂M)δp(~σ −~σ′), (16)
e {eL (~σ),χ(~σ′)} e = −χ(~σ)∂eδp(~σ −~σ′), (17)
M P.B. M
where χ are Φ, Ψ and ∆ econstraints. The complex conjugate relations have to be added
W
to (12)-(17). The remaining P.B.’s of the constraints are equal to zero in the strong sense.
e e e
Havingthealgebra(12)-(17)onecanconstruct BRSTchargeofthetensionless superp-brane.
3 BRST charge and OSp(1|8) symmetry generators
The algebra(12)-(17) has the rank equal unity and may be presented in the generalized
canonical form
{ΥA(~σ),ΥB(~σ′)} = dpσ′′fAB (~σ,~σ′|~σ′′)ΥC(~σ′′), (18)
P.B. C
Z
where fAB are structure functions. Let us note that the algebra (18) generalizes the original
C
algebra [12] by the taking into account ∂ δp(~σ−~σ′) in the structure functions following from
M
the P.B.’s including the Virasoro constraints L (~σ) such as
M
fLMΦα˙βΦγ˙δ = −δγα˙˙δδβ∂Meδp(~σ −~σ′)δp(~σ −~σ′′),
e e (19)
fLMLNLQ = −δNQ∂Mδep(~σ −~σ′)δp(~σ −~σ′′)+δMQ∂N′δp(~σ −~σ′)δp(~σ′ −~σ′′)
e e
e
and other ones.
The canonically conjugate ghost pairs of the minimal sector corresponding to the first-
class constraints may be introduced forming the following triads
(Φαβ,C ,P˘αβ); (Φ¯α˙β˙,C¯ ,P¯˘α˙β˙); (Φα˙β,C ,P˘α˙β);
αβ α˙β˙ βα˙
¯ ¯
(Ψα,C ,P˘α); (Ψα˙,C¯ ,P˘α˙);
e αe α˙ e
(P˜α,C ,P˘α); (P¯˜α˙,C¯ ,P¯˘α˙); (20)
eu uα u eu uα˙ u
(P˜(ρ),C(ρ)τ,P˘(ρ)); (P(ρ),C(ρ)M,P˘(ρ));
τ τ M M
(L ,CM,P˘ ); (∆ ,C(W),P˘(W)).
M M W
Utilizing nonzero structuree functions of theesuperalgebra (12)-(17) one can present the
corresponding BRST generator Ω of the minimal sector [12]
Ω = dpσ(C ΥA + 1(−)bC C f˘AB P˘C)(~σ), (21)
Z A 2 B A C
by the following integral along the hypersurface of the closed super p-brane
¯ ¯ ¯
Ω = dpσ (C Φαβ+C¯ Φα˙β˙+C Ψα−C¯ Ψα˙+C Pα+C¯ Pα˙)
αβ α˙β˙ α α˙ uα u uα˙ u
h
R
+C eΦβ˙α+C(ρ)eτP(ρ)+Ce(ρ)MP(ρ)e+C(W)∆eext+CMLeext (22)
αβ˙ τ M W M
+4i(CeC P˘αβ +eC¯ C¯ P¯˘α˙β˙)−4iC C¯ P˘eβ˙α . e
α β α˙ β˙ α β˙
i
4
∆ext in eq.(22) is the generator of the gauge world-volume Weyl symmetry
W
e ∆ext = ∆ −2C(ρ)MP˘(ρ), (23)
W W M
and Lext is the generalized Virasoero geneerator
M
e Lext = L +∂ C P˘αβ +∂ C¯ P¯˘α˙β˙ +∂ C P˘β˙α
M M M αβ M α˙β˙ M αβ˙ (24)
¯
+ ∂ C P˘α +∂ C¯ P˘α˙ +∂ C(W)P˘(W) −C(ρ)N∂ P˘(ρ) +∂ CNP˘
e eM α M α˙ M M N M N
extended by the ghost contributions.
