JHEP10(2017)144 Springer June 20, 2017 : October 8, 2017 October 20, 2017 : : September 20, 2017 : Received Accepted Published Revised -flux background and arises R c,d,e [email protected] Published for SISSA by , https://doi.org/10.1007/JHEP10(2017)144 [email protected] and Richard J. Szabo , a . 3 1705.09639 The Authors. Emanuel Malek c Flux compactifications, M-Theory, Non-Commutative Geometry, p-branes

We introduce a magnetic analogue of the seven-dimensional nonassociative -flux algebra that describes the phase space of M2-branes in four-dimensional a,b , R [email protected] E-mail: Department of Mathematics, Heriot-WattColin University, Maclaurin Building, Riccarton, Edinburgh,Maxwell EH14 Institute 4AS for U.K. MathematicalEdinburgh, Sciences, U.K. The Higgs Centre forEdinburgh, Theoretical Physics, U.K. Arnold Sommerfeld Center for 37,Ludwig-Maximilians-Universit¨atM¨unchen,Theresienstraße Theoretical Physics, M¨unchen,80333 Germany Department f¨urPhysik, Max-Planck-Institut f¨urPhysik, Werner-Heisenberg-Institut, F¨ohringerRing 6, M¨unchen,80805 Germany b c e d a Open Access Article funded by SCOAP because the smeared Kaluza-KleinU(1)-fibration. monopole is described by a U(1)-gerbeKeywords: rather than a ArXiv ePrint: algebra of electrons inin a M-theory background as M-waves of probingsystem uniform a also magnetic delocalised has Kaluza-Klein charge. monopole, aKlein We and seven-dimensional realise monopole show phase that this is space. this set-up non-geometricis because We the suggest it magnetic that cannot analogue the be of smeared described the Kaluza- by local a non-geometry local of metric. the This by a Spin(7) automorphismeight-dimensional of M-theory the phase 3-algebra space. thatphase provides space We a of argue covariantshowing electrons description that that of probing this upon the a algebra appropriatealgebra smeared also contractions, of magnetic a underlies the spin algebra monopole the foam model reduces in of to three-dimensional quantum quantum the gravity, or gravity noncommutative to by the nonassociative Abstract: octonionic locally non-geometric M-theory backgrounds. We show that these two algebras are related Dieter L¨ust, Non-geometric Kaluza-Klein monopoles andduals magnetic of M-theory R-flux backgrounds JHEP10(2017)144 (1.1) -flux, R ], which 9 – 1 -flux. On the R , ]. -flux and 6 Q = 0  5 j , p i -flux and p 9 -flux alone, and thus only Q  R 12 , i j δ 5 ~ = i  j , p i x – 1 –  11 , -flux background and M2-branes 18 k 1 R p ijk R 3 s ` 12 ~ i =  j , x 16 i x  -flux background. In this case the nonassociative coordinate algebra of the R -form fluxes of supergravity globally — such backgrounds are termed “globally non- p In this paper we will study three instances of nonassociative deformations and the In this context ‘non-geometry’ refers to the feature that one cannot define the metric 3.1 Non-geometric Kaluza-Klein3.2 monopole in M-theory M-waves 3.3 Free M-theory phase space 2.1 Nonassociative M-theory 2.2 Spin(7) automorphisms2.3 and magnetic dual Monopoles background and quantum gravity relations between them. Thea first is constant given by theclosed example string of phase closed space strings is propagating given in by respectively. At the same time,tivity these of also the parameterise space, noncommutativity and withother nonassocia- the hand, noncommutativity spatial related to nonassociativityoccurs both is in parameterised locally by non-geometric backgrounds. the can be quantised in the framework of deformation quantisation,and see e.g. [ geometric” spaces — ormeasure even of global locally and in local the non-geometry is case provided of by non-geometric “locally non-geometric” spaces. The commutative gauge theory. However,geometry it has has any concrete longof realisation been string within unclear theory, a whether whereby similar it noncommutative gravity. context can In in provide recent the an years closed it effectivebackground, string target has where been space sector the description found usual that fornoncommutative and the quantum geometric even closed notions nonassociative string deformations of sector of target in spacetime geometry a space [ non-geometric break down, does reveal 1 Introduction and summary It is well-known that the effective theory of open strings on D-branes with flux is a non- 3 M-wave phase space Contents 1 Introduction and summary 2 M2-brane phase space and its magnetic dual JHEP10(2017)144 ] ], 6 10 ) in (1.5) (1.2) (1.3) (1.4) is the 1.1 m ` . ijk gauge-invariant R , 3 s k ` , where B 3 m = -flux algebra ( ijk
R ] are the ε j N/` i ~ p , x ] = i = i , x  , k j , mag x ρ , p ~ M i i B, M i p x p  ] + [[ m m ∇ · i ~ ~ ) becomes everywhere nonassociative; ~ ` N ` , x ijk ] 1.3 ε k , i 2 j ←→ ←→ ←→ , x δ ~ j ~ i R − x R i ijk x – 2 – p R s s 1 = i ` ]]= ` ] + [[ k  ijk along the momentum space directions; alternatively, j k ε , p j , p , x 1 ijk i 3! ] j x , p R i  = are the electron coordinates and p , x [[ i i R x x is the magnetic charge density. Thus the coordinate algebra [[ , 1 3 mag -flux example does. This analogy can in fact be made more precise: ρ = 0 := R  = j ii ], the violation of associativity can be attributed geometrically to a non- i k 6 , x i B , x i x j ) is isomorphic to the nonassociative string theory ∂  ], and the electron wavefunctions necessarily vanish at their locations [ , x = 1.3 i 16 x ~ hh B ] is the magnetic field, 16 ~ ∇ · B – Now consider a constant magnetic charge density The second nonassociative algebra we shall study is that governing the phase space of 10 empty space. This signals thatgeometry, just the like magnetic the charge examplethe also algebra shows signs ( of localthree non- spatial dimensions with the isomorphism given schematically by monopoles as connections on non-trivial fibre bundles over thecharacteristic excised magnetic space. scale. Becausethroughout all the of magnetic space, chargeif the is one coordinate now wanted algebra uniformly to ( remove distributed the magnetic sources from position space, one would be left with magnetic charges, onetion can space. avoid the Bymonopoles angular nonassociativity [ momentum by conservation, excising themotivating