>K9m 2014, TOM 145, uun, 5, CTP. 825 829
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ALGEBRAIC FORM OF THE M3-BRANE ACTION
H. Ghadjari"*, Z. Rezaeih**
" Department of Physics, Amirkabir University of Technology (Tehran Polytechnic) 15875-4413, Tehran, Iran
b Department of Physics, University of Tafresh 39518-79611, Tehran, Iran
Received December 20, 2013
We reformulate the bosonic action of an unstable M3-brane to manifest its algebraic representation. It is seen that in contrast to string and M2-brane actions, which are respectively represented only in terms of two- and three-dimensional Lie algebras, the algebraic form of the M3-brane action is a combination of four-, three-, and two-dimensional Lie algebras. Corresponding brackets appear as mixtures of the tachyon field, space-time coordinates .Y, the two-form field and the Born-lnfeld one-form bfl..
DOI: 10.7868/S004445101405005X
1. INTRODUCTION
Algebraic reformulation of known actions in string theory and M-thcory shows that string theory is based on the conventional algebra, or a two-dimensional Lie algebra (known as a two-algebra), but a complete description of M-thcory reguires an extended Lie algebra called a three-algebra fl], which was mainly developed in [2 5]. The numbers two and three are respectively-associated with string theory and M-thcory. Two is the string worldshcct dimension and also the codimcnsion of D-brancs in both typc-IIA and typc-IIB supcrstring theories [6]. Three is the membrane worldvolume dimension in M-thcory and the codimcnsion of M2- and M5-brancs. This means that via two-algebra interactions, some D/>brancs can condense to a D(p + 2)-branc [7] and through three-algebra interactions multiple M2-brancs condense to a M5-branc [8 16]. These connections between two and three and, respectively, string theory and M-thcory become obvious by rewriting Nanibu Goto actions in algebraic form.
By analogy, we can expect to describe />brancs applying a (p + l)-algcbra structure [17]. These extended algebras are used to construct worldvolume theories for multiple />brancs in terms of Nanibu brackets that are classical approximations to multiple commutators of
E-mail: h-ghajari'fi'aut .ac .ir E-mail: z.rezaei'&aut.ac.ir
these algebras [18]. Nanibu '/¿-brackets introduce a way to understand the /¿-dimensional Lie algebra presented in [19]. Formulation of the />branc action in terms of a (p + 1 )-algcbra makes it more compact and we are left with algebraic calculations, which are then usually-simpler to handle.
In string theory, we are inevitably- faced with unstable systems, and studying them deepens our understanding of the string theory. In bosonic string theory, the instability- is always present due to the tachyon presence in the open string spectrum. Two examples of unstable states in supcrstring theories are 11011BPS brancs (odd (even) dimensional brancs in typc-IIA (IIB) theory) and branc anti-branc pairs in both typc-IIA and typc-IIB theories [20, 21]. A11 interesting fact about the dynamics of these unstable brancs, generally- obvious in the effective action formulation, is their dimensional reduction through tachyon condensation [22 27]. During this process, the negative energy-density of the tachyon potential at its minimum point cancels the tension of the D-branc (or D-brancs) [28], and the final product is a closed-string vacuum without a D-branc or stable lower-dimensional D-brancs. O11 the other hand, stable objects in string theory- can be obtained by dimensional reduction of stable brancs in M-thcory- (M2- and M5-brancs). Naturally, we can expect to have a prcimagc of unstable brancs in supcrstring theories by formulating an effective action for unstable brancs in M-thcory. Among different unstable systems in M-thcory- [29], the M3-branc is noteworthy
because it is directly related to the M2-brane. Tacliyon condensation of the M3-brane effective action results in the M2-brane action, and its dimensional reduction also leads to a 11011-BPS D3-brane action in type-IIA string theory [30].
Despite attempts made to formulate the M3-brane action consistent with desired conditions [30], there has been 110 algebraic approach towards this formulation. The existence of the algebraic form for the action of the M2-brane, as the fundamental object of M-theory, motivated us to search for the algebraic presentation of the M3-brane as the main unstable object in M-theory, whose instability is due to the presence of the tacliyon.
What distinguishes the present study from conventional algebraic formulations is the instability of the M3-brane. I11 other words, the presence of the tacliyon and other background fields affect the resultant algebra. It is shown that a pure four-algebra does not occur, as expected, and we are encountered with four-, three-, and two-brackets that are mixtures of the tacliyon, space-time coordinates, and other fields.
DtlX
M
= dtlx
M
A,,k
M
—dtlx
M
— (2) K/J2/J3/J4 — °l-t2U> /IslH
km,
■
(2.2)
1
3!'
— CkmnDu2Xk Dß XM£>„ Xa
¡I-, (¿^3^4
The tensor Htll, consists of the pullback of the background metric, the field strength Ftll, of the gauge field Aß, and the tacliyon field T; M and N represent space-time indices and Dtl is the covariant derivative. The field strength itself is expressed in terms of the Born Infold 1-forni bfi and the R R sector field C. The curvature of the 2-forni is denoted by k.
