JHEP02(2017)099 tructure Springer January 19, 2017 February 12, 2017 February 20, 2017 : : : non-geometric M- 10.1007/JHEP02(2017)099 Received roduct reduces to that of Accepted tainty relations as well as -structure which defines a Published duces to the quasi-Poisson 2 a phase space star product doi: -dimensional vector space of G hose deformation quantisation cts and the configuration space .K. mensional configuration space. y fit in with previous occurences lated to the nonassociative alge- idade de Federal do ABC, Published for SISSA by c,d,e [email protected] , -flux model in the contraction of M-theory to R and Richard J. Szabo a,b . 3 -structure to a Spin(7)-structure, we propose a 3-algebra s 2 G 1701.02574 The Authors. c Flux compactifications, Non-Commutative Geometry

We describe the quantization of a four-dimensional locally , [email protected] -structures and quantization of non-geometric 2 E-mail: Maxwell Institute for MathematicalEdinburgh, Sciences, U.K. The Higgs Centre forEdinburgh, Theoretical Physics, U.K. Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U etod ae´tc,Cmuaca on¸˜o Univers eCentro Cogni¸c˜ao, de Computa¸c˜ao Matem´atica, Santo SP, Andr´e, Brazil Tomsk State University, Tomsk, Russia Department of Mathematics, Heriot-Watt University, b c e d a Open Access Article funded by SCOAP Abstract: Vladislav G. Kupriyanov M-theory backgrounds G bra of octonions. The construction is based on a choice of theory background dual tofor a deformation twisted quantization three-torus of quasi-Poisson by brackets deriving re nonassociative deformation of theFourier addition momenta. law on Wethe the demonstrate three-dimensional seven explicitly parabolic that constant this star p triproducts for the nonassociativeBy geometry extending of the the four-di string theory, and use it to derive quantum phase space uncer of 3-algebras in M-theory. Keywords: ArXiv ePrint: on the full eight-dimensionalalgebra after M2-brane imposing phase a spacesimultaneously particular encompasses gauge which both constraint, the re phase and space w triproducts. star produ We demonstrate how these structures naturall JHEP02(2017)099 ] 4 5 7 8 1 4 8 19 21 23 26 31 32 33 36 39 10 11 13 14 15 16 31 16 27 , 25 , 13 s metry fluxes ric backgrounds in flux com- NS sector of ten-dimensional not only because of their po- es beyond supergravity. Many -fluxes. hey make explicit use of string supergravity (see e.g. [ R -fluxes and Q – 1 – -flux background R -flux background R -symmetric star product 2 G -structures and deformation quantization 2 5.2 Phase space5.3 3-algebra Vector trisums 5.4 Phase space triproducts 4.4 Uncertainty relations 4.5 Configuration space4.6 triproducts and Noncommutative nonassociative geometry geo of momentum space 5.1 Triple cross products 4.1 Quasi-Poisson algebra4.2 for non-geometric M-theory Phase fluxe space4.3 star product Closure and cyclicity 3.2 Cross products3.3 Baker-Campbell-Hausdorff formula3.4 and vector star Quasi-Poisson sums algebra3.5 and Malcev-Poisson identity 3.6 SL(3)-symmetric star product 2.1 Quasi-Poisson algebra2.2 for non-geometric string Phase theory space2.3 star product Alternativity and Malcev-Poisson identity 3.1 Octonions G for reviews). They havesupergravity been which studied involves extensively non-geometric for the NS- duality symmetries which allows them toof probe them stringy can regim be obtainedpactifications by of duality ten-dimensional transformations and of geomet eleven-dimensional 1 Introduction and summary Non-geometric backgrounds of stringtential theory phenomenological are applications, of but interest also because t 5 Spin(7)-structures and M-theory 3-algebra 4 Quantization of M-theory 3 Contents 1 Introduction and summary 2 Quantization of string theory JHEP02(2017)099 - 2 G els of space 0 which ]. → 45 ], these two mputations s , g 30 ]. A nonasso- 3) double field 11 ] and in Matrix ] and in double , , 17 eory. Quantiza- 7 44 , 22 , ]. The contraction , 6 16 36 , 14 2 , 9 t of the three-dimensional ntly is seven-dimensional. , ing from this story: firstly, n-geometric strings is that 5 , s of closed string background condly, the role played by the ted. Their approach is based ered in [ ontraction limit 1 ng. ic functions appearing in the tion of the coordinate algebra [ ackets contract to those of the arified in [ ircle to a point; our derivation ysical and geometrical features ory background, which is dual ative algebra. In [ f flat space, that may in future truction [ n quantities are underpinned by f phase space star products were ve dynamics of closed strings in es and is relevant for compactifi- o more general compactifications en be mapped precisely onto the ns of the string theory case. Our sions. They argue that the phase ther than the more conventional on our structure manifolds involve etry, particularly the theory of ]; the physical significance and viability of 43 , -flux to M-theory within the context of SL(5) 8 – 2 – R ], which extends the SL(4) = Spin(3 ], in the sense that it reduces exactly to it in the 10 44 -holonomy. In contrast to the usual considerations of 2 -flux background obtained via T-duality from a geometric G R ], and subsequently applied to building nonassociative mod ] and field theories [ 45 , 45 -flux. The goal of the present paper is to quantize these phase 44 ]. This has been further confirmed through explicit string co , H -flux background [ 40 37 ], and also in a topological open membrane sigma-model [ R , , 4 15 12 ] which both suggest origins for non-geometric fluxes in M-th 19 Until very recently, however, there had been two pieces miss One of the most interesting recent assertions concerning no For this, we derive a phase space star product which lifts tha vector cross products, whose theory we review in the followi calibrated geometry, however, for deformation quantizati corresponding bivectors andthree-forms trivectors, and respectively, four-forms. ra All of the relevant deformatio is based on extendingrelated and elucidating to deformation the quantiza imaginarylimit octonions reduces that the was recently complicatedresulting consid star combinations product of to trigonometr theconstructions elementary exploit relevant algebraic facts functio from calibrated geom appropriate contraction limit which shrinks the M-theory c the M-theory version of thisoctonions deformation which of are geometry the archetypical and, example se ingredients of are a nonassoci treated simultaneouslyon and lifting shown the to non-geometric be string rela theory tion of these backgrounds throughprovided explicit in constructions [ o quantum mechanics [ these nonassociative structures inciative quantum theory mechanics of is gravity cl locally describing non-geometric the backgrounds low-energy is effecti currently under cons in left-right asymmetric worldsheetfield conformal theory field [ theory theory [ exceptional field theory, following [ developments enable an extension of theseof considerations t M-theory on manifolds of structures and Spin(7)-structures, simplified to the case o The corresponding classical quasi-Poisson bracketsLie can 2-algebra th generated by the imaginary octonions. In the c theory of string theory withcations three-dimensional of target eleven-dimensional spac supergravity to sevenspace dimen of the four-dimensionalto locally a non-geometric twisted three-torus, M-the lacks a momentum mode and conseque quasi-Poisson brackets, and to useof it the to non-geometric describe M-theory various background. ph they probe noncommutative andgeometries nonassociative [ deformation reduces M-theory to IIA stringnon-geometric theory, string the theory quasi-Poisson br three-torus with string theory JHEP02(2017)099 we 3 ion M-theory we briefly ing theory erned by a 2 -flux model on a ]. We also derive onfiguration space R -flux, the M-theory ibe various physical 45 R matical feature of the pful in understanding e three-spheres; in the -structure, which deter- In the generalisation to . In section toroidal string vacua. 2 ppears as a phase factor G ticular, we derive quantum ntisation of the 3-algebraic e cross products determine uration space triproducts in y case [ aining of both configuration the classical Malcev-Poisson ambu-Heisenberg 3-Lie alge- r. This parallels the situation he dimensional case, relevant for t in dimensions three (where ive). In the three-dimensional he physical seven-dimensional zation; in particular, we point ing theory of nonassociativity in physics, ly found in studies of multiple ase space, extends to a Spin(7)- he NS-NS nd M-theory backgrounds: they ensions and to the treatment of commutative deformation of the rom the theory of cross products -flux background, the vector cross ct to those on the four-dimensional mal area cells in the M-theory phase R ], which quantize the four-dimensional 2 – 3 – -flux background, the vector cross product determines R 0, these triproducts consistently reduce to those of the str ] for a review). Higher associativity of the 3-bracket is gov 3 → s and suggest an interpretation of the quantum geometry of the g ]. As preparation for the M-theory lift of this model, in sect 4 A 29 , ]. In contrast to the string theory case, this curving of the c 17 14 , 2 The organisation of the remainder of this paper is as follows In fact, in this paper we emphasise a common underlying mathe Armed with the phase space star product, we can use it to descr -flux background as a foliation by fuzzy membrane worldvolum -flux is not a multivector. This point of view should prove hel -flux background arises as a certain gauge constraint, tripl R contraction limit R review relevant aspects of the parabolic non-geometric str the present case isstructure most of naturally the understood eight-dimensional in membrane terms phase space: of qua t by three-spheres also results ingeometry a of novel momentum associative space but itself. non The origin of these config configuration space triproducts, in3-Lie algebra the spirit of [ 5-bracket, but it is notwith related the in lift any simple of way non-geometric to string a theory 5-vecto fluxes: unlike t star products which quantise non-geometric stringall theory originate, a via the Baker-Campbell-Hausdorffon formula, f real vectorthey spaces; are non-trivial associative) and crosscase, seven products relevant (where for only they the are exis quantisation nonassociat of the string theory space, as well asspace minimal and volumes phase demonstrating a space coarse-gr itself, in contrast to the string theor generalisations of these considerationsmissing to momentum both modes higher in dim M-theory backgrounds dual to non- and geometrical features ofuncertainty the relations membrane which explicitly phase exhibit space. novel mini In par three-torus with constant fluxesout and its that deformation star quanti algebraic product structure algebras that of sometimessee functions appears generally e.g. in spoil [ discussions product determines a 3-cocyclein the among associator Fourier for momentathe the that quantisation star a of product, the whereas M-theory in the seven- a nonassociative deformationthe of full the eight-dimensional sum M-theoryR of phase Fourier space, momenta. wherein t mines the star product quantising thestructure determining seven-dimensional phase ph space triproducts thatconfiguration restri space. an underlying 3-algebraic structureM2-branes akin (see e.g. to [ those previous configuration space which quantizebra the [ three-dimensional N JHEP02(2017)099 ⊕ Z that with (2.1) isson ⊕ D Z be a partial -flux and the ) = , coincides with R Z ) rotation of the Z = 0 M, is a twisted three- D ( ∈ ( } 1 j and the non-torsion -field after T-duality. of dimension M O d H . B , p mmation operation on i 4 × M M nd p is technical formalism is ) { ntization of the parabolic we extend the vector cross novel 3-algebraic structure of D produces the quasi-Poisson ( ertainty relations, as well as 5 momentum. The non-trivial the associated linear algebra ction O -flux background originates as ound in the original T-duality frame ompasses both the phase space by the M-theory on R two-form 3 ) define a quasi-Poisson structure ing how 3-algebra structures have y T d and whose degree = -flux model. In this way, the position 2 -flux background, are classified by the along the string winding directions of ) of closed strings propagating in the R ] R i T i j R i p δ = d vol ˆ 40 ∂ = H = ( (with } in the framework of double field theory: it is j -flux background in three dimensions. p in the contraction limit that sends M-theory to -flux background ijk – 4 – R , p ijk 2 i -flux R x R -flux is represented by a trivector or locally by a H R ε { R ) and = i , which is related to an x ijk jk , β k R = ( p ). Since the first homology group is ). The 3 x Z T ijk -flux R M, R to derive a star product quantising the phase space quasi-Po ( ], whose derivation we also review; we demonstrate in detail 2 3 s ℓ ~ 1 4 30 H with the classical brackets [ = } j M ∗ , x T i ] upon imposing a suitable gauge fixing constraint; we descri x { 30 In the parabolic flux model in three dimensions, the backgrou , there are non-trivial windings along all three directions d given by taking suitable “covariant” derivatives generalised vielbein containing the background metric and a globally well-defined bivector torus which is a circle bundle over the two-torus totally antisymmetric rank three tensor In this section wephase review space model and for elaborate the on constant features string of the qua In our situations of interest, a non-geometric string theor 2.1 Quasi-Poisson algebra for non-geometric string theory fluxes the cohomology class of the three-form a double T-duality transformation of a supergravity backgr 2 Quantization of string theory brackets of [ quantisation of this 3-algebra andand show how the it configuration naturally space enc nonassociative geometry from se IIA string theory, and furtherthe nonassociative apply geometry it of to configuration derive space induced quantum unc products to triple crossof products the and full use them eight-dimensional to membrane postulate phase a space which re it reduces appropriately to that of section radius of the M-theory circle.arisen Finally, in after other briefly contexts review in M-theory as motivation, in secti geometric flux; this sendswindings, closed and string hence winding the number momentum into modes in the Z (consisting of a three-torus first homology group of vector crossFourier products, momenta and that use definesthen them applied the in to pertinent section derive star a products. deformed Th su review pertinent properties of the algebra of octonions and winding numbers map to momentum modes of the and momentum coordinates background of a constant on phase space brackets proposed by [ JHEP02(2017)099 ) ) ˜ p , 2.1 ˜ (2.3) (2.7) (2.2) (2.4) (2.6) (2.5) x ]; the , p 15 , , x as n particular 14 , f ebra; in fact, hing brackets ) p , throughout this x 3 R . Strictly speaking, ordinates ( . J ∈ ∂ ) = ( ∂x I }} x x ∧ to a function ame the dual phase space I f, h = ( ˆ ∂ -flux background generate a is convenient to rewrite ( f { ∂x sider x R ometric background, we define ) g, 3) transformation [ or es the standard Nambu-Poisson h , x , ( k , near and satisfies the Leibniz rule. (3 IJ , J g ∂ } − { O . ) with ∂ j Θ -flux background and their dynamics, 3, is the alternating symbol in three x ) ) 1 2 ]. For ease of notation, in this paper , h , ( R x x 4 } 2 f ∂ ( ( i , , but as we are only interested in local ∂ 3 IJ ˆ f, g ⊲ f f ⊲ g ! T Θ i j ijk = 1 I δ ~ 0 x 2 R i − ) = 3 s – 5 – 2 = ˆ ℓ x k ~ + 3 p }} − {{ )( i, j, k I on phase space, the classical Jacobiator is defined as x , ⋆ f = i j ijk I h δ g, h } = ijk R x { f ⋆ g -flux frame and suppress the dependence on the dual ε ( I 2 3 s R ℓ ~ ˆ f, x and { = +1; unless otherwise explicitly stated, in the following