these electrons their never points removal reach from in thefrom position posi- magnetic the space. fact thatthe This the excision loci magnetic is of field of the itself course magnetic and also sources, its motivated and associated is gauge field commonly are used singular in at the geometric description of given by where violates associativity precisely at the loci of the magnetic charges. In the case of isolated an electron in three spatialby dimensions [ in the presence of magnetic charges, which is given where momenta. There is now a non-trivial Jacobiator amongst the momentum coordinates we may regard italong as the a “dual gerbe coordinates”dinate on in algebra, the a like doubled suitable the space solutiondespite others to of its we the nonassociative will double nature. section be field constraints. encountering theory in with This this coor- curvature paper, can be quantised [ As pointed out by [ commutative gerbe with connectivehas structure components on given phase by the space flux whose curvature three-form which has non-vanishing Jacobiator amongst the position coordinates given by JHEP10(2017)144 - ], R , , 19 i k (1.7) (1.6) p x 4 1234 , , ,ijk i 4 ] that the -flux case, 4 x R R R 17 2 3 s 2 s λ s ` ~ ` λ ` ~ , 2 i i λ = = = ]]= 0    i 4 k 4 , x , p , x , x 4 i j ] which parameterises the j p x -flux algebra to M-theory.  , p  18 , x i R i p [[ x ) can be understood by duality  1.6 -flux [ R ), led to the conjecture [ , ,
1.1 j = 0 x -flux algebra, like the other examples con- to distinguish the phase space variables of µ k i R p δ ,
− – 3 – j i p ] on non-geometric backgrounds in M-theory has R/M x , k i ], the constraint ( µ,νραβ k j 17 k δ is the M-theory δ R x 17 − 2 jk -flux algebra ( k i λ 2 s p , R ε ` -flux background, which has also been argued before [ 2 j i k µ,νραβ , ~ λ δ R , p , , s R k ~ ` 4 4 k k + i x p x p 4 4 4 ,ijk 1234 x ijk + 4 , ijk ε 4 k 4 ε ,ijk R ,ijk 4 and 4 3 ij 4 x R 2 s λ ε λ i j ` 2 s 2 s R s R ` δ ~ 2 ` 2 λ 4 s 3 s i 2 s ~ ` λ ` λ ~ ` s ,..., 2 ~ ` i i ` ~ − − − − − = 1 ======     ii ii ii -flux algebra is to be identified with the octonion algebra and hence is given by j j j 4 4 k k k ] it was shown that the M-theory R , p , p , x ]. These observations, together with the fact that the octonion algebra can be , x , x i i is the dimensionless contraction parameter. , x , x , x i j j 21 j j j p x x 20  µ, ν, . . . λ   -flux/magnetic backgrounds, respectively. In this case, just as for the constant , p , x , p , x , x i i i i i In [ The third nonassociative algebra that we consider here is given by the seven imaginary R p p p p x   hh hh hh where sidered here, can also be quantised based on the observation that it is determined by a M-theory by a further duality tothere the Freed-Witten are anomaly. no In D0-branes theand in string which theory the further limit agrees it with implies expectationsspace that from [ quantum mechanics on thecontracted nonassociative to the string theory where local non-geometry. As discussed into [ the twisted torus, wherein a particular two-cycle is homologically trivial, or equivalently units of the octonions.suggested Recent that work these [ Crucially, underlie this identification the relies on the lift factM-theory that of backgrounds in four-dimensional the there locally non-geometric is stringseven-dimensional. a theory The missing missing momentum momentum mode constraint and was conjectured hence to the take phase the space form is the flux example, the nonassociativity cannotcharge be density avoided. can Furthermore, the only constanta be magnetic gerbe described geometrically rather in thanalthough terms in a this of connection instance a on the connective gerbe a structure resides on fibre over position bundle, space. again similarly to the where we have included the subscripts JHEP10(2017)144 , ), ). i , , p 4 . k 1.8 (1.8) p k x µνρσ p 4 ). The 1234 ijk , 4 ε ijk ) and the N ε ,ijk 1.8 ε 2 R 4 = λ 2 2 s 2 s 1.3 R s ` 2 ` λ ` 2 s 2 -structure can ~ 2 2 ` 2 2 ~ λ 2 2 2 G ,µνρσ λ − − − 4 R ) can be understood ] = ] = ] = ] = j 4 4 k 1.5 ,p ,p ,x ,x i i j j ,p ,p ,x ,x 4 i i 4 p p p x [ [ [ ) when the momentum con- [ 1.7 , -flux and magnetic monopole back-
-flux is given by j R R p ik δ , − k k p p , 4 -flux, becomes a gerbe over position space, the ij
j δ R – 4 – ,ijk x 4 k i R δ , 1234 2 s , − k ` i 4 2 x 2 x R λ k j jk 2 s i δ (when the M-theory − ε 2 -flux phase space with that of the magnetic monopole, and 4 λ` 2 2 p R N 2 s 2 s λ λ j i − ` ` 2 δ 2 4 2 , , 2 , ~ p ~ 4 k k and 4 x x p − + ) to a 3-algebra with 3-bracket relations 1234 4 4 4 4 λ , , 4 ,ijk i x x ijk 4 1.7 j i ε x k ,ijk R ,ijk ]. This suggests that the parameters we identify are the Planck length δ 4 ) in turn reduces to the algebra ( R 4 3 3 ij 2 s s 2 s 2 2 s ` ε 2 s 22 λ λ λ R R ` ` 2 ` 2 1.8 2 2 2 2 2 s 2 s 2 s 2 2 λ 2 2 ~ ~ ~ ` λ` λ ` 2 2 2 ~ λ` − − − − − − -structure on the seven-dimensional phase space. In turn, the ) is implemented as a type of gauge-fixing constraint. 2 ] = ] = ] = ] = ] = ] = ] = ] = G j j 4 4 4 k k k 1.6 ,x ,x ,x ,x ,x ,x ,x ,x i i ] it was proposed that the full eight-dimensional Spin(7)-covariant phase space is i j j j j j Given this observation, it is natural to wonder what role the octonion algebra plays In this paper we shall begin by showing that the isomorphism ( ,p ,p ,x 21 ,p ,p ,x ,x ,x and the magnetic charge. i 4 4 i 4 i i i p p p p p p p x [ [ [ [ P [ [ [ [ volves two parameters, whose contraction limits respectively leadnoncommutative to algebra the underlying magnetic a monopolequantum certain algebra gravity Ponzano-Regge ( [ model of` three-dimensional smeared magnetic monopole. Thisgrounds suggests are that related the by a sort of “canonical transformation”, preserving thein 3-algebra the ( case ofdimensional the phase space three-dimensional is magnetic related monopole. to a We magnetic will monopole in argue quantum that gravity this since it seven- in- straint ( as an automorphism ofautomorphism the exchanges Spin(7) the symmetry groupcorrespondingly underlying a the missing 3-algebra momentumlarly, coordinate ( the for gerbe a over momentum missing space, position the coordinate. Simi- The 