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Proceedings of the Corfu Summer InstituteGravity 2011 School and Workshops on ElementarySeptember Particle 4-18, 2011 and Corfu, Greece Centre for Particle Theory and DepartmentDurham of University, Mathematical Durham, Sciences, DH1 3LE, UK E-mail: Chong-Sun Chu Quantum Nambu Geometry in Theory PoS(CORFU2011)043 (1.1) (1.2) (1.3) ) for multiple is identified as Chong-Sun Chu µ 4.4 X , hg f − g f h . ] − λ ] X ν f hg , ν X , − X µ , µ X [ i h f g X 2 [ i − + as a 3-form field strength − = µ = gh f X µν + F µνλ H f gh = : ] h , g , f [ are any three operators and the binary product is the usual operator product. This differs h , g , f Since a system of multiple D4- can be considered as a dimensional reduction of multiple It is generally expected that the usual description of spacetime in terms of Riemannian geometry In the paper [1], a new kind of quantized geometry was discovered. It is characterized by the For the standard Lie algebraic type noncommutative geometry [2–7], a 2-form field strength can To reach the D4-branes system, we use the D1-strings matrix model to derive the Matrix model ) as a 3-form field strength whose Hodge dual would be the Yang-Mills field strength. Physically, 1.3 this means the system ofsystem fundamental of strings multiple has D4-branes. expanded over the quantum Nambu geometry intoM5-branes a on a circle, itbe has identified been with proposed the recently KKtons, that modes the [9, of D4-branes 10] the SYM the theory compactified is M5-branes,natural on in and way fact the to that equal D4-branes include by to can all including the M5-branes the all theory. KK the In modes instan- view into of the this, D4-branes we and suggest a propose the action ( Quantum Nambu Geometry 1. Introduction would break down abovespacetime the coordinates Planck become quantum energy operators. scale.locality In and this causality, A and case, even traditional possibility the spacetime fundamental isString concepts nature theory, such of that as spacetime as geometry itself, a will is candidate havesome for to quantized of be a these and re-examined. theory questions. of , provides an interesting setupquantum to Nambu bracket address be written as from the previous examples of noncommutative geometrycommutator in and string could theory be which referred are to characterized as by of a Lie-algebra type. where when the fluctuation over the noncommutative geometrythe background fluctuation is around taken the into quantum account. Nambubracket What geometry? of about It the is target space suggestive to coordinate interpret fields the quantum Nambu descriptions for the IIB string, IIA stringor and its M-theory uplift in to a corresponding eleven backgroundquantum dimensions. of Nambu For large geometry RR-flux the is IIA again Matrix allowed.a string We Lagrangian also theory, for we find a that find 1-form the that fluctuation gaugePST a around potential action classical the (which whose solution solution describes form of gives a is single exactly D4-) the [8] same if as a the quantum dimensional Nambu reduced bracket of To check this idea, weleads have us to to look consider for the aHodge system dual place of of where a a multiple 2-form non-abelian D4-branes fieldbe 3-form (where strength) self-dual). field and the multiple strength 3-form M5-branes lives. field (where strength the This 3-form would field be strength the would ( PoS(CORFU2011)043 (2.4) (2.5) (2.7) (2.2) (2.3) (2.6) (2.8) (2.1) (1.4) -field on the -field, and the C Chong-Sun Chu B  4 C 4 Φ i F 3 2 λ , − 2 5 4 Ω C , d 2 Φ 2 i f 0 Φ / R , i Φ 3 P 2 , , − 2 ) + λ , , i , , e 2 1 + YM 80 2 S may not obey the desired self-duality condi- dU = µνλ , , 2 0 C = θ k 5 5 + + i , Φ 2 constant U i j 2 0 µνλ M M CS , Φ R i otherwise i dt S ] = H ) = F 3 λ on on − and is formulated in terms of a non-abelian 3-form of unit radius. In addition, there is a nonvanishing + 2 2 , ( X 5 2 X λ , √ i 0 i jk , 5 5 S S ν R 2 ε 2 ε ε , µνλ ( + f X f c c 0 otherwise 0 = + / θ , : 2 / 2 µ χ (        1 ) Φ i 2 X D PC [ specified by: = = = . S − parallel D1-branes in this background. The worldvolume action parameter of the quantum Nambu geometry as a e dX 4 + . 