g f, g, h { , := f 123 ) = } ε x ( live on a three-torus IJ x f, g, h ) one finds { denotes the action of a differential operator on a function. I = Θ 2.1 ⊲ ). } have canonical Poisson brackets among themselves and vanis ˜ p J , which further obeys the fundamental identity of a 3-Lie alg ˜ p , i . Hence the classical brackets of the constant , ˜ , x ∂ x 3 ˜ ∂x I x R x { is the string length and = , and the totality of brackets among the double phase space co i s p ℓ ∂ , x For any three functions can be rotated to any other T-duality frame via an with coordinates paper. From the perspective of double field theory, in this fr considerations we take the decompactification limit and con dimensions normalised as repeated indices are always understood to be summed over. It which identifies the components of a bivector Θ = same is true ofwe the restrict star our product attention reviewed to below the [ in a more condensed form as coordinates ( here the coordinates The operators where one has where For the brackets ( By construction it is antisymmetric in all arguments, trili where the symbol nonassociative phase space algebra. 2.2 Phase space starTo describe product the quantization of closed strings in the as well as the ensuinga nonassociative star geometry of product the by non-ge associating a (formal) differential operat bracket on when restricted to functions on configuration space, it defin JHEP02(2017)099 . I = M x the ∂ l , (2.9) (2.8) ) (2.12) (2.10) (2.11) i . LM k ˆ g Θ , ◦ J . It should ˆ ∂ = ( f ijk n . IJ R k 6= , ) 3 s Θ
ℓ x L ( , the differential , 3 k g f I n ··· − [ f ⋆ g J J k ) and ) . l ∂ ); in particular, the −→ = ∂ of degree , −→ n ) ⋆ f ⋆ J k x t we first observe that n ] 2.1 ( tion ∂ ) k ··· S n ( IJ 1 σ , x . J Θ ··· I j up hich quantizes the standard on the left and on the right I ) = ( 1 ets ( I x ∂ , I −→ J ˆ ( x ∂ ←− , x I k J ∂ i I . n ⋆ ∂ ~ ∂ uantization of the quasi-Poisson k 2 J e, in general i x n ∂x ) i
n J x I k = ( ( − n ··· ∂ I Θ ) e IJ ⋆ k ~ Θ x ) e Θ ( ··· k I (2) f ( k σ ··· 1 + i ˜ I 2 f ~ J and [ 1 1 x k J e I 6 ˆ p 1 ) = ) I k ), with Θ x ⋆ x 6 π · x ( Θ . n by requiring that, for any ( d k I 3 is not an algebra representation: since the star ijk (2 n (1) i f 3, close to an associative algebra of differential – 6 – IJ I ). Weyl star products satisfy σ ⊲ g R ˆ ∂ , R I f ←− − I k Θ I 2 ˆ x 3 s x Z ˆ x , ℓ ~ ~ i 7→ I J ··· n ··· between two arbitrary functions on phase space, we k S = e ) on x 1 1 i f I I ∈ = 1 I X = i ) := − σ k + ˆ ˆ x ∂ ] = i 2.4 f ←− I j I ! J ( k = 1 ˆ x f ⋆ g ) e n ˆ x i = , ⋆ x for k n W i I − ( n
= x x J i ˜
e [ˆ f J (ˆ = ∂ x J ∂ ) can be obtained by symmetric ordering of the operators ˆ ∂p ˆ 1 2 6 f ∂ ⋆ f ) − k −→ Weyl star product IJ 6 = π 2.5
J d n Θ IJ ) = ] = 0 one can write ) as a definition of the star product, one may easily calculate (2 I I J +3 L Θ x ⋆ x i k x ∂ stand for the action of the derivative I ∂ 2.7 I Z I x I ··· ∂ x ←− = 0 it follows that JL 1 ( ∂ I −→ k Θ ) = := and x ∂ W x J ⋆ i ] ) can be written in terms of a bidifferential operator as defined by ( ) and ( )( ∂ lki J , k and ∂x I ˆ R f I 2.1 , x x 2.6 . Then l = I ) denote the Fourier transform of 3 I k ∂ f ⋆ g ←− x i k the standard Euclidean inner product of vectors. By the rela i k ( R [ ( ∂ k · ˜ To define the star product To obtain an explicit form for the corresponding star produc f ∈ 2 3 s ℓ ~ ) i l ( Let For example, = operators; in particular introduce the notion of quantum 3-bracket represents a Nambu-Heisenbergclassical algebra Nambu-Poisson bracket w ( quantum brackets correspondingly. Thus the Weyl star productbracket representing ( q with operator with which thereby provide a quantization of the classical brack where the sum runs over all permutations in the symmetric gro One may also write Taking ( where be stressed that the correspondence product that we consider here is not necessarily associativ since [ JHEP02(2017)099 ]. ]. ; it 45 45 , , ⋆ f (2.14) (2.15) (2.16) (2.13) mation 44 44 , , , , = 1 4 ) 4
′′ f , ′ l 2 k · ε ] for a systematic k 1 = × 18 shows that alter- ′ − k 1 f ⋆ ′ ) is completely an- x ( . ]. ]. This star product k · · p · x p , 45 k · l ) 2 x ′ [ R f, g, h 2 ~ R ( 3 s C 2 k,k ℓ ⋆ 4 6 s ( certain 3-cocycle [ ℓ ~ e Baker-Campbell-Hausdorff A B ) + i = ⋆ h ) restricted to Schwartz func- roducts the Jacobiator is pro- ′ bra is exemplified in [ , ense of [ ) x k , and unital, x s on phase space this property is · ) e · ∗ ε = 2 i ′ )
2.15 on through an associative algebra ′
× k nctions with function-valued (rather than f ⋆ g )) ) vanishes whenever any two of its ⋆ f in integral form through the Fourier ( 2 ( | k g ∗ k,k ′′ ]. The significance and utility of this ( ( x g ) ˜ | r products containing ours. − · B dinate generators, the star product is trivially , k k , 2.16 i 45 best behaved on algebras of Schwartz functions, ′ ) ( p = 2 integrable functions. For some accidental cases, nt for the present paper; see [ f ⋆ g k | ˜ nderstand more precisely the class of functions on f ( ∗ R x ) | B 6 ′ ~ 3 s = e , ) g ⋆ h ℓ 2 k 2 ( k, x | π 6 · ( – 7 – f ⋆ g d x − ′ (2 | k f ⋆ i p ⋆ 6 · − B e ) k ) A 6 π ′ )) ⋆ l ) := d ], whereas the nonassociativity of the star product is ′′ (2 x · + 44 , k k l ) , i Z ′ 4 e f, g, h if the star associator of three functions ( + ( )[ ⋆ ) = k, k x the usual cross product of three-dimensional vectors. Then ( A x · ) is Hermitean, ( 2.9 ′ l B ) )( k ( ′ j B k k 2.12 ] where it was derived using the Kontsevich formula for defor + f ⋆ g alone. The star product of plane waves is given by ijl ( k ε 44 g alternative ) := = ′′ i := ( ) and ˜ , k ′ ′ x ˜ f · k ) ε k, k ′ ( × A k k, k ( For later use, let us rewrite the star product To be more precise, alternativity generally holds only for fu B 1 with ( vanishes whenever any two of them are equal (equivalently In this form, theformula star to product the follows brackets from ( application of th which is antisymmetric in all arguments, and in fact defines a where and general analysis of the non-alternativity of a class of sta arguments are equal, ittions follows is that alternative. the However, forviolated; star generic in product smooth fact, ( function the simple example 2.3 Alternativity andA Malcev-Poisson star identity product is encoded in the additive associator first appeared in [ quantization of twisted Poissonof structures. differential Its operators realisati wasstar first product pointed in out understanding in non-geometric string [ theory nativity is even violated on the phase space coordinate alge distribution-valued) Fourier transform, suchsuch as as Lebesgue linear coordinatealternative. functions The or star powers products ofso considered we in single will this coor often paper make are which this the restriction. star It product is is interesting alternative, to but u this is not releva is moreover 2-cyclic and 3-cyclic under integration in the s transforms It is easy to see that ( tisymmetric in its arguments,portional or to ‘alternating’). the For associator. such p Since the function ( JHEP02(2017)099 , 2 1 = x } and 1 (3.3) (3.2) (3.1) to be 1 = (2.17) , x 1 f R p . a Malcev p , the left- 3 s { 2 ℓ . x f ⋆ g 4.5 ~ 3 4 t alternative R not . − 3 s 2 ℓ ~ 3 = ] is anishing values − 673 } p , , = x [ } onions, in this section ], which we review in C 624 f, g, h , f elevant in later sections, 30 { , } 2 satisfies the Malcev-Poisson 572 ), and consequently no star their own right. The reader om [ raic and geometric features of , } so alternativity can be restored in a . f, g, h . y phase space model of section does not depend on ip ahead to section dating various aspects which will } 516 f, g {{ C 3 ) for the three functions he form that we need in this paper. { e , f , } the Malcev-Poisson identity can be R p A 2.17 2 3 s e 471 h ℓ ABC ~ η A , . We return to this point in section k f, g, h = + 0 and 435 + } {{ 1 = ) one finds 2 ], a star product decribing quantization of the , 1 g p = ) can be alternative. 0 -flux coordinate algebra – 8 – , x is the identity element and the imaginary unit AB , 36 1 k δ 2.17 R , f }} x 2.1 1 − { = 28 = 123 = f, h = X { ). On the other hand, one has B } e ABC can be written in the form 2.4 A f, g, 7, while f, h e { { O ] that a necessary condition for the star product ∈ 36 ,..., X = 1 . Since 2 x A , ) vanishes by ( = +1 for = R of octonions is the best known example of a nonassociative bu h ∈ satisfy the multiplication law 2.17 O ABC ]. For any three functions is a completely antisymmetric tensor of rank three with nonv η A A e and 54 , k , is intimately related to the nonassociative algebra of oct 1 0 4 -structures and deformation quantization p k ABC 2 3 η . It follows that the classical string x Another way of understanding this violation, which will be r G The counterexamples always involve momentum dependence, and R 2 = 3 s 2 ℓ ~ 3 Here where algebra. Every octonion hand side of ( octonions g identity [ 3.1 Octonions The algebra coordinate algebra based on the imaginarybe octonions, important eluci for lateruninterested sections in and these technical which details are may interesting temporarily in sk section alternative is that the corresponding classical bracket consequently for the right-hand side of ( suitable sense by restricting to configuration space In this paper we are interested in the lift of the string theor algebra (beyond linear orderproduct representing in a the quantization phase of ( space coordinates to M-theory. As the conjectural quasi-Poisson structure fr 3 As a simple example, let us check both sides of ( written as In particular we will derive, following [ we shall take a technicaloctonions, detour, together with recalling their some related of linear algebra, the in algeb t is via the observation of [ JHEP02(2017)099 . = O ∈ (3.4) (3.8) (3.9) (3.7) (3.5) (3.6) (3.10) . 1234567 ε ebra of the , X,Y,Z 4567 D e , nishing values BED η 3256 ABCD , η FC δ , , 12 i , e 3247 − − i , 2 , f ; we will use this component i C − e e , ABDE ]] = BFD i η , η A e 1537 ] = ] = 2 i + i , , e ABC = EC η C , f , e . EFG δ 7 7 for any three octonions e 7 as the dual of the rank three tensor BD e η [ e ) can be rewritten as e δ ,
i + 1425 , ) f = 2 B , k , , 3.2 e . BCE e k AE A k η δ f e ) e YZ ECD ABCD k ( ijk η B η − f 1346 ε implies that the Jacobiator is proportional to ijk ]] + [ FD e ijk ε X and [ δ , and [ ε B + ABCDEF G O and ijk FB − – 9 – BE ε − − δ ε − δ , e − 1 B 6 1 7 A k Z e − − e 1 ij e e ) [ δ AD k 7 = A i ij ij , δ e = 1267 e BCF e − f δ δ ijk C η ij ε = e FCD 3, the algebra ( XY ijk δ = = = = η ( , 2 ε of quaternions generated by ] := ED i 2 j j j − ABCD δ , e e f f B satisfy the contraction identity H η EB ]] + [ i i i 7 DEC + δ ABCD e e e , e f η C = 1 ] = ] = 2 ( ] = 2 A ] = 6 j j j = , e e i ABC [ B η ), for the Jacobiator we get , e , f , f ABC e i i i [ η for e e f frequently in what follows. , [ [ [ 3.7 A O X,Y,Z ABCD e +3 is neither commutative nor associative. The commutator alg i η is the alternating symbol in seven dimensions normalized as e O := ] := [ is a completely antisymmetric tensor of rank four with nonva AEF = +1 for i C η f , e B ABCDEF G ABCD ABCD , e ε η η through A e [ The algebra ABC and the alternative property of the algebra the associator, i.e., [ +1. Together they satisfy the contraction identity Introducing η Taking into account ( where which emphasises a subalgebra form of the algebra where One may also represent the rank four tensor which can be written in components as octonions is given by The structure constants JHEP02(2017)099 is ′ and by k ~ cross is the 7 η (3.13) (3.12) (3.11) ), and A e R O × V A k ∈ ~ k states that ) ]: ′ uniquely up and is equivalent A . k ~ ′ := 53 as η
k H k ~ ) , O can be described × ′′ (C3) X , k ~ C ) = ( 2 (C2)
, ′ ′′ X ′ G k ′ ~ k ~ R k ~ . However, unlike the , k ~ η 7 ) ) he star product for the X ) make the vector space A R × ′′ ( · k k ~ k k ~ ~ . · in , y writing oss products [ sions it is the standard one V X They are intimately related ′ ill be the notion of a ion algebra k ~ = ( η + ( k ( ~ (namely dimensions 0, 1, 3 and 7. In − k ~ k × ~ is the Euclidean vector norm. − t, it is useful to note that it can ′′ R η k ~ k ~ k ~ and C , while property × X · ′ ′ ) . k ) ~ ) k k ~ k ~ ′′ ′ ) in a suitable oriented frame.  ] for nice introductions), generalising ′′ ). To help describe and interpret the ′ k ~ k ~ B k ~ k ~ p · identified with the homogeneous space 53 k ) it does not obey the Jacobi identity: X η ′ 3.3 , and 3.11 k ~ , where the action of = k ~ ,X V × | η k ~ ′ X k 33 ~ 3.10 k ~ ( ⊂ k ABC ~ | + ( X × η  6 ′′ 3 2 1 2 k S + ( ~ := – 10 – η ′′ = ] (see [ = k ~ A  × ′ ) η ]. Only rotations in the 14-dimensional exceptional ′′ 32 k ~ ) , where ′ k ~ ′ η 2 , due to ( k × ~ 56 ); ) k ~ × ε ) η introduced in ( ′ ′′ ,X η ′ on an oriented seven-dimensional vector space k ~ ′ k ~ imply that the products ( k × ~ × k k is the stabilizer of a unit vector in ~ ~ · × X η η k 2 ~ k ~ k ~ ( ( ABC ,X ( × G × k η ~ are linearly independent vectors, property k ~ k − ~ (C2) ( X ′ 2 · ;  | k ~ ′ is equivalent to the statement that the cross product ′ 1 4 = 3 , k ~ k ~ -structure k ~ and k ~ | 2 η − ) := ( = 2 G | ′′ ) × k ~ k ′ ~ ′′ (C1) | ) = , A k ~ k ~ ′ , (C1) ′ ′′ = k ~ − k 3 ~ k ~ -flux background, while in seven dimensions it can be defined ( 2 k, ~ | k, η ~ ( = R ( ′ η ′ ~ η × J SO(7) preserve the cross product k ~ k ~ ~ ′ J η X η k ~ ) the Jacobiator is given by ⊂ SU(3). ( × / into a pre-Courant algebra [ on a real inner product space [ 2 × · 2 k ~ 3.7 7 | k k G ~ ~ G R This bilinear product satisfies the defining properties of cr (up to sign) which has appeared already in our discussion of t This means that the Lie group ≃ 3 = ε 6 its norm calculates the area of the triangle spanned by non-zero if and only if to the statement that it is orthogonal to both choice of a cross product that can be written as ( dimensions 0 and 1× the cross product vanishes, in three dimen likewise cross products only exist for vector spaces of real three-dimensional cross product observing that (C3) which can be represented through the associator on the octon As usual property group using ( (C1) underlying geometry of the seven-dimensionalbe cross produc expressed in terms of the algebra of imaginary octonions b the well known crossto product the of four vectors normed in algebras three over the dimensions. field of real numbers as the transitive action on the unit sphere (C2) to orthogonal transformation) in a Cayley basis for vectors string theory An important related linearproduct algebraic entity in this paper w 3.2 Cross products with the structure constants Hence properties V S JHEP02(2017)099 ial and 7 1. To (3.18) (3.15) (3.17) (3.14) (3.16) . , R ′ ). Hence to define k ~  | ≤ ∈ η ′ V ~p k . ~ × | ) η k ~ XX ) A ′ ( k X k × ~ . On the other k, X ~ | k ~ , ( ′ . with V ′ η | k ~ = (  ′ = ~ O | B ~p ′ | k ~ ′ | k ~ Y, X ~p ∈ | k X ~ ′ sin ) , which will be crucial η ) on ) k