3-algebra ( described by lifting ( be naturally extended toframework a for Spin(7)-structure understanding in the eight fullfull dimensions, unconstrained eight-dimensional suggesting M-theory algebra an phase algebraic space ofexpectations in that octonions. terms higher 3-algebra of Based structures the in underlie on the [ this lift of observation, string and theory to the M-theory, general choice of JHEP10(2017)144 - R we (2.1) 3.3 is related to ˆ D ˆ C -flux, the eight- ˆ B R ˆ A φ -flux, the two eight- R , ˆ D 0, and describes the propagation Ξ 0 turns off quantum gravitational ˆ → D ˆ C P ˆ → B , that we show can be reduced to the , ` ˆ A s λ φ g 2 2.2 ~ -flux background. Finally, in section 2 R − – 5 – =  , and argue that they only have a seven-dimensional and then describe its magnetic dual via a Spin(7) ˆ C -flux background and M2-branes ) in section R 3.2 ,Ξ we describe this magnetic dual by a delocalised Kaluza- 2.1 ˆ B 1.8 3.1 ,Ξ ˆ ]. The brackets of this 3-algebra are given by A and the rank four antisymmetric tensor Ξ 21 8  R . In section SO(8) [ -flux background or of an electron in the field of a constant magnetic R ⊆ 2.3 -flux algebra in section ) parameterise R ˆ A Ξ The outline of the remainder of this paper is as follows. We begin by reviewing the While the usual Kaluza-Klein monopole solution is based on Dirac monopoles, in order To explicitly realise this picture and to further understand the missing phase space subgroup Spin(7) where ( the structure constants of the eight-dimensional nonassociative algebra of octonions. The 2 M2-brane phase space2.1 and its magnetic Nonassociative dual M-theory In locally non-geometricdimensional M-theory phase in space four is described dimensions by with a constant 3-algebra structure which is invariant under a show that in thedimensional Spin(7)-symmetric limit 3-algebras where become completely there identicalthe and is unique should no non-associative describe eight-dimensional magnetic phasedecompactification charge space limit of and M-theory, of no possibly the even M-theory in circle. the phase space algebra underlyinggravity a in section certain spin foamKlein model monopole of in M-theory, three-dimensionalprobes which quantum of does this not background in admit section phase a space, local similarly metric. to We M2-branes introduce in M-wave the of a string incharge the density. M-theory automorphism of the 3-algebra ( provides an isomorphism between theflux phase background of spaces M-theory of and M2-branesbackground of of in M-waves M-theory, in which the are the non-geometric both non-geometric providedimaginary Kaluza-Klein by the monopole unit nonassociative octonions. algebra of theeffects, The seven i.e. the contraction effects limit of finite string coupling with to accomodate for the uniformsources, distribution of we magnetic actually charge, or needits equivalently a of metric D6-brane has non-local no variant local definedcall expression, this by which solution is smearing a an the “non-geometric earmarkthan Kaluza-Klein standard of monopole”, local a solution; being non-geometry fibre based and on hence bundle a we as gerbe rather in the standard case. Then the canonical transformation above coordinate constraint, we also embed awhere magnetic it monopole can into be string theory realisedprobes and as M-theory of a this D6-brane background and areperspective its realised we Kaluza-Klein obtain as monopole a D0-branes uplift. natural andM-theory The interpretation M-waves, of lift electron respectively. the as From seventh the phase this momentum space along coordinate the in the Taub-NUT circle. JHEP10(2017)144 , 2 8 
G Ξ ˆ −→ B (2.4) (2.5) (2.6) (2.2) (2.3) ,Ξ ) = ˆ ˆ A F  Ξ ˆ ( Ξ E ,Ξ f SO(7) with
 ˆ ,Ξ ˆ E F ˆ ˆ ⊆ C restricts to A ˆ ,Ξ D 2 8 ,Ξ ˆ ˆ D C  G φ ˆ D ,Ξ ˆ ˆ F E ˆ ˆ C ˆ B B ,Ξ ˆ δ Ξ ˆ D B ˆ C  − . φ ,Ξ ˆ ˆ F +
ˆ A E ˆ A  ˆ ˆ B A Ξ ˆ ˆ C δ D of the eight-dimensional phase Ξ ˆ  E ) = 0 in the remaining 3-algebra ˆ + φ E 4 ,Ξ Ξ + ˆ ˆ ˆ ( D ˆ R F C D  , ˆ ˆ f C ˆ B B ] ˆ ˆ D ˆ × A δ ,Ξ C ,  , ˆ φ , f 4 E ˆ ,Ξ − E 1 φ B R ˆ − E , ˆ = 0 F ˆ D ˆ , ξ ⊕ ˆ ,Ξ A A ˆ ∈ µ A ˆ ,Ξ A ˆ B 7 E Ξ p δ  ξ ) ˆ ], these are just the commutation relations of ˆ D ˆ C µ E ) = 0 = ,Ξ + φ – 6 – ˆ ˆ D 17 2 A Ξ , p ˆ ˆ ˆ F ,Ξ C C ( µ G ] := [ ˆ Ξ ˆ ˆ ˆ B ].

B B B f µ,νραβ x B ˆ δ E 8 (  φ R -flux imposes the constraint 21 ˆ ,Ξ D , ξ ) to vanish represents a “higher non-associativity”, as + − -flux [ ˆ R φ A A 4 R ξ ˆ ~ Ξ 2.2 [ C −→ ˆ A − ) δ ˆ  A ). For the particular choice of constraint function ]]are obtained by setting 4 Ξ 1 C 12 ~ 2.1 , ξ B = := , ξ  ] can be understood by duality from the twisted torus, wherein a two- A ˆ E ξ 17 ] from ( C ,Ξ ˆ , ξ D B ) in defining a Nambu-Poisson bracket is measured by its violation of the ,Ξ , ξ ˆ C ), live in the seven-dimensional representation. The resulting seven-dimensional 2.1 A ) lives in the singlet representation and the remaining coordinates, which we de- Spin(7). The presence of A ξ ξ Ξ ,Ξ ( ˆ ⊆ B f 3 ,Ξ The parameterisation ( Consider any constraint ˆ A . Then the action of the group Spin(7) on the spinor representation 2 -flux in string theory is represented by a rank 3 tensor, its M-theory uplift is a rank 5 Ξ  SU(2) which as argued in [ cycle becomes homologically trivial and thus no M2-branes can wrap this cycle. However, relations [ which arises from turningthe on nonassociative the octonion algebra [ space into position and momentum coordinates further breaks the symmetry group to phase space isbrackets governed by a nonassociative algebra invariant under whose Jacobiators [[ according to the decomposition where note by ( which reduces the phase spaceG by one dimension and breaks the symmetry as Spin(7) The fundamental identity plays thethus higher the 3-algebraic failure analogue of of theexpected the 5-bracket Jacobi in ( identity the and M-theoryR lift of string theory.tensor. This also matches the fact that while the fundamental identity through the 5-bracket failure of ( JHEP10(2017)144 4 , , x 4 , p is a k k with (2.8) (2.7) , x p 3 i 3 ijk T ε ijk ijk T ε 2 R p λ 2 2 s R ε 2 s 2 s 2 ` λ ` ` 2 s 2 ~ 2 2 ` 2 2 λ ~ 2 2 λ − − − ]. ] ) implies that there ] = ] = ] = ] = j 4 4 k 20 . However, as 21 2.6 3 , p , p , x , x i i j j T is a dynamical variable we , p , p , x , x 4 i i 4 4 p p p x [ p [ [ [ ,
