Algebraic Form of M3-Brane Action Hossein Ghadjari Department of Physics, Amirkabir University of Technology (Tehran Polytechnic) 4 1 P.O.Box 15875-4413, Tehran, Iran 0 [email protected] 2 n Zahra Rezaei a J Department of Physics, University of Tafresh 1 P.O.Box 39518-79611, Tehran, Iran 1 [email protected] ] h t - p Abstract e h We reformulate the bosonic action of unstable M3-brane to mani- [ fest its algebraic representation. It is seen that in contrast with string 1 and M2-brane actions that are represented only in terms of two and v three dimensional Lie-algebras respectively, the algebraic form of M3- 1 0 brane action is a combination of four, three and two dimensional Lie- 5 algebras. Corresponding brackets appear as mixtures of tachyon field, 2 . space-time coordinates, X, two-form field, ωˆ(2), and Born-Infeld one- 1 form, ˆb . 0 µ 4 1 : v PACS numbers: 11.25.Yb; 11.25.Hf i X Keywords: M-theory; M3-brane; Lie-algebra; Nambu bracket r a 1 Contents 1 Introduction 2 2 Algebraic M3-brane action 4 2.1 DBI part of M3-brane action . . . . . . . . . . . . . . . . . . . 5 2.2 WZ part of M3-brane action . . . . . . . . . . . . . . . . . . . 7 3 Summary and conclusion 8 A Fillipov n-Lie algebra 9 1 Introduction Algebraic reformulation of known actions in string theory and M-theory shows that string theory is based on conventional algebra or two dimensional Lie-algebra (known as two-algebra) but a complete description of M-theory needs an extended Lie algebra called three-algebra [1] which was mainly de- veloped by Bagger, Lambert and Gustavsson [2–5]. Numbers two and three are associated with string theory and M-theory, respectively. Two is the string worldsheet dimension and also the codimension of D-branes in both type IIA and IIB superstring theories [6]. Three is the membrane worldvol- ume dimension in M-theory and the codimension of M2 and M5-branes. It means that via two-algebra interactions some Dp-branes will condense to a D(p+2)-brane [7] and through three-algebra interactions multiple M2-branes condense to a M5-brane [8–16]. These connections between two and three and respectively string theory and M-theory become obvious by rewriting Nambu-Goto actions in algebraic form. By analogy one can expect to describe p-branes applying p+1-algebra structure[17]. Theseextendedalgebrasareappliedtoconstructworldvolume theories for multiple p-branes in terms of Nambu brackets that are classical approximations to multiple commutators of these algebras [18]. Nambu n- brackets introduce a way to understand n dimensional Lie-algebra presented byFillipov[19]. Formulationofp-braneactionintermsofp+1-algebramakes it more compact and we are left with algebraic calculations that are usually simpler to handle. Since in string theory we are, inevitably, faced with unstable systems, study of them deepens our understanding of string theory. In bosonic string 2 theorytheinstabilityisalwayspresentduetotachyonpresenceinopenstring spectrum. Two examples of unstable states in superstring theories are: non- BPS branes (odd (even) dimensional branes in type IIA (IIB) theory) and brane-anti-brane pairs in both type IIA and IIB theories [20,21]. One of the interesting facts about the dynamics of these unstable branes, generally ob- vious in effective action formulation, is their dimensional reduction through tachyon condensation [22–27]. During this process the negative energy den- sity of the tachyon potential at its minimum point, cancels the tension of the D-brane (or D-branes) [28], and the final product is a closed string vacuum without a D-brane or stable lower dimensional D-branes. On the other hand stable objects in string theory can be obtained by dimensional reduction of stable branes in M-theory (M2 and