~ | | | k ~ ) sin O X k k

~ ~ × ′ | | | -structure on | | ) k 2 ~ ~p k k ~ ~ ′ with XX | | 3.11 XX k ~ G k, sin ~ −  A ( ′ ′ ′ k, e ~ sin η  ( − k ~p ~ ′ ~ ′ . ) = ( A B η ′ ·

1 − k ~ 2 ~ k k ~ B k | ~ ′ · k ~

X k | ~p  ~ X k ′ ~ ′ sin = XY k | | ~ | ( k | ~ | ′ ′ ′ | − | · ′ k k ~ ~ + k ~ k k ~ ~ | | | k | | ~ | X sin ′ 1 k ~ k ~ X 1 | | | | k ~ consisting of vectors | sin k k ~ ~ | | ′

| | | | | p ′ ′ sin sin sin ) k k k ~ = ~ ~ k k | | | | ~ ~ | V ′ | | | + sin k k ~ + ~ ′ ], appendix C to obtain a closed form for the k ~ | | sin ~p k ⊂ ~ X 1 sin 28 k, ~ ). This implies that power series in octonions | | 2 | 7 | − | ( – 11 – | . Multiplying two octonion exponentials of the ′ cos cos ′ ′ k k 1 ~ ~ η k ~ | − | | B k ~ | ′ O )) is well-defined independently of the ordering of ~ ~p | B + +

k ~ | := sin k ~ k ) and ~ ··· − | cos ) 0 | X | 1 cos , implies in particular that ( = cos k ~ k ~ ), one can derive an octonionic version of de Moivre’s | | − | X | | ′ YY = cos O p k ~ k ~ k ~ ( | k | ~ ) | X XX  ) we get ) we arrive at 3.2 ∈ ′ | sin ( ′ cos . By setting e k ~ X , it assigns the vector the octonion exponential function e | k ~ ′ | sin n | 7 k, ~ cos 3.2  ( cos | k ~p,~p k ~ ~ = ··· ′ ′ 3.16 | η | | B X ǫ 1 ( ], and in particular one can introduce the octonion exponent  ~ B k ~ k ! ~ X,Y | 0 (with − 1 | Y 1 ∈ X n = 48 X on the unit ball − ) cos ′ ( ′ define , 0 ≥ cos η cos ~p  ≥ X ) in a more manageable form, we use the cos 39 n ) and ( n η ⊛ =: e p ~p ~p, +  XY sin cos × ( ) to ′ ⊛ P := k ~ = 3.17 ~p X 3.15 ] n ′ is the seven-dimensional vector cross product ( e k ) = ~ 3.14 := 48 ′ X ′ X k ~ k k ~ , ~ e X X η ) together using ( e k, 39 ~ k ~ ( × ′ X η k ~ e ~ 3.14 B One can now repeat the derivation of [ To rewrite ( are readily defined [ parantheses for all hand, the Baker-Campbell-Hausdorff expansion is defined by form ( Baker-Campbell-Hausdorff formula for theorem [ where We can take ( for any pair of octonions for the derivations which follow. The alternative property using the multiplication law ( the quantity By comparing ( Let us now work out the Baker-Campbell-Hausdorff formula for 3.3 Baker-Campbell-Hausdorff formula and vector star sums any pair of vectors a binary operation function e JHEP02(2017)099 ) 3.13 (3.19) (3.26) (3.20) (3.24) (3.25) (3.23) (3.22) (3.21) ; these 3.2 . Using the . we introduce ) . 5.3 ′′ acobiator ( V | k ~ , ~p | ′ , / ) | 0 into a 2-group. . We call the binary k ~ | p ~p ~p, V ≥ ~ V ( η 2 ⊂ ~
⊂ J sin( ′ 7 k ~ 2 3 7 satisfying . ~p = B . 2 B · ~p

~p D 2 , , ~p ′ R · s ) = ′′ ′ e seven-dimensional vector star 3 scribe in section ) can be written in terms of the ~p p 1 ′′ − ~p ∈ ,~p − ~p η 2 ~p 2 η -structure as C ~p. ~p | 1 ′ A 2 η ⊛ ǫ ′ − η p 3.17 × k . It admits an identity element given ) take the form G 3 p ~p, ~p ′ ~p 2 ⊛ ⊛ ~p | ~p ], appendix C we can use the trigono- B ′ ~p η | 2 η ) p 2 = ~p. ′ ~p ~p − | ⊛ 28 ~p ⊛ 3.23 ( | − 2 | s ~p ′ 1 0 − η ~ ,~p 1 ~p η with − | and | 1 ~p = | − p ⊛ ~p 1 = ⊛ of the cross product from section ~p 1 ~p η ǫ ~p ABCD | 2 ~p | − ~p ~p η ⊛ η p | ~ ) over the entire vector space of 0 = ( = ~ ~p − 2 ~p – 12 – 1 ⊛ k = ~ | | sin 3 ′ ′′ − 1 we find ,~p − | ~ 0 = ) ~ (C2) A ~p ~p ~p 3.18 | ) = 2 = ) , following [ η 1 ~p sin k ~ = cos sin η ~p | , η ′′ − | k ~ V ⊛ − | 3.2 7 s p ⊛ ′ ⊛ − k ~ ~ | 1 and ( ~p 1 1 , ~p ∈ ) ~p B ~p ′ ′ ) = ′ η − ǫ η p ′ ∈ = k ~p ~ sin( 2 ⊛ k ~ ⊛ 2 η ,~p ~p p ~p ~p, ~p | k, ~ ~ 1 ( ~p = ′ (C1) ~p ⊛ − η ǫ ~p ~p k, ~ vector star sum ~p sin cos ) through A is indeed also belongs to the unit ball η ~ ( ′ 7 ′ ⊛ η , from section the ~p 3.11 B ~ ~p B η V 7 ) := ( 1 ~ ∈ ′′ ⊛ B − | ~p ~p 1 , ~p (C3) 1 is the sign of ′ – ) := ) on ) we find that the corresponding associator is related to the J ′ ± 1 to find that the deformed vector sum ( k ~ p ~p ~p, = ( k, ≤ η ~ 3.18 ) one immediately infers the following properties: 3.19 (C1) 2 ( ~ ) is nonassociative but alternative, making the ball s A ,~p η 1 ~ B ~p ≤ 3.26 ǫ 3.18 1 To extend the 2-group structure ( − which follows by properties From ( operation ( and so the vector It is non-vanishing but totallysum antisymmetric, ( and hence th vector star sum as metric identities for the cross product ( sign factors have a preciseproperties intrinsic origin that we shall de The inverse map is given by for Then for each pair of vectors the map It follows that the components of the associator ( where It is noncommutative with commutator given by the and using ( by the zero vector in and the inverse of JHEP02(2017)099 , 7 ξ of oc- := (3.29) (3.28) (3.30) (3.27) 4
σ ′′ k ~ X ]. For this, . e | ~ ξ ′ [ k ~ | 3 and k ~
/ C , ) ) X | 2 ] which is isomor- ′′ e k ~ , | ~ k ξ ); ~ ~ [ , , 2 ′ k ~ i ~ C , = 1 k ~ ξ X ( sin( i ( i , 2 k ~ η σ O ) ~ = B − j ~p +

σ for ′ ) = = 2 k i k , ~ and e ) one finds that the left-hand k ~ ′′ δ η η +3 η η i ′′ } } e algebra i i k ξ ~ − , ~p ~ B × ′ 3.30 i X , ξ , , σ k ~ σ − e 4 , := 4 ~ C k j σ
p ~p ~p, ) i σ
ξ δ ( j { 2 { ′′ ′ σ ξ η . k ~ k ~ + ~ − k i ) and ( A . X ) , ′ δ 4 k e ABC )

k k ~ , σ , , σ ′ η − ′′ σ k ~ 4 k k

k k 3.29 ~ + k ~p X jk ξ σ σ ′′ ij i ξ and k ijk e ~ k, and ~ η ε = 2 ε ~p j ( i ε jk 7. This bracket is bilinear, antisymmetric and ), and after a little calculation using the identi- ijk ijk i – 13 – ⊛ δ η η η − ε ε ε ~ } ) B 12 ( 12 ) = that it is violated in general on ⊛ k ′ 4 ′ B ξ ) is satisfied for monomials. However, we can show ) 3.15 − − η σ k ~p k ~ ′ ,..., . Using ( ~ ξ , ξ i j B η 2 ~p 2.3 ijk δ = = 12 ( = 12 = 12 = 12 = A k, ~ ξ ε ⊛ 2.17 ( ξ η ijk η η η η η η = 1 η 2 2 ( { ε ~p = ⊛ } } } } } } ~ ) := ( B 4 4 4 − − ); k k k

) and ( A ~p h ′′ ( 1 k ~ k ,

~ , σ , σ , σ , σ , σ , σ ); = = 2 = − , j j j j j j − ′ R ~ k ~ 3.14 , η η η k ′ , ~ and , ξ , ξ } } } , σ , σ , σ , σ sin i i 0 ∈ k i i ~ ~ i i j j j 1 ξ ξ ( k, ξ ~ ξ σ σ ), − { η ( { σ { , ξ , ξ A { { ( , σ { η i i ~ ξ i B ) in components as 3 η ) = ξ ~ σ ξ 3.5 σ A ~ ) one finds for the associator { B ′′ { = { k ~ = − 3.28 k , ~ ′ g 3.25 k ~ , ) with ) = 1 k, ′ A ~ ξ ( ξ 0 ) = k ~ ~ η ) the non-vanishing Jacobiators can be written as = ~ k, k, ~ ~ A ) and ( ( ( f = ( η η 3.10 ~ ξ ~ ~ B Perturbative expansion: B The associator is antisymmetric in all arguments. 3.20 The Malcev-Poisson identity ( (B3) one may rewrite ( (B2) (B4) in an analogous way as in section where ties ( One can explicitly computetonion the exponentials products using ( satisfies the Leibniz rule by definition. Introducing consider (B1) Consider the algebra of classical brackets on the coordinat 3.4 Quasi-Poisson algebra and Malcev-Poisson identity Using ( phic to the algebra ( JHEP02(2017)099 . ) V f . η Π 3.9 ived ⋆ ∗  1 (3.33) (3.31) (3.32) T ) is the the pre- − = 1 ′ ) k | f ~ ′ k, k ~ ~ ) underlying dentity ( | ( quasi-Poisson η 1 = ~ 3.5 ~ η B ions by defining ~ ξ · f ⋆ cot( , ) | ′ ~ ξ . Hence the Malcev ′ and k ~ · , ]; this is also implied k ) ~ ). Consider the quasi- A 0 O ~ | ′ ( ∂ f ~ . ξ k ~ η [ ∂ξ ~ algebra ( ~ B η k, ~  C i ( 3.28 } = η , we have ′ ~ D 2 B | A s ( i k f ) e ′ f, g ∂ ′ η k ′ A ~ { , and unital, | of the deformed vector addition k ion of the Malcev-Poisson iden- ~ ∗ k ) e ( = it provides a quantization of the ′ ˜ = f g ⋆ f k ~ A η 3.3 g ( 0 ~ − ⋆ ) + ψ ) is alternative on monomials and g | C while for the right-hand side one has = ∗ g ′ ) ˜ k ~ g ,

(B3) k ~ η 3 k | ~ the structure of a symplectic nearly Lie f ∂ ( 3.32 D σ = ). We have ~ ˜ B ) f 3 f ⋆ ∗ V ′ ∂ ξ with ) k ~ 7 Π ′ A g = ) 3.32 ∗ cot( k ~ η 24 k, ~ π | η 7 T ( ∂k ⋆ ′ ABC − d ] η from section (2 f ⋆ – 14 – k η ~ | B B = A 7 ~ ψ ∂ f, g ξ ) η k ~ 7 π } ∧ D giving η d (B2) AD ξ We conclude that the Malcev-Poisson identity for the A (2 } δ η = 2 . 7 ′ ψ } ) ) η + 3 k ~ B Z ⋆ C and π 7 f, h ) on the finite-dimensional algebra ] σ ξ E ~ { d , ξ ′ 3 (2 i ξ A k 3.2 explicitly using ( ) = f, g ξ ]. . [ ): defining [ − ). It can be regarded as a pre-homological potential on { ). By definition it is the Weyl star product. Z f ABC ~ ξ f, g, (B1) 0 18 η 2 { ) and canonical Poisson bracket; then the corresponding der η )( 3.4 − = ADE σ → ⋆ g B ; in particular, by property 3.28 lim η ~ 3.28 η 3.26 ) is violated. This is in contrast to the well-known fact that 2 := = η A ξ O , ξ ~ Θ ξ η f A ]] − f ⋆ Θ η 3.28 B ψ ( ⋆ = implies that the star product ( , ξ stands for the Fourier transform of the function 0 = 24 ( ) of imaginary octonions defines a Malcev algebra, due to the i ~ A A ) on the infinite-dimensional polynomial algebra ξ = ξ ˜ η ) holds for octonions, while it is violated in general for the f k ~ ]. We can extend this structure to the entire algebra of funct } 3.5