3 5 Φ ( 2 4 N -field could be reduced to a system of D4-branes in C F F 1 -field. Our proposed action is living on a 6-dimensional quantum χ µνλ − C 3 = C e µνλ = C S ∑ i F C θ 2 λ and + = ). A priori, such an = R  2 2 2 C C Tr 1.3 S ds µνλ + Z is the metric for an θ 1 χ 0 j µ S dX = = 0 : i : 5 CS dX S 0 S j 0 i × . ˆ 2 2 G c is the Chern-Simons term of the Myers type given by 3 2 = AdS CS 2 5 = S -field is described by a quantum Nambu geometry × Ω 2 Next let us consider a system of We start with the background of [11]: The metric (in the string frame) takes the form f 3 d C R and , axion, RR potentials where for the D1-branes is given by latter has a worldvolume describednection by allows the us standard to Moyal identify typeworldvolume the noncommutative of geometry. the M5-branes. This Therefore con- wein obtain a the result that the worldvolume of the M5-branes where Quantum Nambu Geometry M5-branes in a constant self-dual Nambu geometry with self-dual parameter field strength defined using ( tion. Nevertheless quite amazingly wemodel. find The M5-branes that system the in self-duality a condition emerges naturally from our of with self-dual parameters 2. Matrix Model of D1-Strings in Large RR 3-Form Flux and PoS(CORFU2011)043 2 C S ) is (2.9) (3.3) (3.1) (3.2) (2.10) 1.1 ) that are not ) Chong-Sun Chu i jk ). In other words, 2.10 ε k , 4 X j , 2.10 3 X , i 2 , 1 , iFX ( 0 Tr = σ e 2 , d d , c Z , f ] b , λ , a . X ] ) + i jk , ν ν , αβ X θε X , ε i , k µ µ X X abcde [ X ] = β ε i [ 4 k e i D − j X X − , d j X = = X α X c , limit, there are new representations of ( µν D i i X . b F X ) µνλ N X [ 2 X H a L i jk ε X + 2 1 1 Tr ( L σ ( Tr 2 σ σ d 2 2 d d Z as a 3-form field strength f Z Z N µ f f X = = ≡ IIA ) for the D1-branes. 1 S µ / 2.9 2 C S is a constant and the 3-bracket is given by quantum Nambu bracket. θ In [1], a duality argument was performed and a Matrix IIA that is dual to the above We would like to perform an expansion around the quantum Nambu geometry and ask what kind It was also shown that in the large Places where a non-abelian 3-form field strength lives are, for example, multiple D4-branes (where D1-string was obtained. The action reads a quantized geometry, the quantum Nambu geometry, characterized by the Nambu bracket ( of would comegeometry, out. a 2-form We field recall strength thatas is for obtained the from standard the Lie fluctuation algebraic over type the noncommutative noncommutative geometry dominates and one can ignore allgovern the by other terms in the action, In this limit, the system of D1-branes is given by Lie algebraicfundamental representation, and novel The nature existence of of the Nambu-Heisenberg these commutation representations relation ( thatallowed demonstrates as a the solution in string theory. 3. D4-branes in Large RR Flux Background Quantum Nambu Geometry It was shown in [1] that there is a low energy large flux double scaling limit so that the term where For our quantum Nambu geometry, it isspace suggestive to coordinate interpret fields the quantum Nambu bracket of the target The classical solution to the equation of motion was obtained in [1]. It was found a new solution the 3-form field strength would(where be the the Hodge 3-form dual field toto strength a these 2-form would field systems be strength) from self-dual). anddescriptions multiple our for M5-branes To the D1-branes check IIB system. string the theory,our idea, M-theory description And and ( we to IIA would string do theory like this, in to a let connect large us flux background first using derive the Matrix model and we would like to check this idea. PoS(CORFU2011)043 (3.7) (3.8) (3.5) (3.6) (3.9) (3.12) (3.11) (3.10) . Depending i x Chong-Sun Chu matrices decompose matrices. In general, N N matrices matrix can be expressed × N . 