aa r X i v : . [ h e p - t h ] M a y ITEP/TH-25/08
From Simplified BLG Action to the First-Quantized M-Theory
A.Morozov
ITEP, Moscow, Russia
ABSTRACT
Concise summary of the recent progress in the search for the world-volume action for multiple M2 branes. After a recentdiscovery of simplified version of BLG action, which is based on the ordinary Lie-algebra structure, does not have coupling constantsand extra dynamical fields, attention should be switched to the study of M2 brane dynamics. A viable brane analogue of Polyakovformalism and Belavin-Knizhnik theorem for strings can probably be provided by Palatini formalism for 3d (super)gravity.
In the context of general string theory [1] a variety of string and superstring models, linked by a numberof duality relations, is naturally unified in a hypothetical ”M-theory” [2], which in its most naive perturbativephase is represented by ”fundamental (super)membranes”. The BLG action [3, 4] resolves the long-standingpuzzle [5] of finding the 3d Lagrangian with appropriate superconformal symmetry and thus opens a way todeveloping the first-quantized theory of M2 branes. This implies that M-theory can now be studied in thesame constructive manner as bosonic and super-strings in 1980’s [6, 7, 8]. Naturally, this ground-breakingachievement attracts enormous attention [9]-[25], and some minor drawbacks of original analysis in [3, 4] arenow fully cured.The main obvious difficulty of original BLG formulation was its reliance upon sophisticated 3-algebra (quan-tum Nambu bracket) structure [26] – new for the fundamental physical considerations. The lack of experienceand intuition about this structure caused certain confusion at the early stages: original BLG action was writtenonly for an artificial(?) example of SO (4) symmetry, problems were discovered with straightforward generaliza-tions to other groups and even doubts appeared about the very existence of BLG action for the stack of N M2branes with arbitrary N , which would require promotion of SO (4) to SU ( N ). The key step in overcoming thisproblem was analysis of the M2 → D2 conversion in [10], which linked the 3-algebra structure to conventional Liealgebras, governing Yang-Mills and D-brane theories. Based on this analysis, in [14] a ”simplified” BLG actionwas introduced, which makes use of the Lie-algebra structure only (i.e. is based on ”reducible-to-Lie-algebra”Nambu bracket of [27], see eq.(2) below). The only new ingredient, distinguishing this version of BLG actionfrom the ones familiar from string/brane studies was a pair of extra color-less octuplets ϕ I and χ A . While verysimple, this suggestion had serious problems as it was, originated from degeneracy of the underlying Nambubracket and the lack of total antisymmetry of 3-algebra structure constants: this made original supersymmetryproof of [3] unapplicable and the action in [14] potentially non-supersymmetric. Thus it was meant to be atoy-example, showing the direction to eliminate unnecessary(?) elements of the BLG construction, but stillpossessing some extra fields and requiring some further tuning. A natural next step was to look at a centralextension, lifting degeneracy of Nambu bracket [15] – and this was finally done in a triple of wonderful papers[25]. They resolved the discrepancy between [14] and BLG approach in an elegant way, by changing the natureof the extra fields ϕ, χ : they are actually auxiliary, non-dynamical variables. Kinetic terms ( ∂ µ ϕ I ) and ¯ χ A ˆ ∂χ A of [14] are substituted in [25] by ∂ µ ˜ ϕ I ∂ µ ϕ I and ¯˜ χ A ˆ ∂χ A + ¯ χ A ˆ ∂ ˜ χ A where ˜ ϕ, ˜ χ is still another pair of color-lessoctuplets which do not appear anywhere else in the action and thus serve as Lagrange multipliers, eliminatingthe fluctuations of the unwanted ϕ and χ fields. In other words, the modified version of the simplified BLGaction of [14] is now [25]: −
12 tr (cid:16) D µ φ I − B µ ϕ I (cid:17) + i ψ A ˆ D (cid:16) ψ A − ˆ Bχ A (cid:17) ++ (cid:16) ∂ µ ˜ ϕ I − tr( B µ φ I ) (cid:17) ∂ µ ϕ I − i χ A ˆ ∂χ A − i χ A (cid:16) ˆ ∂ ˜ χ A − tr( B µ ψ A ) (cid:17) ++ 12 ǫ µνλ tr (cid:16) F µν B λ (cid:17) −