) is given by 3.28 D 49 , f (B4) ) 2.17 η -symmetric star product ′ } 2.17 ), this star product is Hermitean, ( 2 k ~ ′ A G k ~ k, ~ ( ∂k Let us calculate Due to the properties k, ~ η ( f, g, h B η ∂ ~ by the general results of [ Property It can be regarded as a quantization of the dual of the pre-Lie {{ identity ( classical brackets ( Let us now work out the quantization of the classical bracket quasi-Poisson bracket ( Schwartz functions, but not generallytity because discussed of in the section violat the octonion algebra a star product through defining the brackets ( structure ( 3.5 Poisson bivector Lie algebra ( and the multiplication law ( and after some algebra one finds deformed vector sum ( where again B side of ( with coordinates ( 2-algebra [ brackets are [[ JHEP02(2017)099 . into
r for ′ 3 ], (3.37) (3.38) (3.39) (3.36) (3.34) (3.35) l · . With R 38 3 ) k according R ⊂ ~ ξ ( − 3 ∈ V ′ B k ) from [ ⊲ f , we arrive at i · ′ ) on this three- is preserved by k ε l  k ~ ) reproduces the
, ⊛ ε 1 l = ( 3.18 ) defines a Courant ). ′ 4 × − · 3.26 k k ) , , ) 2 − 3.30 , whose imaginary units i ε / ′ l m ( ~ 1 ξ l H , throughout. In particular, × 4
, △ k ε = ( , 0 tar sum ( ~ r ( , 0) , l × + 0 0) , SL(3). In particular, reduction ′ ackets ( , , D ctor star sum . This reduction is implemented sional vector space 0 ) ξ k = 0) reduces the nonassociative 0 , ∈ ′ ′ , ε coth( l ′ 7 q g , l with e 2 × l ε ε / ( and integrating over l η ~ 1 ξ ε ⊛ for × = ABCD ), where ], × − B 4 l q η △ 4 i . To see this reduction at the level of our gives ′ 3.3 38 2 k l k f ~ i , k ~ e , ε e = k ]. The cross product 1 7→ ,
× 28 l ~ ijk − ξ ) the seven-dimensional cross product can be 56 0) = ( 0) = ( 4 [ ε k ) now vanishes by a well-known identity for the 0) , , k = 12 △ , , – 15 – 0 0 3 3.6 = ( contains a closed SL(3) subgroup acting on these ′ 0 , ,
η ′ ′ R , ⋆ k l A ′ k ] 3.13 ~ q η ( l ] = 2 and 4 ∂ in this case, acting transitively on the unit sphere ( C B ( ) from [ j η k ) × ′ η ∂ from section , k 2 , ξ l ~ ξ , e × − ⊛ , i C G ⊤ B with l 0) ∇ e ξ ) from ( ( k , 0) 7 (equivalently · 1 , ξ 0) ε , ′ , 4 0 ⊂ ε , ~ ξ − A , 6 0 k B (B2) 3.4 ( 0 g ξ J l , , [ ( , ABC ( l 5 + − η ( q , ′ ( η ~ ξ ~ 7→ is the flat space Laplacian in seven dimensions. In particula k preserving ~ B . ε △ 3 k 2 to the associative algebra of quaternions A = 4 , + i A R × ∂ G l ξ O A A k A g ξ ∂ −  + (2) Lie algebra [ 7→ ′ = l l su = ~ 2 ξ ) is now associative; as a consequence, it makes the unit ball ε f ∇ SO(2) in × = 0 for / η l 3.23 = ⋆ A ~ ξ A e = ξ ′ △ SO(3) k ~ Similarly, the reduction of the seven-dimensional vector s η ≃ generate the × 2 i k ~ the corresponding Jacobiator and so yields the expectedin three-dimensional all cross of product our previous formulas by simply replacing to the components of section S a non-abelian group. The reduction of the deformed vector su vector products, consider the splitting of the seven-dimen e to the three-dimensional subspace spanned by algebra of octonions components as the full rotation group SO(3) three-dimensional vector sum dimensional subspace reproduces the three-dimensional ve cross product in three dimensions; as a consequence, the pai written in terms of the three-dimensional cross product as Setting 3.6 SL(3)-symmetric star product The symmetry group algebra structure on the vector space the Jacobiator one finds which thereby provides a quantization of the classical 3-br where respect to this decomposition, by using ( Taking into account property which by ( JHEP02(2017)099 ) is M e (4.1) 3.32 of ∩ g on this d f, g −→ · · · through the oduct which ) 0) from ( M , Z 0 , f M, ξ ( )( 1 g − k ions sending membrane ε he noncommutative and H he second homology group f ⋆ ! π e theorem K¨unneth there is a ], beginning again with some tion of the quantum mechanics si-Poisson algebra for the four- ct ( , by showing that it reduces to −−→ is dual to M-theory on the total 2 30 whose total space is the quotient ) heory background which is dual to 1 Z M S M, to derive a suitable star product which = 0, the Gysin sequence collapses to a ) of the fibration. For instance, in the ( π ] for the quantization of the dual of the 2 3 Z  e f M M over − 38 k / g ) it follows that, for functions M, , H ( f M 2 28 ), the evident similarity with the terms in the ). More generally, for an automorphism  H 3.39 – 16 – e Z are generally related to those of ∩ ∈ 1 3.36 , wherein -flux background −−−→ S 1 e f M, M ( R S ) -bundle 2 Z × H M of the circle fibre translates into the string coupling con- ⊕ ). From ( M, M ( R in the weak string coupling limit which reduces M-theory to ) k -flux background of section = Z ∈ R H 3.27 2.2 λ f M M, ( ∗ 1 π H −−−→ ≃ ) ]. We demonstrate explicitly that it is the lift of the star pr ) Z are the usual pushforward and Gysin pullback on homology, an ) will be crucial for what follows. Z 30 ! π f M, (2). In the general case ( ( f M, 2.14 k su ( 2 H and H ). The homology groups of ∗ . T-duality transformations become U-duality transformat , where the radius Z s π g M f M, ( 2 · · · −→ collection of short exactsplitting sequences, and in particular by th case of a trivial fibration we can define a twisted lift to an the cap product with the Euler class where over 4.1 Quasi-Poisson algebra forLet non-geometric us start M-theory by fluxes reviewingdimensional the non-geometric derivation of M-theory thegeneral background classical considerations. qua from String [ theory on a background quantizes the string theory the star product of section IIA string theory. We applyof this construction M2-branes to in the descrip nonassociative the geometry non-geometric these background, membranes probe. as well as of t quantizes the four-dimensional locally non-geometric M-t a twisted torus [ reproduces the associative starLie product algebra of [ vector sum ( In this section we use the constructions of section Gysin exact sequence with vanishing associator ( stant wrapping numbers to momentum modes, whichH are classified by t 4 Quantization of M-theory three-dimensional subspace, the corresponding star produ space of an oriented circle bundle JHEP02(2017)099 , Z f M ) = ∗ and ⊕ could (4.3) (4.2) Z T Z g ng the M M, ∈ ( ) = 1 xceptional ) generally x Z H Z M, M, ( ) = ( 1 is the alternating 1 Z H ) where H 4 M,    the remaining indices ( µναβ after a suitable choice , x ) = 2 R ε = 0. As a consequence , which is related to an x Z 3 s H ℓ 4 3.4 0 p of the possibly non-trivial mponents of the M-theory 3 ραβ M, λ ( ) = ( trained to a codimension one ackground metric and three- on 2 µ p , the lift of the non-geometric , H x that would allow for non-trivial 4, where 3
-flux background. ) shows that , 1 2.1 tended application; for example, 3 R , 4.1 R 2 , π n λ 0 0 3 s , -fibre which are dual to momenta along λ ℓ 1 ] that, generally, the phase space + 2 = 1 S ] for the parabolic toroidal flux model in -direction, i.e., p = 0 30 4 , t = +1. Note that this reduction only occurs ) is a twisted three-torus. In the M-theory lift ··· 3 30 x µ x 1 p ( -flux background is only seven-dimensional and M n ~ 0 0 0 1234 = 0 there is no torsion in the homology or coho- – 17 – R µ, ν, g ε , i.e., a choice of linear functions. For this, let us λ d 7 , − R µ,νραβ    R 7−→ µναβ ~ ) 1 : it is derived by taking suitable “covariant” derivatives , with local coordinates ( 2 , wherein 1 , t R ε ] is that the classical brackets of this seven-dimensional S . The Gysin sequence ( x = 2.1 ( = 30 Z
× and duality Poincar´e-Hodge implies µ,νραβ ∈ 3 AB M R 4 ), it is proposed in [ T n Λ ,µναβ Z = 4 = , so that all two-cycles are homologically non-trivial. R Z ) classifies “horizontal” wrapping modes dual to momenta alo = f and M, M M ( Z ⊕ Λ 2 -action R 7 matrix Z Z H M, ∈ × -flux can be described locally within the framework of SL(5) e ( t ⊕ 2 ∈ R , Z H e M by the ) = ∈ R , by the Poincar´eduality second homology group is Z -field after U-duality. Its first index is a vector index while 1 x × C S along the membrane wrapping directions of a trivector Ω This proposal was checked explicitly in [ In the situations we are interested in from section The main conjecture of [ M, The precise sort of automorphism should be specified by the in ( ∈ 4 M -fibre. , whereas 1 -flux in this case are 1 µν 4 ˆ S H introduce the 7 where form SO(5) rotation of the generalised vielbein containing the b M phase space are givenof by affine the structure quasi-Poisson on brackets the of vector secti space to the four-manifold three dimensions from section define a completely antisymmetricEuler rank class four tensor. Because string theory which does not contain thewrapping requisite modes non-trivial dual two-cycle to momenta along the reflecting the absence of momentum modes in the dual of the locally non-geometricsubspace background defined in by M-theory the is momentum cons slice field theory as a quantity be an automorphism preserving some background form field. the phase space of the M-theory lift of the of mology of the torus lacks a momentum spaceR direction. The only non-vanishing co x ∂ in the presence of non-trivial flux: when symbol in four dimensions normalised as classifies “vertical” wrapping modes around the JHEP02(2017)099 . - s R g , -flux (4.5) (4.4) (4.7) (4.6) D in the ′ 1 R − D Λ 2.3 ′ . , D
′ ξ C C ′ 1 , ′ i − C ~ B 0 which reduces p ′ veniently take to Λ λ A ′ = 0 which shrinks η − a, which is another antum level. As in → C ′ ′ 1234 , , s λ , B 4 4 ′ , g CC
i A R j Λ ) of the M-theory η x x 3 s ) of the string theory ′ ′ 2 R σ 2 k ~ i oordinates 3 s λ ℓ 4.7 λ δ , ℓ 2.4 BB BB
3 entity from section = = Λ g limit − Λ j λ ′ ′ i p is central in the algebra defined λ λ x } k } i p 4 i i AA AA δ k j ) and ( x , δ Λ ) and ( Λ , p , x − σ 4 4 4.6 , k λ x x := 2.1 := k p { R , { 3 x j i 3 s k , δ 4 − p is non-degenerate as long as all parameters 4 4 4 λ ℓ ABC , x ,ijk x ABCD λ Λ k 4 4 incorporates the string coupling constant p k λ ,ijk x 1234 R , 4 and ij and λ 4 ~ ,ijk – 18 – 3 s 1 ε R 2 4 ℓ ijk R 2 2 3 s 2 ε k R ℓ ~ 3 s λ λ with = 3 2 x 2 2 3 s 3 . ~ with 3 2 λ λ ℓ λ ℓ ~ ~ ξ ~ jk 3 3 3 k − − i , Λ ] is that in the contraction limit p k 4 = = = = = 3 = 0 = C ) one obtains the quasi-Poisson algebra 30 p λ ε := x D λ λ λ λ λ λ λ ,ijk
+ x } } } } } } } ijk 4 4 4 4 k k k k p 4 3.28 , R x λ ε 4 ABC , p , x , x , x , x , x , x 2 i 3 s j j j j j j j j λ ℓ ~ − δ , x ABCD , p , p , p , x , x , x , x x i i i i i i i λ = = = p p p p = 2 p x x { { { λ λ λ { { { { 12 λ = = 0. } } } } j j j −
λ B A , p , p , x = i i i 3 identity matrix. The matrix x , x p x x λ A × { { { } x = the classical coordinate algebra here is not a Malcev algebr C { ~x , x the 3 3.4 B ) and so may be set to any non-zero constant value, which we con 3 = 1. In the following we will extend this observation to the qu , x 1 4 The crucial observation of [ 4.6 A x x { are non-zero, but it is not orthogonal. Using it we define new c which can be written in components as the M-theory circle to a point, i.e., the weak string couplin M-theory to IIA string theory, the classical brackets ( From the classical brackets ( section be way of understanding the violationcontraction limit of the Malcev-Poisson id where we recall that the M-theory radius background reduce to theflux quasi-Poisson background; structure in this ( limit the circle fibre coordinate with the components with The corresponding Jacobiators are by ( JHEP02(2017)099 f λ 5 ⋆ (4.8) (4.9) ) the (4.13) (4.10) (4.12) (4.11) A x rator as 3.32 ), we can mials and  p . 3.26 , ∇ D .
], eq. (3.30). ~ x ~x , ~x λ 30 ) ˜ 1 al M-theory phase △ − − ~x 2 . ( Λ , ) from ( ~ · g ′ ABCD 4 -symmetric star prod- )  ), we calculate ) k ′ 2 ~ ∂ λ p χ ∂ k ~ ∂x −→ 2 3. We have also corrected the G
k, 4.6 ~ rs △ ~ ], eq. (3.30). ( − A . + i R k,Λ 2 ~ η ˜ 30 ∂ ∂ 3 s ~ ~ ←− Λ  B h those of [ ~x ) ℓ ( 1 λ 1 η in [ ~x 3 = 12 ~ B − )+ i λ − 2 ˜ i + ∇ Λ λ λ ∂ 2 ⋆ · −→ 2 4 = ] p / 2 ~ ξ Λ ~x 1 ) to get ~x ) e ∂

, C ∂x i ( ′ ) ]. For the same reasons as ( ˜ x △ k − ~ , x ~ ξ R − ( 45 ∇ 3.34 B ] by a factor of ∂ , 3 s g ←− ~x )( ⊲ f ℓ ) ˜ 30 Λ Λ R ˜ , x 2 A i coth △ g k ~ 3 s A λ x ( − 2 η A ˜ x ) in ( ( f / ⋆ x η + λ ℓ 1 ~x = ˆ ~ B 7 Λ ′ x 4.4 ˜ – 19 – ) f 2 1 p △ f k ~ ~ − π  7 λ ). For this, we use the  △ Λ d ⋆ ~ 1 ( (2 1 + 2 R · − ~x 4.6 A ) = ( ~x 3 s B and [ 7 x i ˜ ℓ ) = k △ which is missing a factor ∂ ~x ~  7 π λ  )( C d 2 ⋆ ) e g := ] (2 x ~ λ B 4 4 ~x ∂ λ
C ( ∂x ~x , x x Z f j = ˜ f ⋆ ABC △ ( 2 , x ~x λ BA i ˜ ) = x ∇ ) = χ ~ ABC Λ ~x λ ~x ). Using the deformed vector addition ) provides a quantization of the brackets ( = ) may also be written in terms of a (formal) bidifferential ope ~ )( )( ~ ξ + i ~x g g := 4.9 A Λ ˜ 4.9 λ λ
△ ( x ) is unital, Hermitean and Weyl, and it is alternative on mono A f = 2 i ˜ f ⋆ ∂ f ⋆ = λ ( ( 4.9  ⋆ ] A ) := ˆ x B = ~ ξ ) as ~x , x ( ) to define a star product of functions on the seven-dimension ˜ ∇ A Λ 4.8 f x [ 3.32 To show that ( Our definition of the Jacobiator differs from that of [ 5 Written in components, these quantum brackets coincide wit where which identifies it as a cochain twist deformation [ expression for the 3-bracket [ star product ( Schwartz functions. write ( space by the prescription where by making the change of affine structure ( with We will now quantize the brackets ( 4.2 Phase space star product We have also introduced the (formal) differential operator and We thus find for the algebra of star commutators and Jacobiato The star product ( uct ( JHEP02(2017)099
| ) . ). ′ ′
| l
k ~ k ′ ) (4.14) (4.15) · Λ ~ ′ k ~ | ) (4.17) 3.18 l k ~ λ Λ k · ~x. | − − ′ 4 · l ′ k , l
sin − ′ · 4 − ′ k ~ ) and (
l k ′ | ( · k + k ~ R k 4 | k 4 Λ 3 s k ~ x k ( ~ 3.26 | k ℓ ( 4 ~ ry lift of the quan- 3 Λ · | x = λ ) + 2 p ′  , in the sense that the ~x l ~ p sin R 1 4 ~x , 2 = 0; this calculation will 3 s

− k ~ 2.2 1 ′ Λ k ) + λ ℓ λ Λ −

~p ′ − ′ · Λ Λ R η k l omprising ( ~p ·  3 s ε ′ 4 ⊛ ) + η
Λ ′ k . ′ l × ~p ( Λ 0 limit. We do this by carefully ⊛ λ ℓ k ~ ε · ~p k 2 1 ~

Λ ,

( p Λ x × 1 ~p ′ → · Λ ~

η l = λ − ~p ( p ~

λ 

= · × = 0 (4.16) . ′ ) Λ R sin −

k p ~

2 ) + 4 ~p 0 3 s | ~ ′ ′ Λ ′ Λ  1 λ ℓ ) Λ k η ~ ~p ~p 0 k 2 4 = | R k q ⊛ η − ε Λ η ~ → k ~ | 1 3 s Λ 2 ) λ × ⊛ Λ ⊛ ′ ℓ + × R k ~p ~ Λ – 20 – ) in the ~p | l 2 k Λ 3 s ′