4 (3.4) N , N 4 , , 3 × 4 , 3 , , 2 N 3 2 , = 2 = i , i 1 , , , 0 1 . 1 , ραβ . abcde , = 0 ) ε H 0 j e e x = , abcde X , = ε d d α , θ σ 5 α i X ( c i c , de µνλραβ A b , limit, the trace over large X , H ε ] = 0 b + a 4 . The expansion of the dynamical variables around N g X i limit, we get a set of large 5 x , abc K matrices. Then every a , x , − ] = ] 1 3 × N i H X c x N √ K tr , tr X X 6 1 , 2 , 5 i 5 , × ] Σ x -field whose components are defined by x Σ e α b scalars [ among themselves are not constrained. Let us consider the 0) in a large RR 2-form flux background. = Z Z N : i H X X X ) becomes = [ , , = f = = X i N a d i 0 5 a 3.8 X X X µνλ X , [ [ S = ∗ X i i 0 5 H − − S , α ) admits the classical solution: ] = = = β iD 5 3.3 X , is an integral on the quantum Nambu geometry which can be constructed abc de = α x H H α , the action ( R X [ X ραβ and H x R tr. x σ of quantum Nambu geometry R 2 i ) becomes d x = R = 3.3 matrix whose entries are functions of = i do not generate the whole set of cl i K 5 X x Σ R × With the introduction of a three-form The matrix string action ( K In [1], it was argued thatof this multiple matches D4-branes with theory the (where dimension reduced PST action and so describes a sector where from a representation of the geometry.as In usual as the Tr large is the Hodge dual of As before, the commutation relations of solution on the representation chosen, they may or may not generate the entire set of Quantum Nambu Geometry where assume as a We note that the D1-strings action andcoefficient. the This IIA Matrix is string similar action to aremetric indeed the Yang-Mils theory the matrix could same IIA up have string different to stringparameters a of constant with interpretations DVV the depending where string on the theories. how same one This 2-dimensional associate is supersym- its consistent with T-duality. and consider a fluctuation around it. In the large the classical solution can thus be parameterized as The action ( where PoS(CORFU2011)043 B (4.4) (4.6) (4.5) (4.1) (4.2) (4.3) ,  Chong-Sun Chu g − √  5 ab H 5 satisfies the correct equation ab ) was proposed to describe a H direction transverse to the D4- ) µνλ 4.5 is determined from the represen- 5 α x H X R − to a general field. The identification 1 ). We note that the identification for 1 . , ] + ( = ] , ) 3.10 λ 5 5 x 1 ) as saying only the zero mode of the M5- ( abc X X X , , µ H e ν µνλ 4.2 A X X θ abc . , , i + and the trace 1 d µ H K X 1 X × = [ 6 ] = [ 3 i ) and ( i µνλ K − 5 λ 1 ) over the geometry ( − θ x − X -field are related, although one can expect the relation to , µ α x ν = X 3.10 = 4.4 ) x 5 , = + µ de 4.5 5 µ x µνλ [ H de X H H abc H term denotes terms necessarily for the non-abelianization. Thus we are -field and the B abcde ··· ε 1 6  as a scalar field describing the compactified tr matrices. The theory ( ; and the x 5 µ Z K X σ where the 4 1 d × − µ ··· K X ) can be put together as = are arbitrary self-dual constants. This corresponds to a six dimensional quantum Nambu + M5-branes. = θ , are 4.1 5 K X dB µ µνλ M 6) was proposed. The theory has solution S A / θ = 1 can be written as ) for D4-branes in a large RR flux background, how can we incorporate the higher KK modes in The fluctuations around the solution can be written as Recently, it has been argued that [9, 10] the instantons on multiple D4-branes could be identified H ) and ( 5 = de 3.9 α 3.12 theory of ( of motion (i.e. the self-dual equation)(i.e. and three). describes In the [1], correct the number following of action on-shell degrees of freedom ( our description? A possible hint is from the identification ( H branes has been included, i.e.to include a the higher dimensional KK/instantonic reduction modes to by D4-branes. promoting In this picture, it is suggestive where Quantum Nambu Geometry 4. A proposal for a theory of multiple M5-branes using 1-form gauge field with