where Jab is another auxiliar)' melric and f = det(fflb)'. Of course, il is imporlanl lo rcmark that lhe lhrce actions S~I),sf), and S~3) are classically equivalenl [.1].
Revista Mexicana de Física 36 Suplemento
1 (1990) 5204-5207
Weyl invariant null p-branes J.A. Nieto Departamento de Física, Universidad Michoacana, Apartado postal 7,.9, 58000 Mordia, Michoacán. México. Abstract. A WeyI invariant action for null p-branes is derivcd. The central idea of this work is to emphasize the fact that it can be associated a WeyI invariant action to any p-brane; including, now. null p-branes.
There.are two main reasons to be interested in \Veyl invariant null p-branes. First the Weyl invariance plays a central role in string lheories [1]. Second, null p-brane theory [2] is an approach to get a better understanding of sorne subtleties of string theory. Just as a null partic1e is a physical system with zero ma~s a null string [3J is a one dimensional extended objet with zero tension; null p-brancs [2] are straightforward generalizations of null particles and null strings. Consider a p-dimensional extended objed moving in a D-dimensional Minkowski space-time. The motion of such a system may be described by IIsing Dcoordinales Xp. = xp.(~a) where Ji. = 0,1,2, ... D -1 and ~a, with a = O,1,2, ... , p are arbitrary parameters. The pararneler ~o = T is the evolution parameter while the parameters ~j, with i = 1, ... ,p are space.like pararneters used to label points along the system. There are three equivalent alternative actions to determine the equations of motion of a p-brane [4]. The first one is a straightforward generalization of the Nambu-like action for strings [5), 1
(1 ) where h = det(ha!), with
and Op is a constant measuring the inertia of the p-brane. Bere ~"v=diag(-l,l, is the Minkowski metric.
... ,l)
I\'cyi
inl'fll'io1J1
nI/U
p-brfITlC$
5205
The st'Cond alternati,.c action, which is a straigbtfor\\'iHd gr-lleralization of the Brink el al, ¡6} action for strings, is
where 9ab is an auxiliary metric and 9 := det(gab). Finally, tlle third altt'rnative action discovcred hy Nielo [7] is tIte following:
(5C(,
also Hefs. [8])
(3 ) where
Jab
is another auxiliar)' melric and
f = det(fflb)'
Of course, il is imporlanl lo rcmark that lhe lhrce actions S~I),sf), and S~3) are classically equivalenl [.1]. Thanks to lhe mctric 9ab one may be able lo ronsidcr the invariallce of lhe aclion S~2l under \Veyl transformations . (4 ) where /\(0 is an arbitrary function. It tums out lltat lhe action sf) is \Ve)'1 invariallt ollly foe the steillg (1' = 1) (sce \(cf. (4]). It secms that from tItis resu)t il ariscs the general belicf lhat lhe only extended system lo whieh one call assoeialc lhe \Veyl illvariant aetioll is the strillg. (By lhe way, lhis resull provides the usual argument why one shoul'CiHlS(~ it allows -+
O. In this ¡¡mit
OIle
gels thc actions (11 )
corrcsponding lo null
p-branL'S.
Subsliluling L~l), Lf), alld L~3), given in (7), (8), and (9), in (11) lead, lo lhe following thrce alternative actions [oc null p-branes: (12)
sf)
= -~
S~3) = -~(p
J
dP+l~ e-Ig[gabhab - (1'-1)]',
+ 1)-(p+l)
J
d!'+l~
1 (1"bh"b],,+1
( 13)
(14 )
These three actions are, oC course, classically equivalcnt. In recent years, in arder lo study different asrects of nul! p-brancs sorne au. thors [2] have concentrated their attention in the aetion (12). Here, however, I would like to cal! the attention about the action (14), which is in fad \VeyI invariant. In particular the aetion (14) may be used to calculate the critica) dimensions of nuH p-branes. In conc1usion, I derived the Weyl invariant action (l.1) for nul! p--branes. Such a.u aetion may be useful to c1arify sorne aspeets of quantum null p-branL"S, in particular the so caBed critical dimensiono
IVeyi inl'Orionf
mlli¡,
-bmno
5207
References 1.
2. 3. 4. 5. 6. 7.
8.
M.n. Grcen, J .11. Schwa.rz, and E. \vitten, SIlIJ€rstrillg Thmry. Ca.mbridge U. P., Cambridge (1987); J.B. Schwarz, Superslrings [, /1. World SciClltific, Sillgapore (1985). J. Gamboa, C.Ra.mírez, ancl M. Ruíz-Altaba, "Fidd Theory of Null Strings and pbrane,". !'reprin! CEHN.TIl. 545.\/89 (July 1989). A. Schild, Phys. Rev. D 16 (1977) 1722. 1. Cenclejas ancl J.A. Nieto, "'Comments OH Super p-branes". Prcprint, Universidad Michoacana (October 1989). Y. Nambu, Lecturcs at the Copellhagen SymposiuOl (1970). L. Ilrink, P. di Vccchia and P. Howe, Phys. Lett. 65B (1976) .171; S. Deser and n. Zurnino, l'hys. Le/t. 65B (1976) 369. J.A. Nieto "'Classical Test Particles and (4 + N)-Dim{,lIsional Theories of Spacctime", Ph. D. Thesis, University of Texas at Austin (1986); .I.A. Nielo, "" rclativistic 3dimensional extended object: the terrón", Rev. Mir. Fis. 34 (1989) 597; .LA. Nieto, C. Nuñez, and L.C. Shepley, "Dynamical CosOlological Constant from a Four-Index Antisymetric Gauge Ficld", preprint 1986 Austin Texas. U. Linclstrom and G. T1Ieodoridis, "Weyl.Invariant Super p-bralle" Phys. [,eU. B 203 (1988) 407.
Resumen. Se deriva una acción invariante dI' \Veyl para T)-membranas nulas. La idea central de este trabajo es ellf(ltizar el lH'c1l0 d(' q\le se puede asociar una acción invariante de \Veyl a cualquier p-membrana, incluyendo, ahora, p-membranas nulas.