Using the P.B.’s of the superalgebra (12)-(17) one can show that the P.B. of the BRST
generator density Ω(τ,~σ), defined by the integrand (22), with itself is equal to the total
derivative
¯
{Ω(~σ),Ω(~σ′)} = ∂ (CM(C Φαβ +C¯ Φα˙β˙ +C Φβ˙α
P.B. M αβ α˙β˙ αβ˙
¯
+ C Ψα −C¯ Ψα˙ +C(ρ)NP(ρ) +C(W)∆ext +CNLext (25)
α α˙ e eN e W N
+ 4i(C C P˘αβ +C¯ C¯ P¯˘α˙β˙ −C C¯ P˘β˙α))δp(~σ −~σ′),
eα β e α˙ β˙ α β˙ e e
because of the presence of ∂ δp(~σ−~σ′) in the structure functions of the superalgebra (12)-
M
(17). But, the contribution of the total derivative in the r.h.s. of (25) vanishes after integra-
tion in ~σ and ~σ′ due to the periodical boundary conditions for the closed p-brane. It results
in the P.B.-anticommutativity of the BRST charge Ω ≡ dpσΩ(τ,~σ) (21) with itself
R
{Ω,Ω} = 0. (26)
P.B.
The introduction of the ghost variables leads to the extension of the OSp(1|8) symmetry
generators providing the P.B.-(anti)commutativity of the OSp(1|8) generators with Ω (21).
¯
The ghost extended ”square roots” S (τ,~σ) and S (τ,~σ) of the ghost extended conformal
γ γ˙
boost densities K (τ,~σ) and K (τ,~σ) are given by
γγ˙ γλ e e
S (τ,~σ) = (ze −2iθ θ )Qδ +e (x −2iθ θ¯ )Q¯δ˙ +4i(u˜δθ −u¯˜δ˙θ¯ )P˜˜ + 2 P˜˜ f
γ γδ γ δ γδ˙ γ δ˙ δ δ˙ uγ (ρ˜τ)1/2 uγ
e +C βP˘ +C P¯˘β˙ +4iC (θ P˘β −θ¯ P¯˘β˙) (27)
γ β γβ˙ γ β β˙
−8iθ C βP˘ δ +4iθ¯δ˙C βP˘ +4iθ C P˘β˙δ −8iθ¯δ˙C P¯˘β˙ ,
δ γ β γ βδ˙ δ γβ˙ γβ˙ δ˙
S¯ (τ,~σ) = (z¯ −2iθ¯ θ¯ )Q¯δ˙ +(x +2iθ θ¯ )Qδ −4i(u˜δθ −u¯˜δ˙θ¯ )P¯˜˜ − 2 P¯˜˜ f
γ˙ γ˙δ˙ γ˙ δ˙ δγ˙ δ γ˙ δ δ˙ uγ˙ (ρ˜τ)1/2 uγ˙
e −C P˘β −C¯β˙ P¯˘ −4iC¯ (θ P˘β −θ¯ P¯˘β˙) (28)
βγ˙ γ˙ β˙ γ˙ β β˙
−8iθδC P˘ β +4iθ¯ C P˘δ˙β +4iθδC¯β˙ P˘ −8iθ¯ C¯β˙ P¯˘δ˙ .
βγ˙ δ δ˙ βγ˙ γ˙ δβ˙ δ˙ γ˙ β˙
Using the densities Ω(τ,~σ) (22) and S (τ,~σ′) (27) we find their P.B.
γ
{Ω(σ),S (~σ′)}e = −(CMS )(~σ)∂ δp(σ −σ′) (29)
γ P.B. γ M
and conclude that the contribeution of the total dereivative in the r.h.s. of (29) vanishes after
integration with respect to ~σ and ~σ′. Thus, the BRST charge Ω (22) has zero P.B.’s with
¯
the conformal supercharges S ,S
γ γ˙
e e ¯
{Ω,S } = 0, {Ω,S } = 0. (30)
γ P.B. γ˙ P.B.
e e
5
The same P.B.-commutativity
{Ω,G} = 0 (31)
P.B.
between Ω and other OSp(1|8) symmetry charges G ≡ dpσG(τ,~σ) extended by the ghost
contributions will also be preserved, because of the genReral relation (17) for the generator
densities: {L (~σ),G(~σ′)} = −G(~σ)∂ δ(~σ −~σ′).
