CC BY
29
UDC 530.1; 539.1
BRST approach to gauge invariant higher spin theory of conformal field in flat
space
H. Takata
Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia.
E-mail: [email protected]
We propose a closed higher-spin algebra and its representation that reproduces conformal invariant Lagrangian presented by Fradkin and Tseytlin. We use this algebra for constructing gauge invariant Lagrangian by BRST method. Lagrangian constructed by BRST method does not have any off-shell constraints or higher derivative terms as in the non-conformal case. As an example for spin 2 case in four space-time dimension, our Lagrangian agrees with that of conformal gravity by using gauge fixing and equations of motions of auxiliary fields.
Keywords: higher-spin, BRST, conformal.
1 Introduction
2 Higher spin algebra
Higher-spin field theory is expected to include particles with spin higher than two [1]. In particular, the theory we consider here is a spin independent formalism of particles. It is defined in the same form for spin one vector gauge field, spin two gravitational field and higher spin fields those were not observed. To construct Lagrangian of higher-spin field, we here use, so called, BRST method. Free Lagrangian construction is well developed by using this method in the case that the Poincare symmetry is space-time symmetry, for bosonic case, fermionic case and mixed symmetric case [21 [3].
A spin s conformai higher spin theory in a free Lagrangian form was proposed by Fradkin and Tseytlin
[4] in four space-time dimension1
Lagrangian(l) and [4] [5] suggest a set of constraints those the basic field must satisfy. They are d’Alembertian constraint, divergence free constraint and traceless constraint:
i D-4 ,
□ s+ — ^
Ml'"Ms
Ml --'Ms
MlM2---Ms
(2)
Ml "'Ms
□ s pMl'"Ms J.V J Vi •• Vs V'
(1)
where P is a projector for traceless and divergence free field. The structure of P considered there was very implicit.
Our motivation is to reproduce this Lagrangian with explicit form. For that purpose BRST method may be a good choice. In this paper we develop BRST method œ
for constructing gauge invariant Lagrangian with con- |$) = formal symmetry.
In section 2 we start by a set of constraints suggested from (1) and find a closed Higher-spin algebra. In section 3, BRST operator and Lagrangian will be constructed. For example in section 4, the Lagrangian agrees with that of linearized version of conformal gravity in spin two and in four space-time dimension.
Last two constraints are the same to the constraint for higher-spin field for Poincare symmetry. Only difference from Poincare case is in the first one.
In BRST method we rewrite these constraints on a Fock space by introducing several set of création-annihilation operators. In order to find a closed algebra(higher-spin algebra) corresponding to (2), we need to look for appropriate representation that defines algebra. Difference of the first equation of (2) from Poincare case leads different higher-spin algebra and its representation.
As a result, we found them in the following. First define Fock space:
k+ J k-bt kl bt k2 _t .. t |Q\$Ml 'Mfc0
c+ c- b1 b2 aMl aMk0 \Q/^k-,k+,ko,klk2,
{k} = 0
(3)
where
[aM,atv] = —nMV, \bi, bj] = [c_,c+] = [c+,cL] =
1, aM|Q) = bI|0) = c±|0) = Q, (I = 1, 2).
1It is also given in arbitrary dimension by [5].
Then define representation of generators:
Lo
Li
L+
L2
L+
Go
Lc
2
—p2 + c_c_ ap + c_ bi atp + c_b1
a2 , t Í b
c_a2 + c_ ( — ^ + (b2b2 — s — )b2
_ Ÿ + c_ (—f + ^
-ata + bi bi + 2b2b2 — .
t h.. _l_ Oh^ _l_ y>t r> _l_ y->t
— bibi + 2b2b2 + c+c_ + c_ c+ — s + 4 — ~2
These define the following higher-spin algebra.
—c L1
[[L0, ] ] =Q
[L i,L+] = L0
[G0, L i ] = —L1
[G0, L+ ] = L+
[G0 ,L2] = —2L
[G0,L+ ] = 2L+
[L2,L+ ] [L+,Li ]
L
L+]
2, L2
[Lc,L2] [Lc ,L+] [Lc,Li] [Lc, Go]
c_L+ c_c _G0
—L2
L+
Constraints in Fock space are then written as: Lo|$> = Li|$) = La|$> = Gq|$) = Lc|$) = 0.