j -flux background [ p , R , ik δ
= 0, and since j , 4 = 0 − x p k k 4 k i – 7 – p ) reads δ p ). This also agrees with quantitative expectations , ij R p − SU(2), where SO(4) is the subgroup leaving invari- ijk 2.6 k δ 2.6 i x x × R ε k R j jk i δ 2 s -flux wherein the only non-vanishing tensor components 2 s ε ` R 2 2 2 2 2 λ ` 2 s 2 s λ λ SU(2) λ , which breaks the symmetry group of the seven-dimensional ` ` 4 2 2 2 − 2 p − ' ~ , ~ 6= 0, this forces 4 , , 4 λ k p 4 2 p − ~ k + R x i x p j i 4 is related to the radius of the M-theory circle parameterised by 4 , ijk x i x − ijk , the constraint ( λ ijk SO(4) ijk j i ε x k R δ δ R ε 3 3 ij 2 s ⊇ 2 s 2 s ). For ε 2 s ` ) = 2 s λ λ λ R ε R ε ` ` νραβ ` 2 2 2 2 2 2 2 2 s 2 s 2 s λ 2 2 2 2 2.7 ~ ~ ~ ` λ ` λ ` G 2 2 x, p R ε 2 ( ~ λ ` − − − − − − f = ) which thus has to vanish. ] = ] = ] = ] = ] = ] = ] = ] = j j 4 4 4 k k k 2.6 ,νραβ , x , x , x , x , x , x , x , x i i i manifold, the Freed-Witten anomaly forbids precisely this configuration and thus is a 4 j j j j j For the particular choice of In the string theory limit, where we compactify one of the spatial directions on a circle c R , p , p , x , p , p , x , x , x i 4 4 i 4 i i i -flux while the D0-brane would be a D3-brane wrapping this p p p p p p p x [ [ [ [ [ [ [ [ the constraint ( need to implement thisThus we constraint set following thephase gauge-fixing space procedure to described above. ant the split into position and momentum coordinates. The corresponding nonassociative where the parameter and is specified more precisely below. This 3-algebra is understood to be supplemented by are and the corresponding 3-algebra adapted to this frame is given by [ the three dualities, theH locally non-geometric background becomesspin a three-torus dual version of momentum constraint ( from nonassociative quantum mechanics on the out by ( and consider the limitcan where be its no radiusrelated D0-branes vanishes, by in the T-duality the along constraint locally the ( other non-geometric three string directions background. to the This, Freed-Witten anomaly: in after turn, is this would-be M2-brane wrapping mode is related by U-duality to the momentum singled JHEP10(2017)144 , ), , i 2) x , 2.9 (2.9) 2 (2.10) (2.11) (2 λ ). This s , where ` ~ ) on the = 0, ob- 1 i 2.8 4 2.6 p ] = 4 , x i Spin(7), the spinor p [ ⊆ 3 1 x . , k p 3) , , , i ijk 2) (1 ε , R p 2 2 -flux algebra. The commutator of two λ p s 3 , ⊕ R ~ s i ` ~ (1 λ ` i 1) − , ⊕ (1 = 0 explicitly, e.g. by setting ] = ] = i 3 3 4 j SO(4) by identifying two of the SU(2) factors 2) x x p µ , ⊕ , x , p – 8 – p 1 i = 0 in the remaining 3-brackets from ( 4 , p x −→ [ 4 [ 2) p 3 (2 , νραβ ; 4 transform in the fundamental representation . Under the embedding SU(2) , µ (2 = 2 1 3 , p 3 R , G = k 2 x , ⊆ = 1, can be conveniently represented as in figure SU(2) jk s

i ` SO(4) = 1 ε 8

µ 8 λ = 2 s , x , SO(4) ~ ` ~ µ i k x ] p = + ]: 17 −→ λ ijk 4 3 17 x restricts according to the decomposition = i j i . R ε δ 8 3 s − R ` ~ s ~ SU(2) i i ` . Commutation relations of the nonassociative −→ ] = ] = j j , p , x The 3-algebra provides a “covariant” formulation of the phase space algebra ( i i x x [ [ , with their diagonal subgroup. One then obtains the representations Here we recognise thephase anticipated space SO(4) from [ representation content of the reduced M2-brane representation corresponding to the fourphase space position constraint and we break four SU(2) momentum coordinates. To implement the whereas imposing thescures constraint some of theposition symmetry. and momentum UnderstandingSpin(7) coordinates the therefore action requires of the the symmetry constraint ( breaking pattern whose Jacobiators are given byalgebra, setting with each node is labelled by a phase space coordinate. algebra is given by [ Figure 1 phase space coordinates ispoints given from by the the first to third theside, node second element otherwise along of the the commutator, line one connecting has a the factor two. of If i the on the arrow right-hand JHEP10(2017)144 4 x (2.16) (2.15) (2.14) (2.12) (2.13) in the 4 p -flux algebra. We R , µ M x = 0. m ~ ` 4 , p has units of length. As already = 0, the M-theory direction ˆ -flux and magnetic phase space D 4 ˆ , C R 7−→ . λ x ˆ k B ˆ p A ijk R µ φ p ijk s since = -flux background: R ε , -flux by a uniform magnetic charge density , and the remaining momentum 3 s R. s R R ε ˆ i j H ` ` R δ ˆ 3 s 3) G ` , ~ ˆ ~ , = F = i ˆ 7−→ , ` E ii – 9 – 4 (1 φ k ]: x ~ N ˆ distinguish the H ] = ] = i ] = 0 M µ , x 21 ˆ j j j G p j ˆ and define it by interchanging the roles of position F M , p , p m , x ~ ˆ , x i i E i ` i p ˆ x x D x [ [ [ ˆ hh = 0, which reduces M-theory to IIA string theory, the C and ˆ B -flux background. ˆ λ A R R ε 7−→ − ; in particular, the latter can be used to represent a central µ R 1) , x s 1 ` (1 , the nonassociativity of this phase space algebra is due to the locally 1 in eight dimensions [ ], and therefore to impose the constraint through ˆ D 21 ˆ C 3 m magnetic dual algebra ˆ N ` B ) reduces to the commutation relations for the nonassociative phase space ˆ 3 in the chiral representation A , = φ 2 2.9 , ~ B = 1 In the contraction limit ∇ · i , = i This canonical transformation preserves thefour-form M2-brane 3-algebra due to self-duality of the where again thevariables, subscripts together with the replacement ofρ the precisely, this exchange isbetween provided position by and a momentum standard variables canonical of transformation the form of order four 2.2 Spin(7) automorphisms andNow magnetic we dual background wish tocall find this the the magneticand momentum field coordinates analogue in of the 3-bracket the on M-theory the eight-dimensional phase space. More It is important tobecomes note a that central element inconstant of value, the that the contraction here contracted limit mentioned we algebra, in take section which to can be non-geometric therefore nature be of set the to any with the non-vanishing Jacobiators trivial representation element, as in [ algebra ( algebra describing the motion of a string in an p JHEP10(2017)144 , 4 , . p k k (2.18) (2.17) (2.19) x p ijk ijk ijk , ε i N ε p 2 m N ε 3 2 m ` 2 N ` 2 2 , λ 2 m , 2 ~ i λ 2 λ ` i 2 m x ~ p ` 2 − − 2 λ 3 m λ N ] = ] = ] = ] = m ` j 4 4 k ~ ` i , p , p , x , x ) turns into the magnetic i i j j , p , p , x 2.9 , x in the trivial representation; ] = i ] = 4 i i 4 i 4 p p 4 p x [ [ [ [ , p x , p i 4 p x , [ [ transform in the fundamental rep- k x µ = 0 to reduce to a seven-dimensional p , 4 , ijk