Λ 0 Λ = lim = ℓ ~p ε λ ~p 1 ~p

−

→ 2 ~x − lim sin ′ λ 2.14 × 2 + 0 λ ~ l

1 0 2 k ε → Λ − ) in the contraction limit sin ( + lim λ k ~p · → Λ ×

lim 2 0 λ · ) somewhat. For this, we need to show that the quan- p R . Evidently k k -flux background from section → −
2.15 3 s ( lim R λ ′ 2 ℓ Λ R R · 4.9 1 ~ 3 s 3.6 ~p ℓ 3 s x + ℓ η q + 2 λ  l reduces to ( × +

2 = k ~ 2 Λ ′ ) we thus find Λ ~x l ~

~p ~x ~p Λ 1 ′ 2 Λ λ 1 η − ~ ( ~p

− 3.20 | ′ ⊛ Λ Λ λ η λ ) we find the limit k

Λ ~ ( · ~p ) reduces to ( ) one easily finds Λ ′ ⊛ | · Λ ) q λ Λ η ~p k ~ ′ ~p |

Λ ) we compute 2 1 ′ 4.9 ⊛ k ~ 4.14 ~p η ~ q 1 Λ k

3.36 ~ |  Λ − ⊛ ~ ~p Λ 4.17

Λ k, Λ ~ η sin sin sin 1 ~p

Λ − × 0 ~ ( = := k η ~ → lim sin ~ λ B First, let us introduce We will now show that this quantization is the correct M-theo Λ Λ 0 ~p 2 → lim λ in the conventions of section and tization of thestar string product theory ( tity and from ( so that from ( also unpackage the formula ( Next, using ( These limits imply that From the identity ( computing the contractions of the various vector products c JHEP02(2017)099 . ], λ = ⋆ ] λ 44 = 0. } , (4.19) (4.18) (4.20) (4.21) 2 ′ 4 0 limit f, g k f, g . { →
= ) ~x λ is central in ′ 7 4 or 3-cyclic in k d k 4 ): · ory we restrict x l R tive phase space p · 2.16 − ) ′ l . ′ ⊲ f . l · ) uct A k -direction; they reduce λ + . ˜ ( ∂ ( 4 non-geometric M-theory l ]. It is natural to ask for ) 4 x
O 8 x ′′ ~x e + ( ssociative theory of gravity 2 ˜ uction of the M-theory asso- , k ~ △ 4 ′ nassociative theory of gravity tric string backgrounds [ x 2 ], for matching with the expec- . eory [ ~ ) . = 1 + ) + 45 k, k ′ 4 = 1. In this way the ) ′ ( k D x , this expression coincides exactly 4 k 0; as before we may therefore set it A 4 ~xχ x ε )( + x 7 = 4 → × d k ~x λ k ), ( with f ⋆ g 1 Z · + ( − 2 ), and hence the coordinate p Λ x ~ 2.15 λ · · ( ) = ( R ) )
) of Schwartz functions one has 3 s ), in contrast to its string theory dual counter- O ~x ′ – 21 – g ′′ ℓ = 6 k k )( ~ D + 4.5
4.9 g ~ 1 λ + f ) is not closed with respect to Lebesgue measure on ) to the six-dimensional hyperplanes f 2 λ ,Λ ′ 4 A k k − ~ f ⋆ 4.9 x x 4.9 f ⋆ ( D , and no modification of the measure can restore closure. = ( − = 0 1 k, Λ ~ f ~x f → − lim λ 1 λ Λ λ ( D ~xf g ⋆ − ⋆ η 7 ) = 0, this is no longer true for the quantum brackets [ Λ A 4 ~ d = A g · ) a simple integration by parts shows that x x ) g 0 B R ) reduces exactly to ( ′ λ → ~x k ~ lim 6= • λ f ∂ 4.10 7 4.9 d g f C λ x k, Λ ~ ) coincides precisely with the string theory associator ( = 0, has the undesirable feature that it is neither 2-cyclic n Z ]; these properties are essential for a sensible nonassocia Λ ( λ η 45 ABC ~xf ⋆ 3.27 ~ B 7 λ ) we see that d 0 ). In the dimensional reduction of M-theory to IIA string the ) at (2 → R A lim λ 4.10 2.14 2.15 ~x∂ Although for the classical brackets ( Putting everything together we conclude that To overcome this problem we seek a gauge equivalent star prod 7 , i.e., d 7 with ( of the star product ( The issue is that the star product ( In particular, using ( We are not quitebackground done because with the our star phase product space quantization ( of the 4.3 Closure and cyclicity R the sense of [ the Fourier space integrations in ( Up to the occurance of the circle fibre coordinate R tations from worldsheet conformal fieldand theory in in non-geome the constructionunderlying of the physically low-energy viable limit actions ofanalogous for non-geometric features a string no involving th M2-branesunderlying the and low-energy a limit putative of nona non-geometric M-theory part ( formulation of the quantisation of non-geometric strings [ the algebra of functions to those which are constant along th to any non-zero constant value, which we take to be the star product algebra of functions in the limit From ( ciator from ( With similar techniques, one shows that the dimensional red JHEP02(2017)099 D tional (4.22) (4.23) (4.24) . ), gives the 6  4.23
| ) ′ | k ~ ) , but it is no longer ′ , ~x f ) are simply unity in k ~ ) and ( 1 λ ) we find − ), k, Λ ~ • Λ 4.24 · Λ 4.20 k, Λ ~ 4.6 ( ) ⊲ f . ′ 4.10 η Λ . = 1 k ~ (0) = 1 ) still dimensionally reduces ~ ( D B ⊲ f . . | 6 η D f A ets ( m ( ~  ~ k,Λ A implementing this gauge trans- ~ B λ . ˆ | x
˜ } ∂ 4.24 Λ 2 ) 1 ( ′ D / η x − 1 = 1 ~x ~ D sin B f, g ]. Order by order calculations, see i )( λ D ˜ 1 ~xf g , with { △
• ); in particular, the star products of 7 | − 38 stands for the (formal) derivative of ~ ′ = ) as d = f ) e D ′ k ~ ′
f ⋆ g 2 λ D 2.11 k ~ f Z Λ • ~ ( | 1 | ] 4.21 ~ D g ′ 2 sinh ~ = , with the help of ( i ) ˜ − k ~ λ 1 f ) = ( g − f, g t ⋆ k ~ [ − Λ ~x A ( – 22 – λ
f , the extra factors in ( √ A ˜ sin | | f • 2 )( t , where λ x /
k ~ g | 1 ~x ⋆ ~xx 7 4.2 ′ C and unital, D k ) 7 Λ ~ λ 0 ˜ k ∂ ~ ˜ | , A ~xf coth △ d • π 7 ′ 1 ∗ Λ 7 → x ~ t | d ~ s , we have − f f d (2 D R lim =constant ( ,ℓ ~ √ A = 2 λ 2 D ~  0 7 x ~ = • − Z ) f / k ~ → = ∗ 3 s = 7 lim π f D = λ λ ℓ sin = 3 are no longer given simply by the Baker-Campbell-Hausdorff g d f 2 • λ 0, and so the star product ( (2 D A ~x ~ t •  λ D · = A d ′ • 2 d A x k ∗ ~ → ⊲ x Z × i ) − 1 A contains only even order derivatives, and in fact it is a func g x − e D λ ~xx λ 7 D D λ ) = ] = • d • C ~x f ~x R · )( , x g k ~ 1 i λ − ). Since • ~x D ˜ f △ ( 2 ), ~ ], show that ( 36 D We may therefore write the star product ( 2.15 = with respect to its argument, together with the explicit for Using now [ D The requirement formula. However, it is now closed, plane waves e the contraction limit to ( elementary Cauchy problem e.g. [ This star product still provides a quantization of the brack formation is analogous to the procedure used in [ The construction of the invertible differential operator Using the limits computed in section whose solution finally yields It is Hermitean, ( a Weyl star product, i.e., it does not satisfy ( JHEP02(2017)099 ) ) is 4.6 e the . (4.25) λ 4.24 ⋆ ] rovides a h D enable a con- g, λ D • al f, ); dynamics in the D and statistical prob- [ -flux background. It 1 a R − 4.24 ψ . D ) ssociativity (but not non- h f a normalized real-valued ct ( , = ular, the desired 2-cyclicity λ ) us treatments of nonassocia- λ • a ctions ative algebras are not alterna- • ds. ] ψ g t star product rinciples which are heuristically mechanics, along the lines given unctions vanishes, and together ( λ . λ • ) that quantize the brackets ( • λ f . • f, g, h ( ] . 4.13 λ ~xf λ • 7 • ] ∗ a d ~xf S 7 f, g, h , f [ d is then implemented via the time evolution Z ~xψ H 1 6 [ 7 R H = d i ~ ) the star associator and Jacobiator transform = – 23 – h ) = i Z = λ ) and [ f • 4.21 h a are real-valued functions on the seven-dimensional h ) ∂t µ ∂f g D f f, g, h as λ ( g, a ∗ a • λ X • ψ D = 0. The closure condition can be regarded as the absence f A ( λ = λ f, • • i ~x ] D a 7 f ( h d ψ λ ⋆ f, g a [ preserves this subspace it follows that the star product ( µ Z A ) is alternative on the space of Schwartz functions, and sinc 1 ~x a 7 D − d ], section 4.5 for a review). In particular, this framework p 4.9 P D R 46 = ) = S . Expectation values are computed via the phase space integr ) we arrive at the desired 3-cyclicity property a f, g, h µ ( ) capturing the nonassociative geometry of the M-theory λ • 4.25 ] (see also [ 4.7 A Under the gauge transformation ( In this approach, observables 45 equations which using closure andstate cyclicity function can be expressed in terms o abilities phase space that are multiplied together with the star produ sistent formulation of nonassociative phase spacein quantum [ quantum theory with classical Hamiltonian concrete and rigorous derivation ofexpected the to novel uncertainty arise p fromand the ( commutation relations ( also alternative on Schwartz functions, i.e., it satisfies covariantly: States are characterized by normalized phase space wave fun avoids the problems arising fromtive the (which fact has that our beentivity nonassoci the in property quantum usually mechanics). required in previo The star product ( By 2-cyclicity the integrated starwith Jacobiator ( of Schwartz f property follows: of noncommutativity (and nonassociativity) among free fiel which identifies it as the Kontsevich star product; in partic This property can becommutativity) regarded among cubic as interactions the of absence fields. on-shell of4.4 nona Uncertainty relations The closure and cyclicity properties of the gauge equivalen differential operator JHEP02(2017)099 is λ = 0 • , ion of (4.26) (4.28) (4.27) (4.29) (4.30) λ

, i . i k

k , together x i , p . h ing cells in k i| h

4 4 i x 4 D i , h x ˜ p x hile the second
,ijk |h h describes phase ]. This can be ,ijk 4 ional to the mo- λ ijk 4 ], an area uncer- ~ • i ε ] R 45

, j R

A 45 1234 , i ,p 2 , 3 s 3 s ˜ inimal volumes) we x i i ABCD ℓ ~ ℓ 4
, 2 2 2 x x 2 3 2 ) in components. For λ A R h C λ λ λ A

2 ˜ x h x ~ ~ [˜ 3 s th area λ 2 = = = ~ 4.28 • λ what subtle. The final ex- λ ℓ λ i i i on • suggest that there are new rea and volume uncertainty 4 4 4 ero string coupling constant B st the configuration space as = 2 ) as l areas) we obtain = ij = ,x ,x B ˜ x i ct that the star product λ j j i n the contraction limit i x i V • i i ,p ] ,x ,p h − e position and momentum in the i i 4 ,p µ 4.27 + ˜ C p p 4 x B A ˜ x x λ V V h ˜ x A . , • h h A ] h -flux background [ λ B h is B • ˜ x 4.6 R ˜ x , C ) and ( A , = 1) governed by the Heisenberg uncer- A and and and (and to the magnitude of the transverse ˜ x x A x 4

, [˜ x ~ i i [˜ x 4.26 B 1 6 j λ as λ and and x − x ) in the present case we expect in fact to obtain

• h = , B
– 24 – k = i

i C ]. In this prescription, we can define the area A x 4.7 i j δ x
x , k p 45 λ h A x • ABC + ˜ h ] k i x i − V i λ δ B • jk x ˜ ] x i

h . , i C k j i − k

A ˜ x λ ε δ k i p , x h k p [˜ + B h + p 4

h and i x j i i i [˜ 4 δ 4 demonstrates, as in the string theory case [ 4 Im ,ijk λ x x ijk 4 x 6= 0 the area uncertainties h i h • are the operator displacements appropriate to the descript h ε = i j

R k ij 4 i A

δ λ C ~ 1234

ij A ˜ A x , 3 s ˜ ~ h x 4 ℓ AB λ ~ x ,ijk 4 λ ε R A

= = = Re R is also a new area uncertainty particular to M-theory, yield − h 2

gives area uncertainties along the M-theory circle proport 3 s ABC 1 3 i i i ~ 2 2 i 3 s j j i A λ 2 λ λ ℓ ℓ j ij i ,p ,p x = 4 ~ ,p i i A proportional to the magnitude of the transverse momentum, w i p h x A = = = p h A A := i i i A M h h = 2 k k h ABC A ijk ,x ,x V j j x V AB ,p ,x h i i A Let us now write the expectation values of the operators ( For the fundamental volume measurement uncertainties (or m From the non-vanishing Jacobiators ( p p V V , although we will see below that this interpretation is some h h s momentum), a point to which we return in section pression momentum space with area proportional to g limitations to the simultaneous measurementsseven-dimensional of transvers M-theory phase space, induced by a non-z alternative on monomials give the operators ( while the volume operator in directions tainty principle. For space cells of position and momentum in the same direction wi quantum uncertainties. Explicit computations using the fa mentum transverse to the fibre direction. The third expressi The first expression tainty on the fundamental area measurement uncertainties (or minima where ˜ a coarse graining of theit M-theory happens phase space, in ratherquantified the than by ju reduction computing to theoperators the expectation following values string of the theory orientedoperator formalism a corresponding of to [ directions it reduces to the standard minimal area (for with new cells proportional to the transverse directions; i expression obtain JHEP02(2017)099 l ]. 45 ]. To (4.31) (4.33) (4.34) , which 45 = 0. In ) 4 eory [ x ψ ⊲ ψ . λ ning in the ) D , proportional ∂ B ¯ B ∂ . E α ¯ ∂ x A x
A x p , x ν ∇ − n the same direction x the actual quantum CDE ) (4.32) − 2 A , λ ψ ~ ¯ A ∂ . µ ~ 2 ¯ λ ∂ x B • D B 2 i x , λ ∂ in momentum space, as we A x ) reproduce the algebra of red simultaneously to arbi- 4 x ⊲ ψ E − rpreted as the non-existence ∂ particular, there is a volume ( + ( ates + x ) ∂x ψ ordinates, we use the Cauchy- he contraction limit ainties will be provided in sec- λ B C 4.33 D ce quantum mechanics [ • , λ ¯ x ∂ R ∂ the corresponding nonassociating

⋆ B 3 s

λ A E nd momentum measurements; they ℓ C x λ x ABDE x ◦ 3 x ] λ ψ ⋆ ( − − B 2 ) , λ ⊲ ψ ~ A -flux background. From the contraction ] ψ , x CDE x ¯ 4 ∂ R B ABC ABC λ A λ ∇ ˆ x B • λ λ x − [ , x ~ ~