M5-branes). Naturally, one can expect to have a pre-image of unstable branes in superstring theories by formulating an effective action for unstable branes in M-theory. Among different unsta- ble systems in M-theory [29] M3-brane is noteworthy because it is directly related to M2-brane. Tachyon condensation of the M3-brane effective ac- tion results in M2-brane action and also its dimensional reduction leads to non-BPS D3-brane action in type IIA string theory [30]. Despite attempts made to formulate M3-brane action consistent with de- sired conditions [30] there has been no algebraic approach towards this for- mulation. Existence of algebraic form for the action of M2-brane, as the fundamental object of M-theory, motivated us to search for the algebraic presentation of M3-brane as the main unstable object in M-theory that its instability is due to the presence of tachyon. What distinguishes present study from conventional algebraic formula- tions is instability of M3-brane. In other words, presence of tachyon and other background fields affect the resultant algebra. It is shown that pure four-algebra does not occur, as expected, and we are encountered with four, three and two-brackets that are mixtures of tachyon, spacetime coordinates and other fields. 3 2 Algebraic M3-brane action The conventional action corresponding to a non-BPS M3-brane is a combi- nation of DBI (Dirac-Born-Infeld) and WZ (Wess-Zumino) parts [30] S = S +S , DBI WZ (cid:90) S = − d4ξV(T)|kˆ|1/2(cid:112)−detH , DBI µν (2.1) (cid:90) S = − d4ξV(T)εµ1µ2µ3µ4∂ Tκˆ , WZ µ1 µ2µ3µ4 whereξµ withµ = 0,1,2,3labelworldvolumecoordinatesofM3-brane. V(T) is the tachyon potential which is an even function of T and is characterized as V(T = ±∞) = 0 and V(T = 0) = T where T is M3-brane tension. M3 M3 kˆM(X) is the Killing vector and the Lie derivative of all target space fields vanish with respect to it [30]. Other fields in (2.1) are defined as 1 1 H = gˆ Dˆ XˆMDˆ XˆN + Fˆ + ∂ T∂ T, µν MN µ ν ˆ µν ˆ µ ν |k| |k| kˆ2 = kˆMkˆNgˆ , kˆ2 = |kˆ|2, MN Fˆ = ∂ ˆb −∂ ˆb +∂ XˆM∂ XˆN(i Cˆ) , µν µ ν ν µ µ ν kˆ MN 1 (2.2) Dˆ XM = ∂ XˆM −Aˆ kˆM, Aˆ = ∂ XˆMkˆ , µ µ µ µ ˆ 2 µ M |k| κˆ = ∂ ωˆ(2) −∂ ωˆ(2) +∂ ωˆ(2) µ2µ3µ4 µ2 µ3µ4 µ3 µ2µ4 µ4 µ2µ3 1 1 + Cˆ Dˆ XˆKDˆ XˆMDˆ XˆN + Aˆ (∂ ˆb −∂ ˆb ). 3! KMN µ2 µ3 µ4 2! µ2 µ3 µ4 µ4 µ3 The tensor H consists of the pullback of background metric, field strength µν ˆ F of gauge field A and tachyon field, T. M and N represent spacetime µν µ ˆ indices and D is covariant derivative. The field strength itself is expressed µ ˆ ˆ in terms of Born-Infeld 1-form b and R-R sector field C. The curvature of µ the 2-form ωˆ(2) is shown as κˆ. Determinant of the tensor H in DBI action can be decomposed as µν (cid:113) (cid:112) ˜ ˜ −detH = −det(G +F ), (2.3) µν µν µν 4 where ˜ ˆ ˆ F = ∂ b −∂ b , µν µ ν ν µ 1 G˜ = L ∂ XM∂ XN + ∂ T∂ T, (2.4) µν MN µ ν ˆ µ ν |k| and (i|kˆ|CˆMN) kˆMkˆN L = g + − . (2.5) MN MN |kˆ| |kˆ|2 Regarding (2.3), DBI action can be expanded to quadratic order [31] as (cid:90) (cid:113) (cid:18) 1 (cid:19) S = − d4ξV(T) −detG˜ 1+ F˜ F˜µν +... . (2.6) DBI µν µν 4 2.1 DBI part of M3-brane action To find the algebraic form of the DBI action, we start with the first term (cid:113) ˜ in (2.6), i.e. −detG , that is determinant of a 4 × 4 matrix and all µν its elements are sum of a tachyonic part and a space-like part (∂X∂X + ∂T∂T). This determinant is totally consisted of 48×8 terms. These terms can be classified into sixteen 4 × 4 determinants in such a way that the elements of these determinants are only ∂X∂X or ∂T∂T and not sum of them. So each determinant has 24 terms that summing them up leads to the same number of terms (16×24) as the initial main determinant. These 16 determinants can be categorized as: one determinant