the KK modes associated with the compactificationmodes, of it M5-branes was on proposed a that circle. the By lowand including energy all is SYM these actually theory the of theory D4-branes of is( multiple a well-defined M5-branes quantum compactified theory on a circle. Back to our proposed action proposing that there is a dualvariables description of the non-abelian 3-form fieldbe strength very in complicated. terms of To justify the our 1-form proposal, one needs to show that and where geometry parameterized by self-dual parameter tations of the quantum Nambu geometry ( with If we think about brane, then one can understand the relation ( It was proposed inliving [1] on to M5-branes. be a In a different conventional way description, to there write would the be non-abelian a self-dual non-abelian 2-form 3-form potential field strength PoS(CORFU2011)043 ) 5 µν 4.3 θ (4.7) (4.8) = . All in µν using ( µνλ B C µ A Chong-Sun Chu -field -field C ) would then have all C 4.4 -field is related to the 11- B is self-dual suggests to identify it to three. In our formulation, we do . 5 with the constant B µνλ µν θ -field whose components are θ i µνλ B θ ), and the conventional formulation in terms only on-shell. As such there can be a relation ] = ν 4.4 is supposed to be equivalent to the conventional µν µνλ X µ , θ B A µ 7 = X ) indeed provides a dual description of the non-abelian ) reads is constructed from the 1-form potentials µνλ ] = [ 4.4 4.5 C 1 , µνλ ν H X , ) is the result of having a self-dual 3-form ) is equal to the non-abelian form of the PST action when the µ X 4.5 4.4 [ , it is correct to identify 5 µν C -field on the worldvolume of the M5-branes. This identification is further = , then the relation ( C 1 µν B = 5 X -field as C ). In a conventional description of 3-form field strength, a 2-form potential is used. Our analysis -field from our description is important as it would allow us to couple to self-dual strings. equation is crucial in reducing the fifteen components of formulation of using a 2-form gaugebetween potential the 2-form gaugeB field and the 1-form gauge field only on-shell. Identifying such a self-duality condition is satisfied. Thefor agreement of our the formulation actions to on-shell be ismore an a support equivalent necessary that description our condition proposed on-shell. action ( self-dual Therefore 3-form. this agreement provides 4.6 Evidence of this can be seen from the counting of the degrees of freedom of our model. Initially A couple of comments on the dual formulation are in order: In our description, the field strength We note that our M5-branes system has a quantum Nambu geometry as its worldvolume geometry. 3. In the conventional formulation, the existence of the tensor gauge symmetry and the self-duality 2. Our formulation of using a 1-form gauge field 1. As noted above, our action ( -field, one in terms of a 1-form gauge field as in ours ( we have six fields.the If degrees the of self-duality freedomappropriate equation are for is reduced a in description to fact of half a the and self-dual equation 3-form we of field have strength. motion indeed The of three theory the ( degrees theory, of then freedom which is of a 2-form gauge potential. the desirable properties of a theoryis of written non-abelian manifestly self-dual using 3-form a field 1-form strength potential except as that the the variables. theory suggests that there maybe in fact two equivalent formulations for the theory of multiple M5-branes in a What is the physical origin responsibletative for worldvolume this on quantized a spacetime? branein The is emergence typically its of the worldvolume. a result noncommu- The ofwith fact a the that background self-dual gauge the 3-form potential quantization being parameter turned on dimensional all, we conclude that the geometry ( turned on in the worldvolume of the M5-branes. and ( Quantum Nambu Geometry supported by the fact that ifamounts we to dimensionally putting reduced the M5-branes, say, on the 5-th direction, which (we remind the readers we are considering the linearized limit). Since the This is the noncommutative geometry over D4-branes with a C PoS(CORFU2011)043 Chong-Sun Chu (1999) 151 (1998) 008 (1999) 032 550 9802 9909 Yang-Mills gauge symmetry. ) N ( (1999) 030 [arXiv:hep-th/9903205]. U 9906 8 (2000) 447 [arXiv:hep-th/9906192]. 