The determinant of the tion can be decomposed as
The determinant of the tensor Hfll, in the DBI ac-
\J— dot H,u, = y — dot (G
' jtr
F,
in't
(2.3)
whore
2. ALGEBRAIC M3-BRANE ACTION
F ¡xi/ — dßbi,
dj),,,
G ¡lu — L
M NUß
d„xMd,,x
-N
-d¿rdvT,
(2.4)
The conventional action corresponding to a 11011BPS M3-brane is a combination of the DBI (Dirac Born Infold) and WZ (Wess Zumino) parts [30]
S — S dbi + Swz,
1/2
Sdbi = - / éS,V(T)\k\ WdcTW,
(2.1)
Swz —
whore with //. = 0,1, 2,3 labels worldvolume coordinates of the M3-brane, V(T) is the tacliyon potential, which is an even function of T and is characterized as V(T = ±00) = 0 and V(T = 0) = 7ms, 7m3 is the M3-brano tension, and kM (X) is the Killing vector, such that the Lie derivatives of all target-space fields vanish with respect to it [30]. Other fields in (2.1) are defined
Hßl, = um ^ Í)¡, X ^^/v^VW.
A' A'
k2 = kMkNgMN, k2 = |/,f.
i-,,,- = dJh^dl,bll+dllxMdl,xN(iiC)MN
and
Lmn — 9MN '
l\hfMN k\,k\
\k\
u-r
(2.5)
Regarding (2.3), the DBI action can be expanded to the quadratic order [31] as
Sdbi = - / tf^V(T) J-- dot G,u, x
1
x [ 1 + -AFlu,F'11
(2.6)
2.1. DBI part of the M3-brane action
To find the algebraic form of the DBI action, we
start with the first term in (2.6), yj — dot GIU,, which is the determinant of a 4 x 4 matrix and all its elements are sums of a tachyonic part and a space-like part (OX OX + 8TdT). This determinant totally consists of 48 x 8 terms. These terms can be classified into sixteen 4x4 determinants such that the elements of these determinants are only dX dX or dT dT and not sums of them. Hence, each determinant has 24 terms such that adding them leads to the same number of terms (16 x 24) as in the initial main determinant. These 16 determinants can bo catogorizod as: one
determinant with 8X8X elements (four combinations
from the 4 states ( 4 ) =1). one determinant with ele---- —.....................-
rows of dX OX elements and one row of &T&T elements
elements and one row of dX dX elements ( ^ J^j =4),
and, finally, six determinants with two rows of dTdT
elements and two rows of dX dX elements ((^^j =6).
It follows that determinants with more than one row of dT dT are zero. We are therefore left with two kinds of determinants: a determinant consisting of only dX dX entities and those with three rows of dX dX elements and one row of dTdT entities. Because a determinant does not change under exchanging of rows, considering all possible permutations (4!) of rows for each of the remaining determinants yields the form of the four-algebra in accordance with (A.5). Eventually, after a
tedious calculation, the algebraic form of ^J — det G/u, is obtained as
• det G
in'
I — (^LmnLopLqRLst[Xm,X°,XCi,XH] x x [_Ya, Xp,XR,XT] + -t-LmnLopLqr x
x [T, XM, À'°, A'[T, XN, A'p, A'] }• . (2.7
1/2
The 4-brackct of the space-time coordinates A' corresponds to the algebraic action derived in [1,17] for p = 3 case and with the fcrmionic fields turned off. The new term here is the mixed four-bracket of the A' and T.
Presenting a general algebraic form for the term Ftll,F'11' in the DBI action is not possible, but in some special cases it acquires a simple form. For example, one can consider a sclfdual (anti-sclfdual) field strength that corresponds to instant on. An instant on is a static (solitonic) solution of pure Yang Mills theories [32]. They are important in both supersymmetric field theories and supcrstring theories, mostly because of their nonperturbative effects. They also play a role in M-thcory, for instance, in applying the M2-branc actions to the M5-branc [33]. The solution of field equations in the Yang Mills theory corresponding to an instant on has a sclfdual (anti-sclfdual) field strength [32]. Considering this property gives the following expression
for tr Ftll,F'11' in the case of a regular one-instant on solution [321:
tr p pßl< - _nc_t_
111 ~ ((*- .r0)2 + p2)4'
(2
where x0 and p are arbitrary parameters called collective coordinates. Hence, in the instantonic case, the full algebraic form of the DBI part of the action
Sdbi = -J d^V(T) (l-24-
P*
x
"((*-*<) W)4
x < ^ I LmnLopLQRLst[Xm,X°,XQ.Xs] x
x[A'J',A' , A' , A' ] +-t-LmnLopLqh x
\k\
x [T, XM, 1°, XCi] [T, XN,XP, Xn}
1/2
(2.9)
2.2. WZ part of the M3-brane action
The integrand of the WZ action in (2.1) can be divided into three parts by replacing k from (2.2):
^W Z : - uji 1 ' '' / ' 'j / ' : ; / ' 1 —
GkmnD112Xk Dfl3XMD/14XA ■
Aß2 (dßs bIÀ4 dßi bßs
(2.10)
where we now deal with each part separately.
By expanding the first
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