j x ε
, , j x -flux algebra ( i j p k N k i δ R 3 m p δ ~ , ik ` ~ i δ , jk − i k i – 10 – , − x x k λ ε k ] = i ] = ] = 0 p k j p jk , j j j δ i m k ` Spin(7). Now ij ε , ijk , p , p , x δ x i i i ε 2 m k p ⊆ ` x N x x [ 2 [ 2 m [ + i 2 λ N ijk 2 ` m 2 λ 3 . The canonical transformation has a natural realisation 4 ) to 2 ` λ 2 2 ijk p + 2 2 m , λ ε 2 ~ ε ` j , i λ 4 4 2.8 δ m k x , x − N ` − p + 3 m k m i k 4 ~ 4 ` ` 4 in the spinor representation, and ij x ijk x , i p p ijk − ε i i j i j i ε δ x x δ ijk 2 m N ε 3 ijk ` ε ] = ] = i ] = ε λ λ 2 N N 2 m j j j λ 2 2 m 2 ` λ 2 2 2 m 2 m m 2 ` 2 ~ 2 , p , x , x ` 2 ` λ λ λ N 2 2 i i i 2 p p ~ ~ ~ λ ` x − − − − [ [ [ = 0 these are just the commutation relations for the nonassociative algebra λ ] = ] = ] = ] = ] = ] = ] = ] = j j 4 4 4 . k k k 4 p , x , x , x , x , x , x , x , x i i i j j j j j , p , p , x , p , p , x , x , x When Under this canonical transformation, the 4 i 4 i 4 i i i ←→ p p p p p p p x [ [ [ [ [ [ [ [ 4 describing the motion of a charged particle in a uniform magnetic monopole background: which also follows from thediagrammatically nonassociative as octonion algebra, in and figure on can again the be diagram: represented x it acts by exchanging vertices with the opposite faces, and replacing in particular, we canphase now space impose with the quasi-Poisson constraint structurephase which space. is “dual” to that of the reduced M2-brane dual phase space algebra given by In turn, this corresponds tofactors an automorphism in of the Spin(7) embedding whichresentation exchanges SU(2) of two of SO(4), the SU(2) and it maps the 3-brackets ( JHEP10(2017)144 /r i of a p (2.23) (2.20) (2.21) (2.22) 1 λ = i q | ≤ ~q | 0, ≤ = 0 and by taking .  4 4 4 q
R , i , p p k 2 ⊆ q 1 4 r p 3 p  jk S i + = 4  λ ε i p i becomes a central element of the m ) reveals a noncommutative but , p p . ` 2 4 2 r x p 2  ijk ] = 0. We can show that 2 4 2.18 + i λ ε ν p i j ~ δ , p i N + 3 m µ = 1. Under this restriction the remaining i 2 2 ` p | − p ~ , 2 4 ~q i q | k 3 4 3 = p − x p p 2 x 2  + λ i λ – 11 – 2 ]]= ijk − , x ~q as k j := 1 2 λ ε p 3 , p λ 2 j S j p r m p 1 `  x , p m i = 0 the algebra ( . i ` 2 = 0, the momentum i . p , i.e. − [[ λ 3 λ − N − S ] = ] = i ] = 0 = ∈ j j j  ) 2 i , q , q p , x µ i i i q q x , x x [ [ [ 2 4 p  so that ( for dimensional reasons. /r m ~ 4 ` p . Commutation relations of the nonassociative magnetic monopole algebra. The commu- − = = 4 q 4 p 6= 0 by observing that for three-dimensional momentum space We can therefore restrict theand momenta to a unitcommutation three-sphere relations can be expressed on the lower hemisphere λ associative deformation of position space with [ is a central element of this algebra; this follows from contracted algebra, which can thereforebe be set to any constant value, that2.3 here we take to Monopoles andWe quantum can gravity gain some insight into the meaning of the position space noncommutativity for with the non-vanishing Jacobiators Again in the contraction limit Figure 2 tator of two phaseIf space coordinates the arrow is points givenon by from the the the right-hand third first side, node to otherwise along the the second line element connecting of the the two. commutator, one has a factor of i JHEP10(2017)144 . 0 1 λ → (3.2) (3.1) ) is a (2.24) whose λ ~y through 3 R P ` , U( 1 S are, generally, = 0 describes a 1 N S , × 2 3 ) at
to get a canonical six- R ~y 3 ), foliated by momentum d = 0 in the reduced seven- · 2.18 ∈ R ), which both follow iden- ~ A ) N 2.23 ∈ -flux algebra admits an inter- + 2.18 ~q R y z ~y, z d ]: integrating out the gravitational 1 − and ( 22 U 7 11 M is a U(1) gauge connection on U, , R d identified with the Planck length ~y P m 3 ` d ` + ∗ λ · ~y – 12 – = ) = d · ~y λ ( F ~y is the asymptotic radius of the fibre ~ A d 11 -deformation of the Poincar´egroup. U = κ R 11 A varies, the phase space algebra ( R 4 + p ). . 7 r 2 M , and s 3 ]. It is defined by the metric 2.24 R 23 = d 2 ) and the magnetic monopole algebra ( is related by s of radius d A 2.9 3 , so that noncommutativity in this instance is a consequence of the fact 3 1 and decompactifies the momentum space S is given by a locally non-geometric variant of the Kaluza-Klein monopole S is the metric of a seven-manifold ) symmetric under a = d  7 ' F plays the role of Planck’s constant. In this context the contraction limit 2.2 = M 2 2.23 s m 4 ` q Of course, we are not literally allowed to take the limit This suggests that the magnetic dual of the M-theory These are precisely the commutation relations that arise in a Ponzano-Regge spin = 0 in the first place. In the following we will propose a precise M-theory background -flux algebra ( 4 local coordinates onthe a hyper-K¨ahlerstructure; four-manifold hyper-K¨ahler here withharmonic a function U(1) on isometrycurvature that preserves In this section we shall proposein that the section nonassociative magnetic dual backgroundsolution discussed in M-theory [ where d Planck length through ( 3 M-wave phase space 3.1 Non-geometric Kaluza-Klein monopole in M-theory x which realises this picture, andR moreover describes the dualitytically between the from nonassociative the nonassociativealgebras algebra the of the meaning octonions. of the We will contraction show parameter that is in the both dual same, being identified as the dimensional phase space. As cone over the noncommutative spacespace defined spheres by the algebra ( dimensional phase space, as it is the presence of magnetic charge that forces the constraint where corresponds to neglecting quantum gravitational effects,mode which freezes the extra momentum that massive particles coupled toThe three-dimensional additional gravity deformation have momenta ofalgebra bounded the ( by canonical position-momentum commutator makes the pretation in terms ofdimensions, monopoles with in the contraction the parameter spacetime of Euclidean quantum gravity in three foam model of three-dimensionaldegrees quantum of freedom gravity there [ coupledan to spinless SU(2) matter Lie yields algebra anis effective noncommutative scalar position curved field space. theory and on the Theon momentum position SU(2) space coordinates in can this case be interpreted as right-invariant derivations JHEP10(2017)144 3 D : R A ρ (3.9) (3.6) (3.7) (3.8) (3.3) (3.4) (3.5) ) yields orbifold of degree . 