R B C A

3 s x ABC ˆ x x 1 2 + ( ( λ – 25 – 0 the relations ( λ ℓ λ 2 ≥ = 2 i = 2 i • -fluxes, which involves the full seven-dimensional ~ ABC ) = [ˆ → R B A = λ ψ x . x ) one obtains the commutator
~ + 4 λ λ ψ ∆ ) as ⋆ B C := ] to obtain the uncertainty relations ˜ A ∂ ˆ 4.10 x A x ψ 45 x 4.33 ( ∆ λ ] = 2 i BA . • λ B Λ ABC ⋆ λ ˆ x λ ◦ 4.6 , ) for the string theory ] 2 B ~ A ~ B x -flux background due to the Freed-Witten anomaly in the T-dua x 2.8 [ˆ R , x − ]; it would interesting to understand the corresponding mea A ) := 4 proportional to the magnitude of the circle fibre coordinate x ] = 2 i ψ [ 30
B λ , A ˆ x M ⋆ = 1 to the expected minimal volume in non-geometric string th ¯ ∂ , 15 B 4 A ) we can rewrite ( x x x [ˆ ( λ 3.7 ⋆ . There are also phase space cubes for position and momentum i A x 4.5 We first observe that from ( The present situation is much more complicated in the case of Next we calculate -flux frame [ where for any phase space wave function where differential operators ( encode positivity of operatorscalculate in the nonassociative uncertainty phase relations spa Schwarz amongst inequality phase derived space in [ co presence of non-geometric M-theory uncertainty principles imposing limitations to position a as well as in transverse directions,triples reflecting of the M-theory fact that phasetrary space precision; coordinates these cannot newthe be volume string uncertainties measu theory vanish in limitof t the D-particles volume in uncertainties the can be inte A geometric interpretation oftion these position volume uncert identity ( M-theory phase space. However, therediscuss are further no in minimal section volumes H to the magnitude ofuncertainty the on transverse coordinate direction; in It is easy check that in the limit They demonstrate volume uncertainties in position coordin reduces for JHEP02(2017)099 , ]; λ • 14 [ (4.35) polar- M in phase is of order 0
A y that would ψ = ¯ ∂ D p B ⊲ -flux background x )
R − ) B ) reproduces exactly ¯ ∂ ψ B ¯ A ∂ D ⊲ ψ , x 4.35 ) is complicated by the λ A nt uncertainties derived ) s in the phase space plane ⋆ x tate. − -flux compactifications, we B eld theory correlation func- e, and it is now natural to momentum configuration space A ¯ ∂ R duct in our case and derive A 4.31 x ¯ grees of freedom as is neces- ∂ ) for phase space coordinate A ( ]) as a candidate deformation . This geometric structure was unctions of tachyon vertex op- x n the framework of double field λ B 3 ymmetric worldsheet conformal , ) via the closed star product tric 12 ⋆ x mation of the

R − 4.31 is sense, so that the corresponding i n of the four-dimensional M-theory B A C x + ( 4.32 ¯ ∂ x ) seem too complicated to suggest a h 0, the result ( C − B x ) x → λ ψ ). 4.31 ABC ~x λ D + ( λ ψ ⋆ ] (see also [ ˜

λ △ C ⋆ ~ D 2 x 55 = 0, and which obey the “symmetry” condition ~ B λ – 26 – , ≥ ( • x ( D 24 ], eq. (5.33)) where this problem does not arise. If B ABC ⊲ψ ψ ) to obtain λ x ) λ = 45 ⋆ B ~ ∆ ). The differential operator ¯ ∂ A D A 4.22 ABC ; in the limit x 2 i A x λ ) into the definition ( B x 4.35 1 1 x ∆ ~ − − − D 4.34 D A ¯ ∂ and = = = 2 i B A x ψ x ] it was shown that these triproducts descend precisely from 2 λ • λ ◦ ] obey ( B ψ , x ], and in [ A . Although at present we do not have available a quantum theor x [ 15 M ). However, the uncertainty relations ( ∗ T ], then the corresponding uncertainty relations ( The explicit computation of the uncertainty relations ( ), and it can be regarded as a generator of “twisted” rotation 4.29 45 λ ( provide an M2-brane analog oftions the for computation closed of strings conformal propagating fi in constant non-geome with similar interpretationsin ( as those of the area measureme of [ this triproduct originally appearedquantization of in the [ canonical Nambu-Poisson bracket on isation of the phasespace space star product along the leaf of zero look at polarizations whichsary in suitably quantization. reduce From thefield the theory, physical perspective closed de of strings left-right probe as the nonassociative defor universal lower bound which does not depend on the choice4.5 of s Configuration space triproductsThus and far nonassociative all geometry of our considerations have applied to phase spac measurements reads as the corresponding calculation fromwe ([ restrict to states which arewave rotationally functions invariant in th second term in the last line of ( O through phase factors thaterators, turn which up can in be off-shell encoded correlation in f a triproduct of functions on To translate the expression ( spanned by the vectors where in the last equality we used generalised to curved spaces withtheory non-constant in fluxes [ withi we use the gauge transformation ( can imitate this lattertriproducts which reduction geometrically of describe the the quantizatio configuration phase space. space star pro JHEP02(2017)099 ) 4 = R . = 0 0 g by a ∈ 4.29 ~x (4.36) λ λ 1 4 (2) ) − 4 N R of ( Λ f · , k ∈ i ) k 0 ′ ) , k ~ µν ~x 4 . 0 4 A d h ; in particular, , x , k,Λ ~  λ ′ 0 x = ( Λ Λ R ⋆ ( ~x ~p , η 1 k ~ ⊲ f , ~ B 2
− = ( . i

0 Λ 0 0 Λ · 0 = 0. Hence the source , ~x ~p ) ~x = generally the pointwise

) e ′ ation space coordinates, 0 0 ˜ p ′ k ~ △ ~x

); then , of k − ~ ) k, ′ ~ 2 1 ( ( p 1 ~ k not h − g uncertainties , . 4 (2) ) ˜ 4.24 4 Λ Λ q χ ~
T · k ~ i e a commutative and associative
polarization, and using the vari- ( , x λ k
λ + ) is and ˜ µ ′ f x ˜ Λ ∂ √ ( k − g ~ ) e 4 ~p
) ′ ′ , ) k Λ 0 4.36 ) it follows that it reduces at h k f . ~ k O 2 , but this can be rectified by defining ~ f ~x π ′ 4 4 η

( λ ′ ˜ d g Λ ∇ λ + ⋆ (2 (2) × f g , as anticipated from the corresponding ~p ) ˜ · ′ λ k 4.16 ) △ 3

k ~ k 0 0 ~ 4 k g ~ R ) ( + k ~x ~ ~x − Λ λ ˜ 4 + ′ ( 4 π f 1 = 1 d d k k ~ (2 4 − ′ f ⋆ ε ) f q R ( k 0 ~ ) and ( – 27 – ; in particular π 4 Z × ~x  ) = d 6= , i.e.,

˜ (2 k ′ = △ f g 0 , and define the product 1 = ′ Λ g k ~ 4.15 4

0 ~x µ 4 ~p − ) := ′ 6= Λ ) = λ R 1 k x ~ (2) k, 0 λ ~ η (2) ~p p 4 π ( − g △ ~x

△ ∈ ⊛ d 2 η R ( ) 2 Λ (2 (2)
f 3 s ) ~ ~ Λ p λ f Λ ⊛ (2) ), it is unital: 4 h ~ 0 , T ~p △ λ ℓ 0 + ,

4 Λ k ~ Z 1 4 ~x ~p , λ 1 (3) f f −

4 = , x 0 ~ − △ , , x d ( µ ]. However, the product ( Λ ′ ~ x 4 ) = x g k x ~ 0 (2) sin R )( R 14 based on the closed star product ( Λ ~x Λ g = ~ λ T , ( (3) = ( η λ 2
λ △ f (2) 0 ) = = g × N ′ f ~x ) inherits properties of the phase space star product f ⋆ λ k ~ k (2) k ~ ~ λ (2) △ Λ △ k, ~ µ ( 4.36 f x ) := ( 0 ) = ~ (2) 0 Λ ) we see that the cross product of four-dimensional vectors ~ T ) we can write the product succinctly as ~x . k, ~ ( (
f g g (2) 3.36 Λ 4.14 0 ~ T λ (2) ~x Next we define a triproduct for three functions For this, we consider functions which depend only on configur 4 △ d f It has a perturbative expansion given by where we introduced the deformed vector sum similar rule: product of functions on It is commutative and associative, as expected from the area so that off-shell membrane amplitudes in thisdeformation. case experienc Moreover, The product ( and it is therefore orthogonal to in this polarisation; from the limits ( that we denote by instead a product gives ables ( of noncommutativity and nonassociativity vanishes in this since R From ( to the ordinary pointwise product of fields on string theory result [ JHEP02(2017)099 . . Λ ~p 0 ) lives ~x λ 2 (4.39) (4.37) (4.38) 1

( Λ , − ~p ′′ Λ O 0 Λ η ~p · ~x ]; indeed, ) + containing 1 η 3 × ′′ −
7 ′ ⊛ k ~ Λ  Λ ′′ ~p ′′ · ′ R Λ Λ ,Λ ) k ~ ) ~p ~p ′

′′ ct 2 in 2 k k ~ ~ | ,

′ ′ − 4 ′ ). The final result Λ k ~ k ~ . k,Λ 1 ~ ~p R k, ~ ( Λ Λ η ( q ee-spheres [ (3) 3.25 η (3) ⊛ + ′′ Λ σ ~ Λ B ~ ′′ T ( f xpressed in terms of the Λ ,~p k i k ~ ~ η re it describes the polari- ′ ~p Λ ~ Λ ) B

Λ ~p λ i (3) | ′′ X ǫ ) e △ − k ~ e , ′′ + ) and ( β + 1 ) e ′
k β ~ x (2)
( ,Λ ′′ k ′ ~ sional subspaces and we can write x ′ σ q k ~ ′′ ˜ h k Λ ~ f 2 k 3.20 ~ ( ′ Λ ) | Λ ′ ˜ h , ~p X ,~p ′′ µναβ λ ′ (3) k ) ~ Λ Λ µναβ e k λ ~ ( ′ ~p △ k, Λ ~ 2 g ǫ k k , ~p ~ ~ Λ ( ~ ) ˜ Λ R ε Λ Λ ( (1) g + + ~p k 3 s ~ X η σ ) ˜ ( 12 ℓ ′ Λ f e k ~ ~ ˜ η A f k ~ | ~p − ( ~ 1 Λ σ A | ˜ | 4 2 f ′′ − = 3 

= ) = – 28 – k ~ 1) 4 Λ

+ ′′ π ′′ Λ ) ), we can compute the deformed vector addition 4 ) ′′ 2 1 λ Λ (3) − k ~p ~ (3) ′′ d k ~ ) amongst linear functions that encodes the nonas-

π (2 ( △ 4 ~p △ k ~ ] ′′ η ] Λ 2 3 2 d (2 η 2 | α ,Λ 4 α ~p 3.27 ~ ′ S ⊛ ) ) ′′ ′ k ∈ ⊛ to the four-dimensional subspace 4 ~ ). It has a perturbative expansion given by 4.13 η ′ X , x k π Λ ~ , x ) 4 k σ ~ + ) k ~ ν ν 3 ~p ⊛ d π ′ 4 Λ

′′ (2 Λ ) k,Λ ~ R d ~p = 0, after a bit of calculation one finds that the triproduct ), we then define a completely antisymmetric quantum 3- k , x := 3.27 ~ , x (2 ′ Λ Λ = 1, these are just the brackets (up to rescaling) of the 3-Lie − 4 µ + ( µ η 0 ~p ) k 4 η ~ + ′ x 1 x λ ~x λ (3) ) ~ [ ⊛ 4 π [ B k η ′ ~ k ~ △ ( 4.37 1 4 d π ] k ~ η q Λ ⊛ (2 − 3 Λ d ~ B ~p | ′′ (2 Λ + Λ ( Λ X Z , f

,~p ~p · + ), together with the identities ( Z 2 k e ~ ( 1 Λ
4. For

~p − , , f Λ ) = ǫ ~ 1 3 0 ~p ) = ) = 3.15 , f 0 + sin ~x η [ ′′ 2 ( ~x k , ~
( × ,
h ′ ). Exploiting again the property that the vector cross produ ′ ) = Λ h k ~ ~p = 1 ′′ λ (3) ) and ( k ~ k, ~ λ , (3) 4.14 , ( △ 4 ′ △ k ~ g A (3) g Λ 3.14 ~ k, T ~ λ (3) ( λ 1 (3) Using the triproduct ( △ µ, ν, α, β − △ (3) , so that Λ f Λ ~ f Λ T ~p can be written as It reproduces the 3-brackets from ( familiar from studiessation of of multiple open M2-branes membranes in ending M-theory on whe an M5-brane into fuzzy thr algebra for sociative geometry of configuration space, bracket in the usual way by where we defined the deformed vector sum is a bitvariables complicated ( in general, but is again most concisely e from products of octonion exponentials which contains the associator ( in the orthogonal complement As in the calculation which led to ( As before, the Fourier integrations truncate to four-dimen using ( JHEP02(2017)099 ) n d 0 λ (3) → ~x ace △ ) = 1 . In . λ ′ − 4.37 5 k (4.40) f ~ Λ ivalent , 4.3 · ) 0 0 ~ e-sphere n ~x k . The cal- ~ 4 k, ~ and in this ( n d ,Λ k ~ 6 ) (3) Λ 0 R Λ (which do not X = ,... ~ T ) b p 0 whenever any e ~ we set 3 in terms of discrete

k ~ k ~ ) 3
4 Λ is p g , R ,Λ R , ) η
2 4 ··· 0 ) = ~ λ k (2) ~ ∈ ′′ , × Λ terms of an underlying ertheless interpret the
′ △ , x ) 3 a ,Λ k ~ k , ~p 4 ~ 1 x ion of k ~ f ′ k ( ~ Λ Λ ation quantization of the k, ~
, x Λ tions of these constructions Λ ( X = ( n , ~p x sents a deformation quanti- η in the contraction limit f Λ (3) g n be computed from products ~ . Λ B ) e ~p cts λ ( 3 ~ T 2 = ( ⋆ λ k ~ (3) R η ..., ijk 0 (
Λ △ ), we see that the triproduct ( that we derived in section ~ A η ~x X ~ B f λ ( R ε e ··· • η of trivialise on-shell, i.e., 3 s 4.19 ~ λ λ 1 B (3) ℓ k i n ~ ⋆ △ 3 Λ
not 3 X − ) we have ) e f a = 1 = ( e λ ), by correspondingly dropping many higher k ~ , . . . , f ⋆ g ( 0 3.21 1 ): it represents a (discrete) foliation of the M- ]. However, in contrast to the string theory a λ ) ~x (3) f ˜ – 29 – 2 f △ 1 ··· λ ] 14 (3) f k − 4 △ a 4, which in the string theory case would represent , 4.39 λ ) Λ k ~ 2 ⋆ 1 , x π 4 j ≥ 1 d (2 ), together with ( f λ (3) ]. Firstly, one can generalize these derivations to work n ( , x i △ x n =1 15 . We will say more about this perspective in section [ f . Since the associator Y 4.16 a -point correlation functions of tachyon vertex operators i 3 , . In the present case we are in a sector that involves only 4 0 by fuzzy membrane worldvolume three-spheres, ··· n 2 0 limit [ S A → 4 14 ~ lim Z λ 1 = R , ) gives a candidate deformation quantization of the standar ]. For functions → 2 R/ = ) and so we obtain the unital property λ ) and ( (3) 3 s ′ 14 . This is again related to the fact that the precursor phase sp △ λ ℓ ) := k 4.37 ~ 0 3 g 4.15 ~x k, ~ ( ( p λ f g h
(3) ) quantize the standard Nambu-Poisson structure on the thre n (2) △ 0 is not closed, and presumably one can find a suitable gauge equ Λ f ~ ~x T analogous to the closed star product f ) λ 4 4.39 n -triproducts for any ⋆ λ d ( ] for an analogous description of a noncommutative deformat λ (3) n △ ) = R 23 N ′ of radius k we will give a more intrinsic definition of this triproduct in ~ ··· 6= 4 5 ]; in particular ) k, ~ h n , R λ -flux background [ ( 14 0 ~ We close the present discussion by sketching two generalisa From the limits ( △ ( R , λ See e.g. [ ⊂ (3) 1 6 2 △ (3) 3 f Λ ~ triproduct foliations by fuzzy two-spheres. 0 [ of its arguments is the zero vector, using ( of corresponding octonion exponentials and as before the nested compositions of vector additions ca culation simplifies again by dropping all vector cross produ the off-shell contributions to contribute to the inner product with star product out explicit the section Spin(7)-symmetric 3-algebra on the membrane phase space. triproduct, here the M-theory triproduct does Nambu-Poisson structure on reproduces that of the string theory configuration space S theory configuration space sense our triproduct ( membranes of M-theory andnonassociative excludes geometry modelled M5-branes, on but ( we can nev g the brackets ( T Thus while theNambu-Heisenberg string 3-Lie theory algebra, triproductzation its of represents lift the a to 3-Lie deform M-theory algebra repre as expected from the in light of the results of [ JHEP02(2017)099 - 2