+2ncn+ P2 + 2n2ncP2+ + ncn+ P2 + n2^cP2+
in the extended Fock space that is defined by multiplying anti-commuting ghost operators ^’s, P’s on |$o...o) = 1$) of (3):
Q|x0) =0 , ¿|x0) = Q|xW) = Q?lx2), ■
(8)
Ix1 ) =
E
nk3 ntk4 Ptk5 ntk6 Ptk7 n0 ni Pi n2 P 2
|$
k3---k7 )
k3 + k4 + k6 -k5 - k7
(9)
(4)
where ghost number restriction is expressed in above summation. To find Lagrangian we need a treatment of Hermitian ghosts as in the bosonic Poincare case. Our case has an additional pair of Hermitian ghosts. We treat them as we did for Poincare case. Then BRST equation (8) is divided to three parts:
Q|xI ) = |xI ) = ^c|xI ) = Q,
(10)
(5)
(6)
where <rs, ctc and Q are defined by using number operators of corresponding création-annihilation operators:
Q = Q + + nc^c - C-c-nln2PG,Q2|x*> = 0
CTS + s = na + n6l + 2nb2 + nm + np + 2n^2 + 2np2. ^c + s + = n61 + 2nb2 + nc+ + nc_ + n^2 + np2
(H)
Constraint by G0 and Lc are explained as fixing spin and conformai weight of state respectively.
This algebra (5) is closed and found to be a natural modification of massive Poincare higher-spin algebra. Structure constant depends on oscillators c^d c-. It is, however, not harmful because the right hand sides of commutators in (5) do not give new constraints. Algebra itself does not depend on spin or conformai weight. Representation, however, depends on them through two arbitrary parameters s mid D and one can choose them freely.
3 BRST construction of Lagrangian
BRST nilpotent operator is defined in the usual manner [2] [3]:
Q= noLo + n+Li + niL+ + n+ L2 + n2L+
+ncGo + ncLc - nin+ Po - c-c-n2n+ Pg
+ (ncn+ + c-n+ ni)Pi + (nine + c-n+n2)P+
and ac fix spin and space-time dimension. Lagrangian and gauge transformation for fixed spin and space-time dimension are then written, by introducing some operator K for Hermiticity [2], as
L = / dn0(x0|KQ|X0)
¿|x0) = Q|x i ),% i ) = Q|x2).
(12)
(13)
(7)
where {ni, P;'} = {nj,Pj} = 1, ni|0) = Pi|0> = Pj|0> =0, (I =1, 2; j = 0, G, c).
BRST equation and gauge transformation are written
These have the same form to Poincare case. All differences are included in the structure of BRST operator and states.
4 Example: spin two in four space-time dimension
In this section we write down Lagrangian (12) for spin two in four space-time dimension and show it reproduces Lagrangian of linearized conformai gravity in flat space-time.
The second and the third equation of (10) as well as
s=2
D = 4, solved for states as
|x0)
|x )
|x2)
|S 0)2, i + n0|A0)2, i |s i )2, i + n0¡A i )2, i Q
I
2
where lower indexes of states represent eigenvalues of right hand sides of last two equations of (11). We expand these states by ghost’s dependencies as:
|S0>2, i = |Si >2, i + n ÎPlt|S2>0, i
|Aq>2, i = Pt|Ai>i, i + P2t|A2>0,0 . (!5)
|S i >2, i = Plt|Sii> i, i + P2t|S2i >0,0
One can eliminate |A i> i , i by its equation of Motion:
|A i>i , i = L i |Si>2, i - L+ |S2>o, i . (16)
Then Lagrangian and gauge transformations are
L = (Si|K(Lq - L+L i)|Si> - 2(S2|KLq|S2>
+ ((Si|KL+2|S2> - (A2|K (L2|Si> + c-|S2>) + c.c.)
(17)
5|Si > = L+ |S i > + L+|S2 >
¿|S2> = L i |S i> - c+|S2> . (18)
¿|A2> = Lo|Si >
We further expand them by c^d bj:
|Si >2, i = c-|S->2,0 + c+|S+>2,0 + b i |SQ > i,o
|S2>0, i = c-|S2^>0,0 + c+|S2+>0,0 .
|S i > i , i = c-|Si,-> i ,0 + c+|S i ,+ > i ,0 + b i |Si,0>o,o
(19)
Next we use equation of Motion of |A2>0,0, then using degrees of freedom of|Si,+>i ,^d |Si,0>0,0 we eliminate |SQ>i ,^d |S+>0,0 respectively. By using equa-(S +|
L = (S-| (lo - l+l i - 1+212)
x (1 - 3/+/2) (lo - l+l i - Z+/2) |S-> ,
¿|S-> = 1+|S i ,-> + 1+|Si >,
(20)
where lower case Z’s are the generators of massless Poincare one. We can write usual fields instead of states and operators:
|S-> = ^ a+Ma+v |0>hMv (x)
|Si,-> = -*a+M|0>AM(x), |Si> = |0>J(x) .