1 3.1 2 3 − S R N A → 3 S 0; we can shift this . with the lens space 3 } ≤ | k 0 ~ k y ~y | y \{ ijk 3 0 by the gauge transforma- . -fibration over , R , 1 ≥ k ) N ε S y 3 m k ~y → , N . ` ( y , 2 | . The appropriate magnetic field, ijk ) = 3 (3) i = ε S ~y 11 y 1 ~y R and the Taub-NUT circle bundle is , ( d R 3 m U − | ` D 2 3 ij N 11 k N 3 + 1 π N δ S  y R 2 | | = . The asymptocially (locally) flat harmonic j ~y + 2 ~y ) = | y | | ) = 4 F S ~y ,F ( 3 m d ~y ~y N | | ` k ( ijk N 3 – 13 – ~y ε 2 ∗ ~ | B B = N ) = = + ijk ~y ∇ · k ) = χ ε ~ B y fibre shrinks to the fixed point of an ~y U( | j of the form 1 + d , and the usual point-like Dirac monopole with magnetic ) = ) = is homotopic to U( ~y ∇ · y 6 | S , ~y D D ~y to get the connection 1 } ( ( = A F D R 0 = 1 this is just the familiar Hopf fibration
ijk ~ ij ρ i ρ j ε F y y = N \{ 7 3 1 ) = − M R ~y by (generalised) Dirac charge quantization of the corresponding dual ( , but it is flat near the origin of , which is homotopic to 0 D i } ρ tan . The hyper-K¨ahlerstructure will ensure that the metric ( , is also easily determined: > 0 A ~y ~ ~y k Z that is associated to a minimal locally well-defined gauge connection y \{ ∈ 0 3 > ijk R ε N Z ∈ is the restriction of the 11-dimensional Hodge duality operator to the space ) = over 3 as total space; for N ~y ∗ ( N N χ In the present case of interest we require a constant magnetic charge distribution Let us begin by reviewing the standard Kaluza-Klein monopole solution. It is based Z / . More precisely, in the region outside the degeneration locus at the origin, where the 3 which is of coursecharge not distribution asymptotically (locally) flatwhich is since linear we in consider a uniform magnetic where again fluxes in string theory. This is solved by singularity, the space homotopic to the Seifert fibrationS over the embedding and it realises theN Euclidean Taub-NUT spacemonopole as is a local situated and the and the Dirac monopole canclass be described asfunction a connection is on a U(1)-bundle of first Chern tion It indeed leads to a point-like magnetic monopole source with charge density and corresponding curvature This gauge field is singularcoordinate along singularity the to negative the coordinate positive axes coordinate for axes for parameterised by half-BPS solutions. on the seven-manifold charge where JHEP10(2017)144 ~ ). A , 0 3.1 ~y 3 (3.13) (3.10) (3.11) (3.14) (3.12) in the d ). This ~ A
| 3.1 0 ], which is ~y 25 , − is non-zero, its ~y 24 | H ) 0 j ) + . y 0 k k y y − is contractible the gerbe − j -flux background there is Dirac monopoles situated y k ijk 3 . While ( R . ε y ρ R K ( 3 3 m 3 m -fluxes are generated by linear ∗ | N ` ` N 0 3 R . ~y = = . = 0 − 11 F ) 0 1 ~y ~y a R | 3 ~y ~y 3 = d + , and since − ) d Z 0 | } ) d ) in this case? Let us first give a heuristic 0 ~y a H ~y 0 ~ ( ~y ~y ijk D 3.1 − ε N \{ − − ~ A ~y 3 ~y 3 m ( ~y is nowhere closed, the associated gauge field | – 14 – ( R D N =1 π` K D ρ ij F a X 4 =1 K F a X , and hence no local expression for the Kaluza-Klein Z = ~ , which has exactly the local geometry needed to insert ) = A 0 Z 1 3 m ~y ) = ~y S ( 3 1 3 m ~y π` ~ A 1 × 4 π` 3 ) d U( 0 4 R whose curvature is = ~y ρ F − ) = ~y ~y ( ( i D ij A F is simply in the large radius limit, or alternatively by formally applying Stokes’ 3 , with gauge field Z 3 ~y R R 3 m 1 ∈ π` a 4 ~y , rather than a U(1)-bundle over 3 ) = , so that now there is no need to excise the origin. The solution describes a U(1)-gerbe R ~y It is instructive to approximate the constant magnetic charge distribution by a finite What is then the meaning of the metric ( 3 ( i R A It corresponds to a magneticat charge points distribution consisting of sum of Dirac monopolessolution and hence based to on compare the withdefined multi-centred the by version the standard of asymptotically Kaluza-Klein the (locally) monopole Taub-NUT flat geometry harmonic [ function singularities. The factlocal that expression for for the the casemonopole gauge of metric, field constant is magnetic analogousalso to charge no the density local observation there expression that is for for the no the metric. theorem. Therefore we can express the gauge field in the non-local smeared form where here the integration should be additionally defined so that it avoids the Dirac string The same is true for the magnetic field: The integration here canball be defined centred more at precisely by using distributions supported on a argument, which producesfollowing at way. least We a knowvia formal that an non-local the integral over uniform smeared infinitely-many magnetic expression densely charge distributed for distribution Dirac can monopoles: be obtained is necessarily trivial.is However, given the by trivial the gerbecohomology two-form class can vanishes still because have thetrivial a gerbe gerbe is non-zero on trivial. curving Nevertheless,a which the Kaluza-Klein total type space monopole of solution the into the four-dimensional part of the metric ( multi-vectors. Now since thedoes two-form not even exist locally.is Hence due there is to also the no fact localof that expression the for constant the magnetic metric charge ( over distribution is globally defined over all This is analogous to the fact that the dual non-geometric JHEP10(2017)144 ] , D a = ρ ~y 26 3 D ∗ ρ (3.16) (3.15) rather vanish . 3 , where 3 = ∗ 3 3 3 R R D R R = ∈ H 0 D with singular , or inside its ~y , 3 F 3 1 homologically R Z R − = K
. For this, we observe ); this means that we Z ; Z 3 ; -fibration over 3 S