M P.B. M
The above mentioned expressions for the generator densities of the generalized conformal
e
transformations extended by the ghost contributions take the form2
K (τ,~σ) = 2z z πβδ +2x x π¯β˙δ˙ +z x Pδ˙β +x z Pβ˙δ
γλ γβ λδ γβ˙ λδ˙ γβ λδ˙ γβ˙ λδ
+θ (z πδ +e x π¯δ˙)+θ (z πδ +x π¯δ˙)+(u˜δz −u¯˜δ˙x )P˜˜ +(u˜δz −u¯˜δ˙x )P˜˜
λ γδ γδ˙ γ λδ λβ˙ δλ λδ˙ uγ δγ γδ˙ uλ
−2i(u˜δθ −u¯˜δ˙θ¯ )(θ P˜˜ +θ P˜˜ )+ 2 (θ P˜˜ +θ P˜˜ )f − 1 P˜˜ P˜˜
δ δ˙ λ uγ γ uλ (ρ˜τ)1/2 λ uγ γ uλ ρ˜τ uγ uλ
+θ C βP˘ +θ C βP˘ +θ C P¯˘β˙ +θ C P¯˘β˙
λ γ β γ λ β λ γβ˙ γ λβ˙
−(z β +2iθ θβ)C P˘ −(z β +2iθ θβ)C P˘ −(x +2iθ θ¯ )C P¯˘β˙ −(x +2iθ θ¯ )C P¯˘β˙
λ λ γ β γ γ λ β λβ˙ λ β˙ γ γβ˙ γ β˙ λ
−2(z β +2iθ θβ)C P˘ δ−2(z β +2iθ θβ)C P˘ δ
γ γ λδ β λ λ γδ β
+(z +2iθ θ )C P˘δ˙β+(z +2iθ θ )C P˘δ˙β
γβ γ β λδ˙ λβ λ β γδ˙
+(x +2iθ θ¯ )C P˘β˙δ+(x +2iθ θ¯ )C P˘β˙δ
γβ˙ γ β˙ λδ λβ˙ λ β˙ γδ
+2(x +2iθ θ¯ )C P¯˘β˙δ˙+(x +2iθ θ¯ )C P¯˘β˙δ˙,
γβ˙ γ β˙ λδ˙ λβ˙ λ β˙ γδ˙
(32)
for K (τ,~σ) and respectively for K (τ,~σ)
γλ γγ˙
e K (τ,~σ) = z z¯ Peδ˙δ +x x Pδ˙δ +2(z x πδλ +x z¯ π¯δ˙λ˙)
γγ˙ γδ γ˙δ˙ γδ˙ δγ˙ γδ λγ˙ γδ˙ γ˙λ˙
+θ (z¯ π¯δ˙ +ex πδ)+θ¯ (z πδ +x π¯δ˙)+(u˜δx −u¯˜δ˙z¯ )P˜˜ +(x u¯˜δ˙ −z u˜δ)P¯˜˜
γ γ˙δ˙ δγ˙ γ˙ γδ γδ˙ δγ˙ δ˙γ˙ uγ γδ˙ γδ uγ˙
−2i(u˜δθ −u¯˜δ˙θ¯ )(θ¯ P˜˜ −θ P¯˜˜ )+ 2 (θ¯ P˜˜ −θ P¯˜˜ )f + 1 P˜˜ P¯˜˜
δ δ˙ γ˙ uγ γ uγ˙ (ρ˜τ)1/2 γ˙ uγ γ uγ˙ ρ˜τ uγ uγ˙
+θ¯ C βP˘ −θ C¯ β˙P¯˘ +θ¯ C P¯˘β˙ −θ C P˘β
γ˙ γ β γ γ˙ β˙ γ˙ γβ˙ γ βγ˙
+(z¯ β˙ +2iθ¯ θ¯β˙)C P¯˘ +(z β +2iθ θβ)C¯ P˘ +(x −2iθ θ¯ )C P˘β +(x +2iθ θ¯ )C¯ P¯˘β˙
γ˙ γ˙ γ β˙ γ γ γ˙ β βγ˙ β γ˙ γ γβ˙ γ β˙ γ˙
−2(z β +2iθ θβ)C P˘ δ−2(z¯ β˙ +2iθ¯ θ¯β˙)C¯ P¯˘δ˙
γ γ δγ˙ β γ˙ γ˙ γδ˙ β˙
+(z β +2iθ θβ)C¯δ˙ P˘ +(z¯β˙ +2iθ¯ θ¯β˙)C δP¯˘
γ γ γ˙ βδ˙ γ˙ γ˙ γ δβ˙
+(x +2iθ θ¯ )C P˘β˙δ +(x −2iθ θ¯ )C P˘δ˙β
γβ˙ γ β˙ δγ˙ βγ˙ β γ˙ γδ˙
−2(x +2iθ θ¯ )C¯δ˙ P¯˘β˙ −2(x −2iθ θ¯ )C δP˘ β.