112 tr (cid:16) ϕ I [ φ J , φ K ] + ϕ j [ φ K , φ I ] + ϕ K [ φ I , φ J ] (cid:17) ++ i IJAB ϕ I tr (cid:16) ¯ ψ A [ φ J , ψ B ] (cid:17) + i IJAB tr (cid:16) ¯ ψ A [ φ I , φ J ] (cid:17) χ B − i IJAB ¯ χ A tr (cid:16) [ φ I , φ J ] ψ B (cid:17) (1)1t essentially differs from eq.(21) of [14] in the second line. Here φ I and ψ A with I = 1 , . . . , A = 1 , . . . , SO (8) group respectively(related by octonionic triality to the second spinor representation, where the N = 8 SUSY transformationparameter takes values). They are also N × N matrices, i.e. belong to adjoint representation of the gaugegroup G = SU ( N ). A µ is the corresponding connection, also in the adjoint of G , D µ φ = ∂ µ φ − [ A µ , φ ], F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ], and B µ is an auxiliary adjoint vector field (not a connection). The otherauxiliary fields ϕ I , ˜ ϕ I and their superpartners χ A , ˜ χ A are G -singlets (are color-less), possibly fragments ofKac-Moody extension of G . See [14, 25] for further details.As explained in [25], • The action (1) is N = 8 supersymmetric due to original BL theorem [3], because it is now based on the3-algebra with totally antisymmetric structure constants, which is a central extension of the degenerate one [27]used in [14]: [ X, Y, Z ] = tr X · [ Y, Z ] + tr Y · [ Z, X ] + tr Z · [ X, Y ] + ζ · tr ( X [ Y, Z ]) (2) ζ is a central element, different from unity matrix I and related to it by non-trivial scalar product < I, I > = < ζ, ζ > = 0, < I, ζ > = < ζ, I > = −
1, so that the 3-algebra metric is − − h and f abcI = − f abcζ = − f abc = − f Iabc . The last term of (2) was absent in [14] and this made the 3-bracket degenerate and thestructure constants (with the forth index raised by 3-algebra metric) not totally antisymmetric. The ϕ, χ fieldsare associated with the I (matrix-trace) generator, while ˜ ϕ, ˜ χ – with the central element ζ . Non-trivial 3-algebrametric implies the ϕ - ˜ ϕ and χ - ˜ χ mixing form of the kinetic terms in (1). • The would-be coupling constant in front of non-quadratic terms in (1) can be absorbed into rescaling of ϕ and χ , accompanied by rescaling of ˜ ϕ and ˜ χ in the opposite direction. This lack of this feature was one of theproblems in [14], and in (1) we have an action, which has no dimensionless coupling constants, as required inM-theory. • All the unwanted extra fields ϕ, χ, ˜ ϕ, ˜ χ are auxiliary: they do not propagate and contribute only throughboundary terms (i.e. to correlators) and zero-modes. • The fact that Lagrange multiplier ˜ ϕ nullifies only ∂ ϕ rather than ϕ itself is very important, because thisallows the zero-mode ϕ = const . Among other effects, this zero mode can form a condensate, producing a term < ϕ > tr B from the first item in (1), which, after auxiliary field B µ is integrated away, converts the Chern-Simons interaction tr F ∧ B into kinetic Yang-Mills term < ϕ > − tr F µν for connection A µ . This means thatdespite ϕ fields are now auxiliary, the crucially important possibility to use them for the M2 → D2 conversion a la [10] is preserved.All this means that today we possess a perfectly simple version (1) of the BLG action for arbitrary numberof M2 branes, there are no longer doubts about its existence for arbitrary gauge group SU ( N ), there are nocoupling constants, no extra dynamical fields, and it is clearly related to the other brane actions, as requiredby embeddings of d = 10 superstring models into the d = 11 M-theory. The road is now open for building upthe first-quantized theory of M2 branes (supermembranes). This implies that attention can now be shifted fromthe study of 3-algebra structure (where a lot of interesting questions still remain) to the other issues: we knowwhat should be the crucial next steps from the history of first-quantized theory of superstrings.Constructing the action (1) can be considered as the very first step, corresponding to substitution of Nambu-Goto action for bosonic strings by a σ -model action, of which (1) is supposed to be a (super)membrane analogue.In the case of membranes the problem was more complicated, because Nambu-Goto action is ill (does not dampfluctuations) from the very beginning, no approach to bosonic membrane is still available (problems look moresevere than the tachyon of bosonic string) and one should begin directly from the supersymmetric case, moreoversupersymmetry should be immediately extended to N = 8. Thus it may be not too surprising that we had towait till 2008 to have this action written down...In the case of strings the next big step was consideration of world sheets with non-trivial topologies, with twocomplementary formalisms finally developed for this purpose (and still not fully related, see [29] for descriptionof the corresponding problems). One is the Polyakov formalism [6], promoting the σ -model action to