1 S coincide, the corresponding Z 3 S ( ∪ {∞} ; 3 a H } 3 ~y H ) = 0. However, since the two- 0 ~ R ] . To see that this description of Z ' ; = \{ N 27 3 −−→ 3 3 , we use the long exact cohomology . R
S D ( Z , the class is non-trivial as an element R Z 3 3 , ; H 3 R } H R ) = 0 = ~ Z 3 0 then means that the cohomology class of ; D ~ \{ } H F ) generated by three-forms on 3 are generic, the space has = 0 ~ of the monopole is entirely encoded in the con- Z a . For this, we appeal to a construction from [ – 15 – 0 ~y ; ,S ' ~y \{ } 3 } ~y −−→ 3 0 S ~ 0 ~
R orbifold fixed point over each monopole location 3 Z ( \{ \{ ; 1 2 3 -type orbifold singularities develop. Hence the smeared 3 H ) and the fact that all cohomology groups of } ) with − A R H } R 0 ~ , N ' is the magnetic charge 0 ~ 3 . When two or more of the A
3 3.10 \{ R induces an isomorphism [ Z Z ( \{ R 3 to the class of d 3 3 ; 3 R } S D H R ) = 0 ~ 2 , F 1 in Z 3 → , ), can be equivalently described by a U(1)-gerbe with connective ; H \{ R } − Z 3 3 -bundle projection, of paths connecting neighbouring singular points ; 0 ~ R 1 } R is contractible. In other words, the usual geometric description of the , S 0 ~ \{ 3 } 3 R 0 \{ ~ R ,...,K 3 3 0 by virtue of Maxwell’s equations with magnetic sources: d , ~ , which is a class in \{ 3 . When the position moduli } R fiber shrinks to an H ]). The crucial observation is that the magnetic charge of a Dirac monopole can 0 ; the expression ( ( ~ 3 R = 1 0 ~ 2 1 ( S 27 3 cannot be extended through the origin of a S H \{ , H D 3 ,...,K F +1 R in a The information about the location In this interpretation we can move the Dirac monopole around in To understand more precisely the geometric meaning of the smeared Kaluza-Klein ; this class trivially vanishes as an element of , ~y D D = 1 a now deal directlywhich with is the exactly nonassociativity whatif we induced we need by move for the the our location purposes. point-like of magnetic In the particular, monopole charge, the fromnective the same structure on origin class this to is gerbe, any obtained which other can point be described explicitly in terms of the monopole where the first isomorphismfact follows that from theDirac excision monopole theorem as and a U(1)-bundleclass the with in second connection, from whose curvature the structure two-form whose represents curvature a three-form represents a class in which sends the class of one-point compactification which is thethat three-sphere the embedding over sequence for the pair ( to show that there is an isomorphism form of the relative cohomology support at H the monopole charge is equivalent to that above as the first Chern class of a U(1)-bundle than simply on the excised(see space also [ be interpreted as the cohomologysupported class at of the closedH three-form delta-function two-cycles collapse and additional solution defined above, which useshighly a singular continuous because distribution any ofin pair Dirac particular, monopoles, of there becomes monopoles are can no come M2-brane wrapping arbitrarily modes close inmonopole, to this we one background. need another; a geometric interpretation of the Dirac monopole over all of now the a non-trivial two-cycles on whichimages, M2-branes under can the wrap; they are constructed as the inverse The multi-centred Taub-NUT space is again locally the same ~y JHEP10(2017)144 , - 3 is 3 ∞ ∞ z S R ~ . In A , χ ,U 3 0 0 = (3.17) (3.18) (3.19) ~y ~y carries R ): they χ U . ∈ F 3 0 realised as , the usual 3.12 R . From the z ~y 3 \ {∞} ~ A 3 R )–( S 3.10 on 0 , whose intersection D 3 . This interpretation ~y supported in } F R 0 0 is a closed two-form on 3 ~y of the radial coordinate, ∞ ~y = of the gauge bundle of a 0 S b D 2 χ ~y 0 \{ r F D ~y 3 = = d parameterised by A R 0 1 ∞ ) is reflected in the feature that D ~y \ {∞} b S = by degrees of freedom that are F 3 S ∞ ∧ 3.12 , . Since . 0 0 and π N , } ], and indeed we shall suggest below 3.1 = ~y 0 ~y N } . Choose a partition of unity 0 . U χ ~y . Transforming to spherical coordinates 28 ∞ } , which appears as a metric component = 4 0 } ~y 0 U ~ 0 A \{ ~y are functions on ~y D = d ~y \{ = 1 3 0 3 F \{ D . For this, we take a stereographic cover of \{ ∞ R ~y and S ∞ 0 } 3 3 F 0 ] 2 ~y χ , χ } = R – 16 – S R 0 0 ~y ∧ on 26 ; it follows that the Dixmier-Douady class of this Z ~y + ~y 0 3 ∞ 0 = , \{ χ D ∞ 0 ~y = ~y R 3 χ \{ ~y F 0 χ R d 3 ∈ U ~y ∞} ∞ 0 h − S , , i.e. χ 0 on ~y 3 on = = ∞ ~y − S ~y 0 0 0 Z d D ~y , which is defined on ~y ~y ,U · = 0 \{ 0 can we identify the connection b ) A 0 U 0 ~y ~y 3 ~y 3 ~y U b S R = d but supported in so that it depends only on the square = d is just the magnetic charge − 0 0 3 = 0 ∈ ~y ~y Z ~y D ( S ~y 0 h χ ∞ D F ~y U ~ A ) = = ∩ Z 0 = ; 0 ~y 3 0 ~y b D ~y S U ( − A 3 is a sphere centred at 0 , the curvature three-form := ∞ H 2 } ~y b 0 S and choosing ∞ ~y , 0 3 For our non-geometric Kaluza-Klein monopole, the absence of a local gauge field This construction suggests a geometric meaning for the expressions ( ~y , which is magnetically dual to a D0-brane and couples magnetically to R \{ U 7 3 M perspective of the 11-dimensionalgraviton momentum theory, mode. the electric dual to thenot NUT-charge an is issue thus on a reduction to IIA string theory, because only the magnetic field One now needs toelectrically charged probe under the the backgroundin U(1) the of gauge 11-dimensional section field solution.direction That can this be should seen be as theKaluza-Klein follows. M-wave monopole By (or of reducing pp-wave) on M-theory along the becomes the circle a D6-brane of IIA string theory wrapped on is similar in spirittions to of the noncommutative global and descriptiona nonassociative of more tori non-geometric precise [ string interpretation backgroundsnonassociative of as geometry this fibra- adapted singular to Kaluza-Klein the monopole family solution of in3.2 Dirac terms monopole of gerbes over M-waves describe the connective structure ona the family constant of magnetic Dirac chargeparticular, monopole gerbe the gerbes, on inherent parameterised non-locality of byonly the the gauge at monopole field each locations ( point Dirac monopole located at where gerbe in is globally defined on on by using