γβ˙ γ β˙ γ˙ δ˙ βγ˙ β γ˙ γ δ
(33)
2The discussed formulae would look more compact in the Majorana spinor representation. However, it
makes more obscure the contribution of the TCC coordinates z which presence is crucial for the exotic
αβ
supersymmetry protection and generation of the new bosonic gauge symmetries (see [13]) generalizing the
wellknownsymmetriesofthePenrosetwistorapproachoriginallyformulatedinD =4[24]. IntheMajorana
representationzαβ areencodedinthesymmetric4×4matrixYab ≡YadCdb =(cid:18) xz˜αα˙ββ zx¯αα˙β˙˙ (cid:19)togetherwith
β
the space-time coordinates xαα˙ (see [8], [17]).
6
The remaining 16 generator densities of OSp(1|8) supergroup extended by the ghosts are
the following
Lα (τ,~σ) = x Pβ˙α +2z παγ +θ πα +u˜αP˜˜ −2C γP˘ α +C P˘γ˙α +C P˘α,
Leαββ˙(τ,~σ) = 2xββγ˙β˙παγ +z¯β˙γγ˙βPγ˙α +θ¯ββ˙πα −u˜αP¯˜˜uuββ˙ +2Cγββ˙P˘γγα +C¯ββ˙γγ˙˙P˘γ˙α −C¯ββ˙P˘α. (34)
e
The adduced expressions should be complemented by their complex conjugate.
Note that supersymmetry and generalized translation generator densities do not contain
any ghost contribution. One can check that the P.B.-commutation relations of the OSp(1|8)
superalgebra extended by the ghost contributions coincide with the P.B.-commutation rela-
tions of the original OSp(1|8) superalgebra [17].
4 Quantization: nilpotent BRST operator and quan-
tum OSp(1|8) algebra
Upon transition to quantum theory all the quantities entering the converted constraints and
OSp(1|8)generatordensities aretreatedasoperatorsthatimplies achoiceofcertainordering
for products of noncommuting operators. At the same time the canonical Poisson brackets
{PM(~σ),QN(~σ′)}P.B. = δNMδp(~σ −~σ′) used in [17] transform into (anti)commutators
[PˆM(~σ),QˆN(~σ′)} = −iδNMδp(~σ −~σ′). (35)
It is necessary to provide further nilpotence of the BRST operator, fulfilment of
(anti)commutation relations of the OSp(1|8) superalgebra and its generator (anti)commu-
tativity with the BRST operator ensuring the global quantum invariance of the model. In
addition, the Hermiticity of the quantum BRST operator and OSp(1|8) generators has to
be supported. The Hermiticity requirement may be manifestly satisfied if we start from the
above constructed classical representations for the OSp(1|8) generators and BRST charge in
which all coordinates are disposed from the left of momenta, i.e. in the form QP, where Q
and P are the products of the coordinates and momenta contained in Ω and the generators.
Then the operator expressions for the latter are presented in the manifestly Hermitian form
composed of the operator products 1(QˆPˆ + (−)ǫ(Q)ǫ(P)(QˆPˆ)†), where ǫ(Q) and ǫ(P) are
2
Grassmannian gradings of these coordinate and momentum monomials.