R into nΛ ~p λ (2) η n △ f ⊛
1) − ,  n λ 0 ( ···
~x △ Λ η 1 ) by introducing p −  ⊛ aΛ , 4 Λ ···

· ~p ′ Λ
) λ a 4.36 1) n cΛ n ǫ c − k ~ k ~p ~ √ λ p

n λ 2 theory phase space by ( η

,..., − △ nΛ O 1 ⊛ ~p k ′ Λ ~ nΛ a nΛ ), and by making repeated ,..., (
b + η ~p f p ) ~p a b n
4 η k ~ ⊛ ( η 3.9 1) Λ ··· k Λ ~ mation of the string theory T
⊛ − can likewise include such phase ⊛ i ′ η 4 n λ aning of these non-zero constant
(
λ p ⊛ △ ··· ity ( ,..., λ k )). This turns the product
to obtain suitable noncommutative 1 ) e η + ··· ··· a ) k ~ − η bΛ ··· ⊛ ′ Λ n η k ~ λ c ′ ( ~p ( p
⊛ 2.14 1) ⊛ △ k 1) a η ε ˜ Λ −
f − 4
3 n ⊛ n ( λ × Λ ~p ( Λ ; here we abbreviated signs of square roots 4 aΛ 3 a
~ ) △ λ k T η d Λ ~p ~p k ~ π p 1 4 η ⊛ ] (see ( η + = ··· f d ]. 2 (2 ) ′ ⊛ · ⊛ η
. Again we see that Λ – 30 – k ], and it would be interesting to understand their =
) 14 n ¯ , . . . , n p 2 ⊛ 2 -triproducts reduce to those of the string theory ε n k =1 Λ ~ ~p , abc n Y
n 2 a R ǫ 2 = 0 [ f ··· × -triproducts η ~p = 1 2 3 s ) n η λ k ⊛ aΛ ℓ η . Here we only quote the final result: 0 [ ~ n λ a Z . This would modify the product ( ( d i ~p ⊛ 2 ,..., Λ ⊛ · and ¯ p △ 1 η )

→ 3.3 ¯ Λ ~p p a 0 ) ~ Λ for ′ Λ ⊛ 1 = ( ) = ǫ 2

λ ~p ··· = ~p 0 R
′ ~p Λ ( p a ) aΛ ~x η 3 s ~p η k ( n ~ ~p ~ λ ℓ ( ··· ( ⊛
··· η 2 ⊛ i ) by △ n η ··· Λ ⊛

 f Λ ~p -triproducts obey the expected unital property  1 1 ⊛

) Λ

~p n 4.38 n − ,..., ~p λ 1 ( Λ ( =1) ~ cΛ

1 1 − a △ k ~ ~p ~ sin f , ~p ( ) sin ··· = exp bΛ n ) ··· ( n Λ λ ··· ) (

. As previously, these  ) = , ~p ~ T n λ △ n (

− k ~ aΛ △ ~p exp 1 − 1 ··· 1 f η ) q , . . . , n n ~ λ q denotes omission of ( ,..., A 1 △ n =1 k ~ a = 1 X aΛ 1 ( abc X ) d ~p f Secondly, one can consider more general foliations of the M- ǫ  aspin group Spin(7) tion ( Spin(7) is isomorphic tosponding the to double the cover upper of and lower SO(7), hemispheres with of the t product, where the actionsphere of Spin(7) can be described as the t S JHEP02(2017)099 , ] . 4 ]), 52 ) is n a A 5.1 – 55 -flux , 4.2 R 50 , 23 23 ] of the Lie ] that closed α [ 44 2 es of freedom of tivates an appli- G -field background ]: in some systems gebra under SL(4) B (see e.g. [ (2), representing the 20 ce of frame wherein -branes to the maxi- 3 F su S in that gauge is simply -field background of 11- racket (see e.g. [ ent, but related, explicit G C ction that induces a phase ring theory to the SO(4)- ing theory to M-theory, 2- e choice, such that for any iew of [ } een observed previously on wo-form gebra he introduction of some sort a similar point of view in our tures introduced in section f, g ]. It is also reminescent of the o has a corresponding lift to M- ne induces a 3-bracket structure 3-Lie algebra type deformations { e-spheres ], the SL(3) symmetry is restored 42 30 , and indeed the string theory quasi- , is replaced by the 3-Lie algebra . This is prominent in the lift via T- and the momentum constraint ( are SL(3)-invariant, being based on the 3.6 2 F S 2 3.5 = 2 worldvolume theory on the M-theory N ,µναβ of fields can be defined without gauge-fixing, 4 R – 34 – }} f, g, h . Based on this, the BLG model uses a 3-algebra for the = 6 Chern-Simons theories, while the Moyal-Weyl type {{ 4.6 N ] for reviews); it is naturally implied by the Lie 2-algebra . This is analogous to the procedure of reducing a 3-Lie }} 31 ]. If we moreover adopt the point of view of [ and , 21 3 = 0 the quasi-Poisson bracket ]; however, as also noted by [ 4.5 ] it is suggested that this constraint could be implemented i 30 G 30 f, g, G {{ ], regarded as the worldvolume of M2-branes in the M-theory 47 = G } ). This exhibits the non-invariance of the quasi-Poisson al = 0. In [ 2 ) between cross products and triple cross products, which mo ] for a description of the corresponding regular subalgebra -flux compactifications should be regarded as boundary degre 4 f, g G p R 41 { 5.7 -flux has non-vanishing components This symmetry breaking may be attributed to the specific choi Taking this perspective further, we will adapt the point of v R representing the polarization of M2-branes into fuzzy thre the on the boundary [ open membranes whose topologicalspace sector quasi-Poisson is structure on described the bytheory: boundary, then an in this a that to case the action for an open topological 4-bra of 10-dimensional supergravity lifts toof Nambu-Heisenberg the coordinate algebra of M5-branes in a flat three-form and it can bemally used supersymmetric to Yang-Mills theory dimensionallyrelation of reduce ( D2-branes the [ BLG theory of M2 with gauge symmetry, 3-brackets strings in underlying gauge symmetry to construct the symmetric M2-M5-brane system wherein the underlyingpolarisation Lie of al D1-branes into fuzzy two-spheres for reviews in the presentconsiderations context); of sections we have already adapted The impetus behind thisbrackets proposal should is be that, replacedmany in occasions by the (see suitable lift e.g. 3-brackets,structure from [ as discussed str has at b theduality beginning of of the section SO(3)-invariant D1-D3-brane system in IIB st solved by covariant fashion on theof eight-dimensional “Nambu-Dirac phase bracket”. space byproposal Here for t such we a will construction based offer on a the Spin(7)-struc slightly differ in the contraction limit by thePoisson algebra discussion and of its section quantization fromthree-dimensional section cross product. dimensional supergravity [ background. in contrast to quasi-Poissongauge-fixing brackets condition which depend on a gaug membrane, which is related to and SL(3) observed in [ deformation of the coordinate algebra of D3-branes in a flat t (see e.g. [ algebra of given by algebra to an ordinary Lie algebra by fixing one slot of the 3-b JHEP02(2017)099 , η
of 0 ξ
(5.8) (5.9) 2 λ , − ) is now k , C ˆ F ~ ξ, ∂ SO(4) i x 2.3 ∂ξ , p Ξ 4 Λ undamental ˆ × F φ
∧ ˆ , . B , ˆ ,ijk }} φ F ˆ 4 B 4 1234 D ˆ φ k 4 , ∂ A p ˆ }} 4 C p p R ∂ξ ˆ ) = }} D ′ φ R 3 s ˆ µ C SO(4) ijk 2 -flux background is ℓ C ∧ ) to get ˆ ijk , k ′ 3 s E ~ φ 2 2 ) to define an affine ℓ ε 2 ˆ , p 2 R , h, k B A A ~ λ ˆ λ ε ′ 2 µ ∂ ∩ space; this breaks (at E 2 2 δ 5.3 φ . 2 λ , f A ∂ξ ˆ λ x ] with 3-brackets − 4.4 B -symmetric bivector Θ 1 }} η
+ δ 2 f Ξ ′ D = = 2 ˆ [ = = ˆ E ξ , g F G − {{ ˆ φ -flux. One can compute it 2 φ C ˆ φ D φ B CC ˆ ˆ F ˆ ) = ( C R }} C }} . , f }} }} Λ ˆ ˆ 4 A 4 ˆ A k j 1 4 ′ E M-theory ˆ ], so the Jacobiator ( g, h, ˆ ons C f φ D φ ABCD , x -theory phase space subjected to ˆ , p , p , x E Ξ i ˆ i x p ~x, BB j Ξ η B ˆ [ , j D φ ˆ Λ ˆ k Ξ C A , p , x D − {{ , p ˆ + , x ′ D p i δ 4 4 ˆ i , C = ( φ , ˆ η p − p p B 4 − {{{{ ˆ x
B + δ k AA }} j ˆ ˆ Θ φ {{ {{ {{ E A p {{ X ˆ Λ x ,ijk F ˆ − φ D ∧ 4 }} , k 4 ˆ k 2 i λ B ˆ φ C φ ˆ , we extend the 0 F δ ˆ R E ˆ 8 ˆ ∂ B . ,ijk A }} }} ˆ − ∂ξ 3 s = 2 D 4 φ 2 ˆ − ℓ R E ~ C φ i 2 ˆ D φ R ˆ , h ) to a subgroup Spin(7) 2 A D = x λ ∈ and 2 and ˆ x x and – 35 – , C and 3 s 2 φ Ξ

ˆ ˆ ~ k ) j C k A given by 5.8 , f ˆ ˆ g, h, k + 2 λ ℓ δ C C δ ∂ x 1 ~ ξ ˆ 4 B 8 ∂Ξ {{ f ABC , p + δ 12 2 , jk λ ,Ξ 0 k λ R i {{ 2 j ABCD i ξ ∧ ˆ 4 p B ε − − 2 λ δ 4 λ p − 4 ˆ g, , f B 2 x by using the contraction identity ( 4 2 4 1 − − λ ∂ ,Ξ = ( = 12 4 f ˆ x ˆ ∂Ξ ,ijk A A k 1234 φ 4 , ,ijk = = k + {{ − {{ Ξ x Ξ Ξ 4 4 } ,ijk ∧ ij 4 R 4 φ φ ˆ E R R ε ˆ x  i 3 s 2 A ijk := R 3 s 3 s 2 ~ j

i ∂ ε x 2 2 2 ℓ ℓ 2 ,Ξ 2 λ λ ℓ δ ~ ~ φ 3 s ∂Ξ C B 2 2 3 3 ~ ˆ 2 2 ℓ 2 2 } D 2 λ λ 2 λ λ λ − − ˆ D , x , x ,Ξ A B ======Ξ ˆ C ˆ ). Writing φ φ φ φ φ φ φ φ D , x , x ) is a Nambu-Poisson tensor if these 3-brackets satisfy the f ˆ 4 C A , g, h, k ,Ξ }} }} }} }} }} }} }} }} p ˆ 4.2 2 B j j x 4 4 4 ˆ k k k 5.8 B ˆ A  , f  , x , x , x , x , x , x , x , x φ 1 i i i ,Ξ j j j j j f ], thus defining a 3-Lie algebra structure on 3 1 ˆ A , p { , p , x , p , p , x , x , x 4 i 4 i 4 i Ξ i i 55 p p p p p p { p x {{ {{ {{ {{ {{ {{ {{ {{ Φ := To write the 3-brackets of phase space coordinates, we use ( = 1) the symmetry of the trivector ( which generates a 3-algebra structure on coordinate functi Altogether, the phase space 3-algebra of the non-geometric SO(8). Then the 3-brackets are given by The trivector ( summarised by the 3-brackets which is natural from the index structure of the M-theory transformation of the vector space identity [ replaced by the 5-bracket explicitly on linear functions to the Spin(7)-symmetric trivector which we have chosenλ to preserve SO(4)-symmetry of momentum the constraint ( cation of this construction to the full eight-dimensional M JHEP02(2017)099 ; ) ) ) ), , 4 W 4.2 k R 4.1 4.7 = 0 ( 2.3 x ∈ 3.18 (5.10) (5.11) − 4 ) with ′ jk p 4 S i ε R 2 ; the more 2 ) and ( P,P λ , , k k 4.1 4.6 + x p 4 hing 3-brackets x ijk ijk j i ε ε ) describe the free . δ 3 2 ′ 2 2 2 λ λ λ ~p tors from section 5.9 η ). Although at present 0. Quantization of this = = = × tar product of section ~p 5.9 natural ternary extension }} }} }} → ory phase space 3-algebra. ly, we observe that in the ). This framework thereby j j 4 X i-Poisson brackets through λ , p , x , x + anishing 3-brackets ive geometry of the M-theory ra ( t 2.2 i i j ~p ) (which in the gauge , p , p , p X = 0 in the remaining 3-brackets 4 i 4 ′ 0 p 4.2 p p on of the binary operation ( 4 these brackets reproduce the quasi- . p naturally arise from the 3-algebraic p {{ {{ {{ 4 φ + p ′ }} 2 λ 4.5 ~p = 0, the brackets ( ] for details). − X and and and 34 . To motivate its construction, let us first 0 p = 1) of the M-theory brackets ( f, g, G
) = 3.3 j ) = 0 + µ,νραβ λ {{ x X ( – 36 – X R 1 k = 1 reproduce the string theory Jacobiator ( i ( . In the following we will provide further evidence δ
:= G 4 ′ G x G − (see e.g. [ (which is a quaternionic line bundle over ~p 4.6 } i · 4 ) can be alternatively modelled on the space 4 -flux, x ~p constrains the 3-algebra of the eight-dimensional phase R R R k j f, g 5.8 δ − µ { ⊕ = 1 reproduce the bivector ( . For this, we note that for arbitrary vectors p ′ 0 4 4 2 λ p W R x 0 , which at − p φ = ⊂ 4 4 µ,νραβ of the seven-dimensional phase space from section }} x p = k V W R k -direction). Moreover, setting λ ′ 4 } ij , x P ijk p ⊂ ε j i X ) = 2 x 7 2 λ ε f, g , which at λ , x P i 3 X { φ B 2 ( λ − x ]: the trivector ( X }} G {{ J = 2 = = 30 , x 0 reducing M-theory to IIA string theory, the only non-vanis }} }} }} ] I 4 k k ) are ) reduces them to the seven-dimensional phase space Jacobia 53 , x , p → , x , x 4 i j j p 5.9 5.9 λ , p , p , p {{ i These consistency checks support our proposal for the M-the For any constraint 4 i p p p {{ {{ {{ respect to the splitting from ( space to the codimension one hyperplane defined by ( In particular, for the constraint function one has [ is orthogonal to the on the eight-dimensional phase space, one can now define quas of the vector starprovide sum an described alternative in eight-dimensional section on characterisati the unit ball 5.3 Vector trisums At this stage the nextwe do natural not step know how is to to do quantise this in the generality, 3-algeb we canstructure show how of the s the full membrane phase space; this is based on a for these assertions. phase space 3-algebra structure of M2-branes with the non-v described in [ It suggests that in the absence of we shall see thislimit feature at the quantum level laterand on. Final naturally explains the SO(4)-invariance (at general choice Poisson brackets of negative chirality spinors over which as expected all vanish in the weak string coupling limi and the configuration space triproducts of section from ( 3-algebra then provides a highermomentum version space of discussed the in noncommutat section JHEP02(2017)099 , . 2 ,  | by .
0 ~p ′ V W ~p (5.13) (5.14) (5.12) ≥ − | ⊂ η ′ ⊂ 1 2 7 . ~p × ), ensures 2  B 2 , p ~p ) Z triple cross