Then Lagrangian is written by familiar form:
2L = - 2(hd)Md2(dhf
2 2 1
+ 3h*d2(ddh) + -(hdd)(ddh) - 3h^d4^. (22)
This agrees with the Lagrangian of linearised conformai gravity in flat space-time and it can be written as (1) with the projector.
5 Conclusion
We proposed a closed higher-spin algebra and its representation that reproduces the conformai invariant higher-spin Lagrangian presented by Fradkin and Tseytlin [4].
We started a set of constraints (2) suggested by (1) and found a closed higher-spin algebra (5).
Some properties of the algebra (5) are the following. It looks like a natural modification of massive Poincare higher-spin algebra. It does not depend on spin or conformai weight. It is defined for arbitrary spin in any space-time dimension. It may correspond to a sub algebra of full conformai algebra because, compared to Poincare case, we have add only one additional con-
Lc
did not consider one for special conformai transformation, full conformai Higher-spin algebra will possibly be modified from (5). For spin one or two case, however, differences will not be important. In fact, for spin two, our higher-spin algebra is enough to reproduce known Lagrangian of conformai gravity.
We have used this algebra for constructing free gauge invariant Lagrangian by BRST method. The Lagrangian (12) constructed by BRST method does not have any ofl-shell constraints or higher derivative terms as in the all Poincare cases. The structure of states in our Lagrangian is similar to that of Metsaev [6]. The number of terms in our states, however, are initially not limited by definition (3). It is determined by BRST equation of motion from Lagrangian.
In order to check the validity of higher-spin algebra we proposed, we studied an example of spin two in four space-time dimension. We wrote down generally defined Lagrangian for this case. By using gauge fixing and equations of motions of auxiliary fields, it was reproduced that the known form of Lagrangian of linearized conformai gravity (22) or of Fradkin and Tseytlin’s. It is also possible to study spin one case in arbitrary space-time dimension and the known form can be given too. We expect that the algebra (5) also reproduces Lagrangian (1) and that of [5] for general integer spin in arbitrary space-time dimension.
One of interesting future work is to clarify the relation between the full conformai algebra and higher-spin algebra.
References
[1] Singh L. P. S., Hagen C. R., Phys. Rev. D9 (1974) 898; Fronsdal C., Phys. Rev. D18 (1978) 3624.
[2] Buchbinder I. L., Krykhtin V. A. and Pashnev A., Nucí. Phys. B 711 (2005) 367; Buchbinder I. L. and Krykhtin V. A.,
Nucí. Phys. B 727 (2005) 537; Buchbinder I. L., Krykhtin V. A., Ryskina L. L. and Takata H., Phys. Lett. B 641 (2006)
386.
[3] Buchbinder I. L., Krykhtin V. A. and Takata H., Phys. Lett. B 656 (2007) 253; Moshin P. Y. and Reshetnyak A. A., JHEP 0710 (2007) 040.
[4] Fradkin E. S. and Tseytlin A. A., Phys. Rept.119 (1985) 233.
[5] Segal A. Y., Nucí. Phys. B664 (2003) 59.
[6] Metsaev R., JHEP 1201 (2012) 064; JHEP 1206 (2012) 062.
Received 01.10.2012
X. Таката
БРСТ ПОДХОД К КАЛИБРОВОЧНО-ИНВАРИАНТНЫМ ТЕОРИЯМ КОНФОРМНЫХ ПОЛЕЙ ВЫСШИХ СПИНОВ В ПЛОСКОМ ПРОСТРАНСТВЕ
Предлагается замкнутая алгебра высших спинов и ее представление, которое воспроизводит конформно инвариантный лагранжиан, полученный Фрадкиным и Цейтлиным. Эта алгебра используется для построения калибровочно инвариантного лагранжиана с помощью БРСТ подхода. Лагранжиан, построенный с помощью метода БРСТ-конструкции не имеет ни связей вне массовой оболочки ни членов с высшими производным по сравнению с неконформным случаем. На примере спина 2 в четырех измерениях наш лагранжиан согласуется с тем, что следует из конформной гравитации с использованием калибровки и уравнений движений на вспомогательные поля.
Ключевые слова: высшие спины, БРСТ, конформная симметрия.
Таката, X.
Томский государственный педагогический университет.
Томск, 634061, Россия.
E-mail: [email protected]