In particular, we obtain the following Hermitian operator representations
ˆ
S (τ,~σ) = 1(zˆ −2iθˆ θˆ )Qˆδ + 1Qˆδ(zˆ −2iθˆ θˆ )
γ 2 γδ γ δ 2 γδ γ δ
e +1(xˆ −2iθˆ ˆθ¯ )Qˆ¯δ˙ + 1Qˆ¯δ˙(xˆ −2iθˆ ˆθ¯ )
2 γδ˙ γ δ˙ 2 γδ˙ γ δ˙
+2iθˆ (uˆ˜δPˆ˜˜ +Pˆ˜˜ uˆ˜δ)−4iuˆ¯˜δ˙ˆθ¯ Pˆ˜˜ + 2 fˆPˆ˜˜ (36)
δ uγ uγ δ˙ uγ (ˆρ˜τ)1/2 uγ
+Cˆ βPˆ˘ +Cˆ Pˆ¯˘β˙ +2iθˆ (Cˆ Pˆ˘δ +Pˆ˘δCˆ )+4iˆθ¯δ˙Cˆ Pˆ¯˘
γ β γβ˙ δ γ γ γ δ˙
−4iθˆ (Cˆ βPˆ˘ δ −Pˆ˘ δCˆ β)+4iˆθ¯δ˙Cˆ βPˆ˘ +2iθˆ (Cˆ Pˆ˘β˙δ −Pˆ˘β˙δCˆ )−8iˆθ¯δ˙Cˆ Pˆ¯˘β˙
δ γ β β γ γ βδ˙ δ γβ˙ γβ˙ γβ˙ δ˙
7
for the classical density S (τ,~σ) (27) and
γ
Lˆα (τe,~σ) = 1(xˆ Pˆβ˙α +Pˆβ˙αxˆ )+(zˆ πˆγα +πˆγαzˆ )
β 2 ββ˙ ββ˙ βγ βγ
e +1(θˆ πˆα −πˆαθˆ )+ 1(uˆ˜αPˆ˜˜ +Pˆ˜˜ uˆ˜α) (37)
2 β β 2 uβ uβ
ˆ ˆ ˆ ˆ ˆ ˆ
−(Cˆ γP˘ α −P˘ αCˆ γ)+ 1(Cˆ P˘γ˙α −P˘γ˙αCˆ )+ 1(Cˆ P˘α +P˘αCˆ ),
β γ γ β 2 βγ˙ βγ˙ 2 β β
for the generalized Lorentz density Lα (τ,~σ) (34).
β
ˆ
By the same way can be constructed quantum Hermitian generators Lext of the ~σ-
e M
ˆ
reparametrizations and the Weyl symmetry generator ∆ext
W e
∆ˆext = 1(uˆ˜ Pˆ˜˜α +Pˆ˜˜αuˆ˜ +uˆ¯˜ Pˆ¯˜˜eα˙ +Pˆ¯˜˜α˙uˆ¯˜ )
W 2 α u u α α˙ u u α˙ (38)
−ρˆ˜τPˆ˜˜(ρ) −Pˆ˜˜(ρe)ρˆ˜τ −ρˆMPˆ(ρ) −Pˆ(ρ)ρˆM −Cˆ(ρ)MPˆ˘(ρ) +Pˆ˘(ρ)Cˆ(ρ)M ≈ 0,
τ τ M M M M
entering the BRST operator.
Because other converted first-class constraints are Hermitian by construction, the quan-
tum Hermitian BRST generator
1
Ωˆ = dpσ(Ωˆ(τ,~σ)+Ωˆ†(τ,~σ)) (39)
2 Z
will coincide with its classical expression Ω (22) after the substitution of (38) and the Her-
ˆ
mitian representation for L (~σ) originated from (24) in eq. (22).
M
Now we are ready to prove that this realization of Ωˆ preserves its nilpotency and
e
(anti)commutativity with the Hermitian operators (36), (37) and other ones generating a
quantum realization of the classical OSp(1|8) superalgebra. The proof is obvious and is
based on the observation that Ωˆ and other considered Hermitian operators are linear in the
momentum operators PˆM(τ,~σ) of the original coordinates and ghost fields. The remarkable
property of the ordered polynomial operators composed of QˆM(τ,~σ) and PˆM(τ,~σ), which
form the Weyl-Heisenberg algebra (35), and are linear in PˆM is the preservation of the cho-
sen ordering in course of calculations of their (anti)commutators. As a result, the transition
from the P.B.’s to (anti)commutators will preserve all classical results obtained in the P.B.
realization of the extended algebra of the OSp(1|8) generators and classical BRST charge of
the super p-brane. So, the quantum Hermitian BRST operator (39) occurs to be nilpotent
{Ωˆ,Ωˆ} = 0. (40)
However, the Hermiticity of Ωˆ and the OSp(1|8) generating operators by itself is only
a necessary condition for the quantum realization of the physical operators, because the
relevant vacuumandphysical stateshave alsotobeconstructed. So,theproblemofexistence
of the selfconsistent quantum realization of the exotic BPS states by the states of quantum
tensionless super p-brane is reduced to the proof of existence of the relevant vacuum and
the corresponding physical space of quantum states. At the present time we investigate this
problem. However, we should like to discuss here a possible way to solve this problem based
on the consideration of the QˆPˆ-ordering studied in [10].