568 ). It is interesting that the gauge fixed theory has precisely (2012) 055401 [arXiv:1110.2687 [hep-th]]. G 45 × G ’s; and this symmetry is needed to derive the self-duality equation -field. For our case, it is possible that there is a counterpart of the B X -field potential [13], it was shown that the tensor gauge symmetry (part B symmetry structure constructed there) could be gauge fixed to an ordinary gauge (diagonal part of G G × G get the desired number of degreeneed of of freedom a (modulo the tensor issues gauge discussedtheory symmetry. above) using Curiously, and a in there 2-form is a no recent construction of the non-abelian 3-form of the symmetry (and hence the Bianchi identity).exists in It our is model, important and to if so, understand how whetherAnother it such possible acts. a way symmetry to really settle since the supersymmetry would issue require istomatically. the Supersymmetrization 3-form to of field supersymmetrize our strength system our to iscase be action also supersymmetry self-dual needed with for au- is describing (1,0) an M5-branes. or importantfor In topic (2,0) any some and recent we related hopestrength works to multiplet. on return (2,0) to it supersymmetry in of future a work. non-abelian self-dual See [12] 3-form field the reduced matrix description. However tois fully the justify only our nontrivial proposal, solution. we need Wecondition recall to as that establish the in that equation it the of PST action motionsymmetry , immediately. which one acts To does do on not this, the get one the needs self-duality to make crucial use of a the same gauge symmetrysupport as to our both the proposed description description proposed here. in [13] and This the coincidence description proposed provides here. some symmetry which acts on the M5-Branes in C-Field,” J. Phys. A A [arXiv:hep-th/9711165]. [arXiv:hep-th/9812219]. C. S. Chu and P. M.noncommutative Ho, D-brane,” “Constrained Nucl. quantization Phys. of B open string in background B field and [arXiv:hep-th/9908142]. C. S. Chu, “Noncommutative open string: Neutral and charged,” arXiv:hep-th/0001144. More recently, an non-abelian action for multiple M5-branes was constructed [14]. In this formu- 4. In the above we have obtained the Bianchi identity and the self-duality condition as a solution of [1] C. -S. Chu and G. S. Sehmbi, “D1-Strings in Large RR 3-Form Flux, Quantum Nambu Geometry and [2] M. R. Douglas and C. M. Hull, “D-branes and the noncommutative torus,” JHEP [3] C. S. Chu and P. M. Ho, “Noncommutative open string and D-brane,” Nucl. Phys. B [4] V. Schomerus, “D-branes and deformation quantization,” JHEP [5] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP Quantum Nambu Geometry lation, the tensor symmetry is abelian but the theory carry with it a It is not clear yetsame the gauge connection group between emerges. these different formulations, but it is encouraging toReferences see the PoS(CORFU2011)043 (2010) 1008 Chong-Sun Chu (2011) 011. 1102 9 (1999) 023. [hep-th/9908040]. 9909 (2011) 083. [arXiv:1012.2882 [hep-th]]. (2011) 099. [arXiv:1103.6192 [hep-th]]. 1101 (2000) 010. [hep-th/0003187]. 1105 0005 (2011) 020. 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Ko, “Non-abelian Action for Multiple M5-Branes,” arXiv:1203.4224 [hep-th]. Quantum Nambu Geometry [11] C. -S. Chu, P. -M. Ho, “D1-brane in Constant R-R 3-form Flux[12] and Nambu N. Dynamics Lambert, in C. String Papageorgakis, Theory,” “Nonabelian (2,0) Tensor Multiplets and 3-algebras,” JHEP [10] N. Lambert, C. Papageorgakis, M. Schmidt-Sommerfeld, “M5-Branes, D4-Branes and Quantum 5D