with ∂X∂X elements (cid:18) (cid:19) 4 (fourcombinationsfrom4states = 1.),onedeterminantwithelements 4 (cid:18) (cid:19) 4 of ∂T∂T ( = 1), four determinants with three rows of ∂X∂X elements 4 (cid:18) (cid:19) 4 and one row of ∂T∂T elements ( = 4), four determinants with three 1 (cid:18) (cid:19) 4 rows of ∂T∂T elements and one row of ∂X∂X elements ( = 4) and 1 finally six determinants with two rows of ∂T∂T elements and two rows of (cid:18) (cid:19) 4 ∂X∂X elements ( = 6). It is obtained that determinants with more 2 thanonerowof∂T∂T arezero. Soweareleftwithtwokindsofdeterminants: a determinant consisting of only ∂X∂X entities and those with three rows of 5 ∂X∂X elements and one row of ∂T∂T entities. Since determinant does not change under exchanging of rows, by considering all possible permutations (4!) of rows for each one of the remaining determinants, the form of the four- algebra, in accordance with (A.5), emerges. At the end of the day after a (cid:113) ˜ tedious calculation the algebraic form of −detG is obtained as µν (cid:113) (cid:26) (cid:18) −detG˜ → − L L L L [XM,XO,XQ,XS][XN,XP,XR,XT] µν MN OP QR ST 4 (cid:19)(cid:27)1/2 + L L L [T,XM,XO,XQ][T,XN,XP,XR] .(2.7) ˆ MN OP QR |k| The 4-bracket of spacetime coordinates, X’s, corresponds to algebraic action derived in [1,17] for p = 3 case and with the fermionic fields turned off. The new term here is the mixed four-bracket of X’s and T. Presenting a general algebraic form for the term F˜ F˜µν in DBI action is µν not possible, however in some special cases it finds a simple form. For exam- ple one can consider a selfdual (anti-selfdual) field strength that corresponds to instanton. An instanton is a static (solitonic) solution to pure Yang-Mills theories [32]. They are important in both supersymmetric field theories and superstring theories mostly because of their nonperturbative effects. They also play role in M-theory for instance in applying the M2-brane actions to M5-brane [33]. The solution to field equations in Yang-Mills theory corre- sponding to an instanton has a selfdual (anti-selfdual) field strength [32]. Considering this property gives the following expression for tr F˜ F˜µν in the µν case of regular one-instanton solution [32] ρ4 tr F˜ F˜µν = −96 , (2.8) µν ((x−x )2 +ρ2)4 0 where x and ρ are arbitrary parameters called collective coordinates. 0 So for the instantonic case the full algebraic form of the DBI part of the action reads as (cid:90) (cid:18) ρ4 (cid:19) S = − d4ξV(T) 1−24 DBI ((x−x )2 +ρ2)4 0 (cid:26) (cid:18) × − L L L L [XM,XO,XQ,XS][XN,XP,XR,XT] MN OP QR ST 4 (cid:19)(cid:27)1/2 + L L L [T,XM,XO,XQ][T,XN,XP,XR] . (2.9) ˆ MN OP QR |k| 6 2.2 WZ part of M3-brane action The integrand of WZ action in (2.1) can be divided into three parts by replacing κˆ from (2.2) S → εµ1µ2µ3µ4∂ Tκˆ WZ µ1 (cid:18)µ2µ3µ4 = εµ1µ2µ3µ4∂ T ∂ ωˆ(2) −∂ ωˆ(2) +∂ ωˆ(2) µ1 µ2 µ3µ4 µ3 µ2µ4 µ4 µ2µ3 1 + Cˆ Dˆ XˆKDˆ XˆMDˆ XˆN 3! KMN µ2 µ3 µ4 (cid:19) 1 ˆ ˆ ˆ + A (∂ b −∂ b ) , (2.10) 2! µ2 µ3 µ4 µ4 µ3 and each part is dealt with separately. By expanding the first part, three terms of ωˆ(2) derivatives, and con- sidering all possible permutations of four-dimensional Levi-Civita symbol, εµ1µ2µ3µ4, we come to a view of two-algebra. The reason is that according to (A.5) having two derivative factors signals a two-algebra which carries its two-dimensional Levi-Civita symbol. But since here only different permuta- tions of εµ1µ2µ3µ4 give correct signs to the terms, multiplying the resultant two-algebra by another two-dimensional Levi-Civita symbol and using the relation εαβε = δαδβ −δαδβ, γδ γ δ δ γ conduct us to the correct form. So the first part of WZ action is reformulated in terms of two-bracket as S → εµ1µ2µ3µ4∂ T(∂ ωˆ(2) −∂ ωˆ(2) +∂ ωˆ(2) ) WZ,1 µ1 µ2 µ3µ4 µ3 µ2µ4 µ4 µ2µ3 = 3εµ1µ2µ3µ4ε [T,ω ]. (2.11) µ1µ2 µ3µ4 In the second part of WZ action, three X derivatives, ∂X, and one tachyon derivative, ∂T, appear in a way that obviously form a four-algebra 1 S → Cˆ εµ1µ2µ3µ4∂ TDˆ XˆKDˆ XˆMDˆ XˆN WZ,2 3! KMN µ1 µ2 µ3 µ4 (cid:32) (cid:33)3 1 kˆPkˆ = Cˆ εµ1µ2µ3µ4 1− P ∂ T∂ XK∂ XM∂ XN 3! KMN |kˆ|2 µ1 µ2 µ3 µ4 (cid:32) (cid:33)3 1 kˆPkˆ = Cˆ 1− P [T,XK,XM,XN]. (2.12) 3! KMN |kˆ|2 7 Substituting A in the last part of WZ action we are faced with terms µ consisting of ∂X, ∂T and ∂b that according to (A.5) indicate a three-algebra. Similar to the argument made for the first part of WZ action, multiplying this three-bracket by a three-dimensional Levi-Civita symbol and using the identity εαβγε = δα(δβδγ −δβδγ)−δα(δβδγ −δβδγ)+δα(δβδγ −δβδγ), δηλ δ η λ λ η η δ λ λ δ λ δ η η δ givetheconvenientthree-algebra. Differentpermutationsoffour-dimensional Levi-Civitasymbolareresponsibleforcorrectsignsofdifferenttermsinthree- algebra. So the algebraic form of this part is 1 S → εµ1µ2µ3µ4∂ TAˆ (∂ ˆb −∂ ˆb ) WZ,3 2! µ1 µ2 µ3 µ4 µ4 µ3 ˆ k = M εµ1µ2µ3µ4ε [T,XM,b ]. (2.13) 2!|kˆ|2 µ1µ2µ3 µ4 Therefore WZ action of M3-brane is presented in terms of two, three and four-brackets as (cid:90) (cid:26) S = − d4ξV(T) 3εµ1µ2µ3µ4ε [T,ω ] WZ µ1µ2 µ3µ4 (cid:32) (cid:33)3 1 kˆPkˆ + C 1− P [T,XK,XM,XN] 3! KMN |kˆ|2 ˆ (cid:27) k + M εµ1µ2µ3µ4ε [T,XM,b ] . (2.14) 2!|kˆ|2 µ1µ2µ3 µ4 It is seen that tachyon field couples with spacetime coordinates, Born-Infeld one-form ˆb and two-form ωˆ(2) through four, three and two-brackets, respec- µ tively. 3 Summary and conclusion In this article we presented an algebraic form for bosonic M3-brane action by reformulatingthisactionintermsofbrackets. Sinceintheliteraturep-branes aredescribedbyp+1-algebra[17]oneexpectsafour-algebrastructureforM3- brane. But it was shown that the algebraic representation of M3-brane is a combination of four, three and two-algebras. Generally this difference stems 8 from the instability of the system that tachyon is responsible for. Except of a four-bracket of spacetime coordinates in DBI part, tachyon field is present in all other brackets and forms four, three and two-brackets with spacetime coordinates, two-form, ωˆ(2), and Born-Infeld one-form ˆb , respectively . In µ future we try to study the dimensional reduction of this algebraic action. A Fillipov n-Lie algebra Fillipov n-Lie algebra [19] as a natural generalization of a Lie algebra is defined by n-bracket satisfying the totally antisymmetric property [X ,...,X ,...,X ,...,X ] = −[X ,...,X ,...,X ,...,X ], (A.1) 1 i j n 1 j i n and the Leibniz rule n (cid:88) [X ,...,X ,[Y ,...,Y ]] = [Y ,...,[X ,...,X ,Y ],...,Y ]. (A.2) 1 n−1 1 n 1 1 n−1 j n j=1 n-Lie algebra is equipped with an invariant inner product (cid:104)X,Y(cid:105) = (cid:104)Y,X(cid:105), (A.3) as well as the invariance under the n-bracket transformation (cid:104)[X ,...,X ,Y],Z(cid:105)+(cid:104)Y,[X ,...,X ,Z](cid:105) = 0. (A.4) 1 n−1 1 n−1 When n = 2 the definition reduces to the usual Lie algebra and the inner product can be given by ”Trace”. n-Lie algebra can be realized in terms of Nambu n-bracket defined over functional space on an n-dimensional manifold [18] 1 [X ,X ,...,X ] ⇔ {X ,X ,...,X } := √ (cid:15)l1l2...ln∂ X ∂ X ...∂ X . 1 2 n 1 2 n N.B G l1 1 l2 2 ln n (A.5) G is determinant of the metric of the manifold and can be chosen arbitrarily since the properties (A.1)-(A.4) hold irrespective of the presence of the local factor [1]. 9 References [1] K. Lee and J. H. Park, JHEP 0904:012,2009, [arXiv:0902.2417 [hep-th]]. [2] J. Bagger and N. Lambert, Phys. Rev. D 75 (2007) 045020 [arXiv:hep- th/0611108]. [3] J. Bagger and N. 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