8
ˆ ˆ
5 QP-ordering and regularization
It was motivated in [10] that the coordinate and momenta operators have to be used for the
tensionless string quantization instead of the creation and annihilation operators relevant
for the tensile string quantization. This motivation is physically justified by the absence of
oscillator excitations for the tensionless string which makes its dynamics resembling that of
collection of free particles and results in the choice of the physical vacuum as a state anni-
hilated by the string momentum operator. In that case the coordinate Qˆ and momentum
Pˆ monomials forming the above discussed Hermitian operators have to be ordered by the
shifts of all the Qˆ monomials to the left of Pˆ. To achieve that QˆPˆ-ordering we have, in
particular, to permutate some noncommuting coordinate and momentum operators in the
QˆPˆ-disordered Hermitian expressions of the generator densities (36), (37) and others, con-
ˆ ˆ
straints ∆ext (38), Lext and the Hermitian BRST operator. In view of that permutations
W M
divergent terms will appear in some monomials composed from canonically conjugate oper-
e e
ators at coinciding points of p-brane. A typical form of such divergent terms is illustrated
by the relation
1 3i
(zˆ (~σ)πˆδε(~σ)+πˆδε(~σ)zˆ (~σ)) = zˆ (~σ)πˆδε(~σ)− δεδp(0) (41)
2 λδ λδ λδ 4 λ
encoding the permutation effect of the TCC coordinates with their momenta. The r.h.s. of
ambiguously defined relation (41) includes the divergent term δp(0) = δp(~σ−~σ′)|~σ=~σ′ and the
ˆ ˆ
problem appears how to deal with the QP-ordered representation of the symmetric operator
in the l.h.s. of (41). Such type a problem is typical in quantum field theory due to its
inherit divergencies and the regularization procedure should be applied. Thus, in general,
the ordering problem has to be analyzed together with the divergency problem. It might
have happened that a regularization prescribes to use the image of the delta function at the
zero point (here δp(0)) to beequal to zero. Then thechoice of that regularization would solve
the ordering problem. Taking into account such a possibility requires study of the structure
of the divergent terms appearing in the total quantum algebra of our model. To this end we
firstly analyze the QˆPˆ-ordered realization of the OSp(1|8) algebra.
We find that the application of the described QˆPˆ-ordering procedure to the Hermitian
OSp(1|8) generator densities (36), (37) and others yields the relations
Sˆ (τ,~σ) = Sˆ −2θˆ δp(0), Sˆ¯ (τ,~σ) = Sˆ¯ −2ˆθ¯ δp(0),
γ γ γ γ˙ γ˙ γ˙
ˆ ˆ ˆ¯ ˆ¯
Kγλe(τ,~σ) = Kγeλ +izˆγλδp(0), Keγ˙λ˙(τ,~σ) =eKγ˙λ˙ +iˆz¯γ˙λ˙δp(0), (42)
ˆ ˆ
e e Kγγ˙(τ,~σ) = Kγγe˙ +ixˆγγ˙δp(0)e,
ˆ ˆ ˆ¯ ˆ¯
Lαβ(τ,~σ) = Lαβe+ 2iεαβδp(0)e, Lα˙β˙(τ,~σ) = Lα˙β˙ + 2iε¯α˙β˙δp(0)
e e e e
connecting the Hermitian and QˆPˆ-ordered representations for the generator densities. The
ˆ ˆ¯ ˆ ˆ ˆ ˆ ˆ¯ ˆ
operators S , S , K , K , K , L , L , collectively denoted by G , in the r.h.s. of
γ γ˙ γλ γ˙λ˙ γγ˙ αβ α˙β˙ qp
ˆ ˆ
(42) coincide with the classical QP-ordered representations (27), (28), (32)-(34), where the
e e e e e e e e
corresponding operators are substituted for the classical coordinates and momenta. These
ˆ
QˆPˆ-ordered operators G form another representation of the quantum algebra OSp(1|8)
qp
similarly to the Hermitian operators, because they originate from the classical generators
e
9
and preserve the QˆPˆ-ordering in the course of the calculation of their (anti)commutators.