arbitrarycurved 2 d geometries and generalizing the treatment of relativistic particle in [30]. Another is equilateral-triangulation approach, nicely expressed in terms of matrix models [31] and formally equivalent to substitution A trivial mistake of [14] in the φ term (omitted item 2 P I 2f smooth 2 d -geometries by Grothendieck’s dessins d’enfants [32]. In the case of membranes this step is going tobe a hard exercise, already because the topological classification of 3 d world volumes is far more complicated thanin 2 d . Still, the very first movement – introduction of 3 d geometry into (1) by both above-mentioned methods– should be straightforward, and undoubtedly very interesting. For a variety of reasons it seems natural todo this in the modern BF-version of Palatini formalism, which is now widely popularized by controversial, butinspiring papers of G.Lisi [33]. Of certain help can be also comparison with the Green-Schwarz formalism forthe superstrings [34], where world-sheet action has some common features with (1): it also looks non-linear,but actually non-linearities concern only the zero-modes and boundary effects.Of crucial importance should be identification of the relevant world-sheet-geometry degrees of freedom(moduli), which the action is going to depend upon. This is not the 3 d metric or dreibein and spin-connectionthemselves – already because of the general covariance. However, as we know from experience with strings, theremaining degrees of freedom (Liouville field) can also be irrelevant (or identified with the other physical fields[35]), so that the only remaining moduli are those of the 2 d complex structures: finitely many for any given 2 d topology. It is the analogue of Belavin-Knizhnik theorem [7, 8] that formulates this statement for strings, whichshould be the next big discovery in the story of BLG actions. Again, there are many complications in the caseof membranes: as already mentioned, from the very beginning we need supersymmetry (and the correspondingproblem for superstrings was partly resolved only quite recently! [36, 37]). Moreover, the analogue of Riemanntheta-function theory [38] in 3 d is not yet at our disposal – and here we should face the same problems asthe other approaches to 3 d topological theories [39]: there are no conventional terms to express our answersthrough...In this short summary we do not speculate about the resolution of all these problems, i.e. about filling theempty spaces in the right column of the following table:(super)strings (super)membranes2 d Nambu-Goto action → d σ -model action spirit of membrane → simplified BLG action (1)Polyakov formalism: BF-version of Palatini formalism in 3 d introduction of 2 d metric,critical dimensions (where massless excitations occur),sum over geometries,sum over topologies,relation to equilateral triangulations approachBelavin-Knizhnik theorem:reduction of sum over metrics to sum over modulitopology of world sheet:spin structures and GSO projection,string field theory,boundary correlators and AdS/CFT correspondencetopology of the target space (compactifications):generic 2 d conformal theories, T -dualities,other dualities. . . 3ur goal is to emphasize that we are now in front of the new and interesting breakthrough into the unknown –the possibility opened to us by the timely formulated problem [5], a brilliant insight [3, 4] and qualified polishing[9]-[24], culminated in [25] in the elegant formula (1), which is going to be – perhaps, in some reshaped andredecorated version – a new focus of attention in string theory in the coming years. Acknowledgements This work is partly supported by Russian Federal Nuclear Energy Agency and Russian Academy of Sciences,by the joint grant 06-01-92059-CE, by NWO project 047.011.2004.026, by INTAS grant 05-1000008-7865, byANR-05-BLAN-0029-01 project, by RFBR grant 07-02-00645 and by the Russian President’s Grant of Supportfor the Scientific Schools NSh-3035.2008.2 References [1] M.Green, J.Schwarz and E.Witten, Superstring Theory , Cambridge University Press, 1987;A.Polyakov, Gauge Fields and Strings , 1987;J.Polchinsky, String Theory , Cambridge University Press, 1998;A.Morozov, String Theory: What is It? Sov.Phys.Usp. (1992) 671-714 (Usp.Fiz.Nauk 83 - 176)[2] P.K. Townsend, The eleven-dimensional supermembrane revisited , Phys.Lett. B350 (1995) 184-187, hep-th/9501068;E.Witten, String Theory Dynamics In Various Dimensions , Nucl.Phys. 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