′ ′′ ) / ~p ± = 1 2 ~p 7 P 5.11 ′′ | · η 2 φ S ′′ ~p = ~p

~p ×
× 0 − 2 = ′ ′′ | ( p ¯ ~p P ′′ 7 ( = 1, we can easily − | P η = 1 and φ . · ~p B | 1 | ⊛ X , × ′′ ). ) P P ~p
′ P ~p p − | | ′

we reproduce the seven- X = 1 P 1 X | P − + − 3.20 2 2 ¯ X sphere. This interpretation

′′ X p + 1 ) ies. For example, since the , P ~p ~p
P /G P −
, i.e., · q η ¯ P X ′′ , X 2 sum remains in the same hemi- X ~p ( ′′ | ′ ) × W ′ X P ~p 2 ′′ − ′′ ′′ ) | P − ′ ′ X ~p ,~p,~p ⊂ ~p P SO(7) ~p, Im P ) ~p X 0 ~ , ~p

ǫ ′ satisfying ( ′ ǫ 7 X 2 − | X ≃ P = 1 we have ′ |
+ + S − | ′ ′′ 1 = ′′ | Im 2 P X 2 ~p ~p 1 p ~p ~p, P ): we fix a hemisphere

′

P P Z ( p X P
| 2 ,~p X η / p

+ 2 X ′′ 0 − | ~ 7 | ~ X A ′ ) ′′ 2 1 2 ~p, 3.18 ( P ′ 1 S −

ǫ 2 + ( ~p ~p )

+ indeed also belongs to the unit ball P X ′′ = ) of the vector cross product with the defini- ′′ p η ′′ P = ) = ′′ – 37 – ~p ′′ X ′ − |
Im P P ⊛ · X 7 + ~p 0 ~ ~p 5.4 , P P ′ ′ 1 P ′ ′ X X
2 B φ ~p ~p := )

X X P X ) ′′

p ′ ′ ), which is the sign of the real part of ( ′ ( ′ ~p,~p ⊛ 2 ′′ 2 P X ~p ǫ | = 1 we have P ′ ~p | P ~p P − ( η | ~p φ X ~p ~p η X X 1 = X ( ). We define a ternary operation on Re 3.18 P ⊛ × ′ φ Re ′ ⊛ P P

′ − | q ~p X − | ⊛ | ~p 5.5 φ ~p X X 1 ′′

~p,~p Re 1 Im = ( from the unit sphere ~p ( ⊛ ǫ

,~p ~p 2 2 ′ p − in the ball p | =

from ( 1 2 ~p = P . To obtain an explicit expression for it we observe that X Re 1 ′ ′ ~p ǫ ′ 

− P 2 ~p  =

is a normed algebra, for q η X ~p,~p ′′ = = ′′ − | ′ ′ ) we then immediately infer the identity ( ′′ ⊛ p ~p ~p, ǫ ~p P O 2 ,~p 1 P P ′

~p,~p X X ′′ ǫ X X p 5.12

) ~p,~p 1 is the sign of ~p ) ′ + + ǫ ′ φ P ± P = ⊛ X X = ′′ ′ vector trisum . As before, since P ~p ~p 7 ′′ P ) and ( ) of the vector star sum, we can apply the same reasoning to the X S φ φ ,~p X ( ′ (

⊛ ⊛ 5.11 ′ ~p,~p ~p 5.12

ǫ ~p By comparing the representation ( φ − ⊛ 1 ~p dimensional vector star sum through and call it a sphere of which as previously ensures that the result of the vector tri such that for vectors This implies that Here From ( where the sign factor tion ( product represented through ( that the result of theof vector star the sum vector remains star inalgebra the of sum same octonions is hemi useful for deriving various propert which leads to By restricting to vectors which shows that the vector translate this identity to the vector star sum ( JHEP02(2017)099 ′
in ~p
′′ η ; ) k ~ 5
⊛ 3 (5.15) (5.16) repro- k 2 5 ~

~p S , k ~ ′′ 2 k ~ , ∈ ~p k ~ 4 , φ + σ 1 k ~ ′ k ⊛ ~ , ( ′ k ~ ) .

φ ons one sees 3 ~p ~ , we again apply | B k ~ φ − k ~ , , | ′ . 2 ⊛ / W 4 ); this is analogous ) k k ~ ~ | ~p ˆ 0 k ′′ p . ~ ~ , k ~ 2 ⊂ | 1 , P

φ φ ~ m k ~ φ 3
′′ V ⊛ ( × k ~ contains terms involving ) it easily follows that × ′′ k φ ~ , ~p sin( ′ ~p k ~ ~ ′′ B
φ k ~ P φ φ + η ′′ ~ ~p B = φ 5.13 φ ~p ⊛ k ~p ~ φ × ×

~ × ′ φ ′ B − ′ ⊛ ′′ ~p P ~p k × ′ ~
and the vector star sum − − ′ ~p 5 0 ) ; ~ ~p X ′′ ~p k ) for all permutations ~ = +
k = ~ ( φ φ ) φ 2 ~p k, , ~ ) and ( k ~ 5

~p ⊛ (3) ~p , ⊛ × ) φ ′ k ~ ′ η σ Im ′′ 4 φ φ ~p ~p , k ~ k ⊛ ~ ~p k k ~ × 4 ( ~ ⊛ ′ 5.12 X × , − φ k ′ ~ φ ′ ′′ 2 ~p , + , ~ B k ~p ~ k ~p ~ ⊛ 3 ′ φ , φ k ~ (2) φ Im ~p k 1 ~ + ( ⊛ σ

′ k ~ × φ | ⊛ ) = k ~ ~p ( 6= k ~ ~ ′′ B p − 0 φ k ~ ~ | −
η – 38 – , , ) ~ ~p , B ′ ′′ ′′ = ˆ = 2 × ′′ φ k , ~ P ~p ′ ′ k (1) ~ , k 3 k ~ φ ~ ⊛ , the corresponding mapping of the vector trisum is σ φ ~p ~p 0 , ~ , ) over the entire vector space k ~ ′ ′ ( k × ~ φ 1 V ⊛ φ ′ ~ φ ~p k ~ k φ ′ ~ − P ~ × ⊛ B ( ∈ ~ φ B ~p + 5.14 φ k, ˆ φ φ ~ p 0 ~ ′′ ( ⊛ ′′ φ ~ ~ × B B − φ k η ; ~ φ k ~ | ~p ) = ⊛ , P ~ | ′ σ A ′
× ⊛ | there is an analogous relation ~p 1 + k 3 ~ X k ~ 2 | ~p ~p ′ − , ~ 1) ) := ~ k k, ~ 0 ~ ~ W 5 2 = = − 2 ~ k sin ~ O ′ Im ′ + k, ~ ∈ , ( ~p ~p 4 + k + ~ φ k η ~ η ) between the vector star sum and the cross product. However, 0 ) ~ ) = ( ~ , B , 3 ) := 3 × ⊛ k ~ k ~ ′′ ) = ~p ~p . , , 3.22 k ~ 2 ′′ ) = 2 2 | , ), for the antisymmetrization of the product of three octoni ′ k = (1 k ′ ~ ~ k ~ ~p , ). Then for , k ~ , k ~ ′ p 1 1 5.4 k ~ k k, ~ k ~ ~ k, − | ~ ) one immediately infers the following properties: ( ( ( ( 3.24 k, ~ ) it follows that the antisymmetrization of the vector trisu η φ 1 φ φ ( ~ ~ ~ Perturbative expansion: B The higher associator B ~ φ B A ), for ˆ p 5.16 5.6 ~ B is antisymmetric in all arguments. ± 5.7 To extend the vector trisum ( The relation between the vector trisum = 0 because the antisymmetrization of the vector trisum the map ( (TB3) p (TB2) (TB4) contrast to ( From ( By ( given by From ( (TB1) to the relation ( duces the imaginary part of the triple cross product can be described as follows. From the definitions ( JHEP02(2017)099 p · ) ) upon ′ ′′ (5.17) (5.21) (5.19) (5.18) (5.20) k l ) is the x · , ; indeed ′′ + l ~x . k ~ ′ 1 , ) 4.5 ): from the − l ′ . − ′ k ~ Λ l , λ + · · 5.2 ~ ) l k, ~ 4.40 ( ( ′′ k ) φ k ′ ~ O + ( ~ + B k ,Λ 4 ′ + ε ) implies to the configuration k ~ k x , λ · × ) . 0 } and ). In particular, it is ′′ k,Λ ′′ k 4 ~ (3) 1 l 5.19 f = k σ Λ ously to the star product λ ( + fixing of 3-brackets to 2- f + φ ⋄ p implies that the triproduct 4.38 + . λ l ed in section ~ ribed in section ′′ B l perspective of deformation f, g, h f i · ⋄ ′ 4 ital property k { y circle, as in ( k ′′ λ ijk 5.3 ε 2 ⋄ k (2) ~ ) e + × σ g , 3 ′′ R ε . f − 4 k ) k ~ 3 s k λ ) with ( = λ ( − ′′ ′′ ℓ ⋆ ⋄ + ˜ h ]
k k 3 f reduces at + ( · λ ) ′′ ε λ − ′ (1) } 5.16 ′ λ k l x ⋄ σ f, g × ) reproduces the string theory 3-bracket k ⋄ ~ ε · ′ = f ( g − , in precisely the same way that the triple ) 3 [ g f, g | k ) contains terms without derivatives can be × 0 from section λ g ( ′′ σ { ′′ ) ˜ ′ − | ⋄ = l 5.19 · λ k h k k p ~ · 1)

( ( k ′ = ) and ( 5.20 ˜ · + ), which together with ( f – 39 – + λ 4 k − f ⋆ λ = 1 ⋄ ( ( ′ p x ⋄ ) for the configuration space coordinates λ ]] 7 ′′ 3 4 g k } ) k (TB2) 5.14 R R S 1]] k ~ ~ x 5.18 2 π λ 7 3 s 2 3 s ∈ + , x 2 X ℓ ℓ ~ ⋄ 5.19 σ d j (2 h, f k 1 { − − + f, g, 7 , x ′ λ := [[ i g ) ⋄ k ~ = ( x λ π 7 [[ + ⋄ f d ]] ~x 0 (2 λ 3 1 = } → 7 lim − ). Property λ we find , f ) g k ~ it has a perturbative expansion given to lowest orders by 2 Λ 7 π λ g, h · d , f { 4.2 (2 ) by comparing ( 5.16 1 f ⋆ f ′′ f [[ k ~ λ Z (3) ~ △ stands for the Fourier transform of the function (TB3) i ,Λ ˜ ′ f ) = − k ~ yields ~x , by setting = )( 0 is related to the star product λ h k, Λ ~ ⋄ 4.2 = ]] h λ Λ ⋄ ( λ p ⋄ φ g ~ B λ g All this generalises the properties of the triproducts desc ⋄ f, g, h 0 λ [[ f ⋄ → ( lim λ space triproduct where as before deformed vector sum ( calculations of section it is easy to see that the phasestraightforward space to triproduct show thaton the configuration 3-bracket space ( in the collapsing limit of the M-theor easily understood from the property ( f cross product is related to the cross product, through the un so that in this limit the 3-bracket ( setting Although this feature may seemquantization, unusual from it the is conventiona brackets exactly on the the reduced quantum M-theory version phase of space that the we gauge discuss then by property We now define a triproductof on section the M-theory phase space, analog 5.4 Phase space triproducts If we define a quantum 3-bracket in the usual way by The fact that the perturbative expansion ( JHEP02(2017)099 0) , , and -flux 0 , , ε R q J ] ommons = ( ~p d strings ]. ic paradigm of ) to functions (2016) 234 5.17 , 106 SPIRE ]. ring in ]. IN ence and Technology ience and Technology ][ cknowledge support by ESP, Brazil), the Grant ) to vectors duct which quantises the SPIRE , Brazil), and the Capes- arXiv:1504.03915 SPIRE [ IN IN redited. 5.14 ][ ][ ]. ]. ]. m Grant 2016/04341-5 from the homomorphisms Multiple membranes in M-theory J. Geom. Phys. with vanishing associator , ]. 3 SPIRE SPIRE R SPIRE IN IN arXiv:1405.2283 IN ∈ (Non-)commutative closed string on T-dual [ ][ (2015) 012004 ][ SPIRE ][ q IN 634 ][ arXiv:1211.6437 [ arXiv:1309.3172 Nonassociative geometry in quasi-Hopf Nonassociative geometry in quasi-Hopf Working with nonassociative geometry and field [ – 40 – (2015) 091 ) also naturally quantises the 3-algebraic struc- . The restriction of the triproduct ( 06 ε 5.17 (2013) 021 × arXiv:1409.6331 (2014) 171 [ arXiv:1505.04004 ]. ), which permits any use, distribution and reproduction in [ arXiv:1601.07353 [ 06 JHEP Triproducts, nonassociative star products and geometry of 01 , arXiv:1203.3546 [ J. Phys. Conf. Ser. SPIRE , 3-cocycles, non-associative star-products and the magnet T-duality, quotients and currents for non-geometric close IN JHEP JHEP , ][ (2014) 111 , (2015) 543 CC-BY 4.0 (2013) 1 89 replaced with 63 This article is distributed under the terms of the Creative C Non-commutativity and non-associativity of the doubled st η 527 × ). ]. PoS(CORFU2015)081 5.10 , ) of the membrane momentum space. Restricting ( SPIRE -flux string vacua 5.10 arXiv:1507.02792 IN non-geometric backgrounds toroidal backgrounds [ theory representation categories I: Bimodules andJ. their Geom. internal Phys. representation categories II: Connections and curvature Phys. Rept. R Fortsch. Phys. string compactifications [ alone thereby produces a non-trivial momentum space tripro The phase space triproduct ( p [1] D. Andriot, M. Larfors, D. and L¨ust P. Patalong, [3] J. Bagger, N. Lambert, S. Mukhi and C. Papageorgakis, [8] G.E. Barnes, A. Schenkel and R.J. Szabo, [9] C.D.A. Blair, [6] G.E. Barnes, A. 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