ˆ
But, the QˆPˆ-ordered generators G are nonHermitian and their (anti)commutation with
qp
the divergent nonHermitian terms in the r.h.s. of the representation (42) contributes to
e
the closure of the OSp(1|8) algebra presented by the Hermitian generators in the l.h.s. of
(42). So, we obtain two quantum realizations of the OSp(1|8) algebra and one of them
ˆ
G is nonHermitian, because of its QˆPˆ-ordering. But, namely, action of the nonHermitian
qp
ˆ
generators G on the corresponding vacuum state is well defined in accordance with [10].
e qp
Thentheregularizationassumption, whichprescribestoaccepttheregularizedimageofδp(0)
e
ˆ
as equal zero, removes the nonHermiticity of the QˆPˆ-ordered generators G . In the result
qp
we obtain the desired vacuum state for the discussed Hermitian realization of the physical
e
operators. So, the choice of such a regularization would allow to overcome the problem of
construction of the quantum space of physical states. To realize such a scenario one needs
to analyze the QˆPˆ-ordered realization of Hermitian BRST generator (39).
ˆ ˆ ˆ
The correspondent QP-ordered BRST operator Ω turns out to be nonHermitian, as
follows from the relation connecting Ωˆ with the Hermitian BRST operator Ωˆ (39)
Ωˆ = Ωˆ −i dpσ[Cˆ(W) +(p − 7)CˆM∂ + 1∂ CˆM]δp(0), (43)
Z 2 4 M 2 M
and is supplemented by three antiHermitian divergent additions in the r.h.s. compensat-
ing the nonHermiticity of Ωˆ. The first of them, proportional to δp(0), follows from the
ˆ
QˆPˆ-ordering of the Weyl symmetry generator ∆ext (38) and is contributed only by the aux-
W
iliary variables ρˆ˜τ,uˆ˜α and uˆ¯˜α˙ partially cancelling each other during the ordering with their
e
momenta. The contribution of other auxiliary pair (ρˆM,Pˆ(ρ)) in here is cancelled by the
M
ghost pair (Cˆ(ρ)M,Pˆ˘(ρ)) contribution. We observe that only the auxiliary twistor-like fields
M
and the component ρˆ˜τ of the world-volume density ρˆµ, introduced to provide the Weyl and
reparametrization gauge symmetries of the brane action, contributed to the first singular
term. The similar story concerns the second divergent term proportional to ∂ δp(0) 3. This
M
ˆ
term appears from the QˆPˆ-ordering of the extended Virasoro operators Lext and only the
M
ˆ ˆ
above mentioned auxiliary fields together with the ghost pairs (CˆM,P˘ ), (Cˆ(W),P˘(W)) and
M e
the auxiliary fermionic field fˆ contribute to here, because the contributions of the propa-
gating phase-space variables are cancelled by the corresponding ghosts. This cancellation
illustrates the boson-fermion cancellation mechanism provided by the BRST symmetry. The
third singular addition, proportional to the total derivative, restores Hermiticity of the cubic
ˆ
term CˆM∂ CˆNP˘ but, it vanishes in view of the periodical boundary conditions. The latter
M N
could contribute in the case on a nontrivial topology of the ghost-field space.
ˆ ˆ ˆ
Now let us note that the QP-ordered operator Ω in the r.h.s. of eq.(43) is nilpotent
{Ωˆ,Ωˆ} = 0, (44)
3In the symmetric regularization, where δp(−~σ) = δp(~σ), the derivative ∂ δp(~σ) vanishes at ~σ = 0. As
ǫ ǫ M ǫ
a result the symmetric regularization does not capture the divergencies following from the reordering of
terms with derivatives like πˆδε(~σ)∂ zˆ (~σ) (cf. (41)). To capture such a type singularity one can use more
M λδ
general regularization of delta function considered by Ho¨rmander [25]. As a result, we find the r.h.s. in
the regularized(anti)commutators [PˆM(~σ),∂MQˆN(~σ)}=iδNM∂Mδp(~σ,~σ), where ∂Mδp(~σ,~σ) is a regularized
analogue of ∂ δp(0), to be non zero.
M
10