Stokes’ theorem one computes [ Define curving two-forms so that R with contractible open sets is subordinate to the cover which satisfy connection JHEP10(2017)144 i - ρ 4 y 3 x and ∗ = 0. Its P (3.20) (3.21) i = ` x → F 0 should 11 → 0 if we make R 11 → ). The reduction 0. The standard R λ → 2.18 ). However, the wave 0 of weak string cou- 3 11 ). The disappearance of → , x 3 2 ,R s , p 11 g , x 2 1 . is given as ,R , p x s ). This agrees with the seven- 1 s , which implies that the string 2 g 4 p g / 3 ; 3 / finite, the limit , p , and therefore the configuration 3 1 11 3  4 s R x , x and ` 11 , p P 2 ]. In these settings a dual “smeared” 2 ` s ∼ R ` , x , p , 30 1  P , 1 ` x p e 11 = ; 29 ~ 3 R 0 with the limit s , x = 2 → – 17 – 4 p λ , x 1 is frozen and the Kaluza-Klein momentum quantum , g x 4 11 3 P . All of this suggests that one should identify p ). ` R 3.1 ). Our approach to the current problem at hand is differ- = 2.24 2 s . This corresponds to the contraction limit ` 3.1 11 R ∼ s g 0 the momentum ). Let us now provide some evidence for this proposal. → in the M-theory direction disappears from the phase space and we end up -manifold in ( , with the string theory parameters 2 4 11 11 2.18 p G R , such that our proposed phase space algebra of the non-geometric Kaluza-Klein R becomes the electric charge of the D0-particle. z e = 4 x Let us also see what happens to the phase space variables of the M-wave probe in this As the appropriate probe of this background is a momentum wave along the to be a can be easily seen by noting that 7 4 p so that for number limit. In fact, themomentum M-wave becomes an electricallywith charged the D0-particle six-dimensional phase when space ofground, the correctly electric particle described in by the the magnetic coordinates monopole ( back- It follows that,be in taken order with to thecoupling keep vanishes Planck as the length string vanishingthe length as identification proposed in ( pling and vanishing radius oflimit the where eleventh the dimension. contraction Henceidentification parameter of we M-theory would like parameters, to whichthe are identify radius the the 11-dimensional Planck length space of the M-wave iscan described have by momenta the three inseven-dimensional coordinates with all ( coordinates four ( dimensional directions, phase and space that hence appearsfrom the in M-theory phase the to nonassociative space string algebra theory of ( is the described M-wave by is the limit and monopole background in M-theory, which followsis from the given nonassociative by octonion ( algebra, direction, it has no local position with respect to try by torsion, i.e. geometricmatch flux, under which M-theory/IIA leads duality; to for source-modifiedM instance, equations in of certain motion cases that ent, the in uplift that involves taking we interpretdeformation the of smeared Kaluza-Klein phase monopole spacegeometric as geometry description sourcing of that a nonassociative section we propose to be equivalent to the non-local is now accounted foring by a of uniform the distribution upliftan of open of smeared problem such D6-brane which continuous sources.of has distributions supergravity been The of equations mean- investigated of D6-branes onlyKaluza-Klein motion, for to monopole see some in M-theory special e.g. M-theory is is [ instances interpreted generally at as the a level deformation of target space geome- physically relevant degrees of freedom, and the violation of the Bianchi identity d JHEP10(2017)144 , ) ) , ). s P k g ` = 0 2.6 x 2.7 (3.23) (3.22) jk , both 3 m N i ` P ε 0. Then ` 2 ) describe = becomes a 2 s )); in this ) collapses λ -flux which → ` ρ 2 , R 2.8 . 2 R P ~ 3.1 ]; this implied k k ` 3.22 ∈ x p − 17 4 4 p x ijk ijk 0 as j i ε ε δ 3 2 . Recall that one way 2 s 2 s 0 with finite → 2 s λ λ λ = 0 again fails to hold ` ` ), so that the same is ` -flux simply appears as 2 2 2 2 2 λ 4 2 ~ ~ ~ R → p of the M-theory circle, so → ∞ 2.14 − − − ρ -flux. Thus we find that the = 0, there is no longer any s g R R ] = ] = ] = j j 4 → ∞ 0 or 1 as -flux background [ , p , x , x to vanish 11 i with i j → R R 4  → , p , , p , p s 4 P i 4 R T ) (or equivalently the Planck length ` ` p p p s [ [ [ g  λ . This is the strong coupling limit of IIA = 0 ( ), and since the λ λ µ p -automorphism ( 2.7 = , 4 → ∞
λ Z j – 18 – x V by ( k µ,νραβ i 0 where quantum gravitational effects are turned 4 . Hence we can postulate that the constraint ( δ p R − → i 1 V (0) = 0 and x P λ → ∞ k j ` δ -flux on a non-compact space is not related by U-duality . From the M-theory perspective we can describe this more V R R 2 2 s . Thus in either limit λ , ` 1 ∈ 2 4 2 S 4 ~ p p = 0 is based on the Freed-Witten anomaly, which in the present × + the volume of the four-torus 4 ijk 4 , 3 p ε i x V R x 2 k λ 2 ij ) should satisfy ], in the absence of non-geometric flux, 2 s ε s 2 λ 2 s ` g ). However, in the decompactification limit the Freed-Witten anomaly and also ~ 21 ` 2 2 s λ ( 2 2 ~ ` , with 2 λ R 2 2 ~ − − / 3.21 ∈ 1 6= 0 forces − 4 ] the homology of the twisted torus was also used to argue for the constraint ( ] = ] = ] = x R V 4 k k 17 , p , x , x i j j In [ Let us now consider the decompactification limit = 0 by ( , p , p , p i 4 i p 4 p p [ [ [ to the twisted torus.in This the means decompactification that limit themay argument in used fact to be set more generally given by electric Kaluza-Klein charge upon compactification,dynamical and variable the again. momentum However, from the duality withscales the as twisted torus, oneconfiguration necessarily of obtains an non-vanishing the parameter to argue context forbade the existence of D0-branesp in the string argument no longer applies as the D0-particle number is only derived from M-theory as an that now covariantly as the limit ofstring infinite theory, volume wherein the contractionshould parameter nevertheless remain finite; this means that, as a function of the string coupling (with the roleslimit of the position four-dimensional and metricto momentum in that coordinates of the interchanged flat Kaluza-Kleineight-dimensional in space monopole Spin(7)-symmetric ( solution 3-algebras ( arenonassociative eight-dimensional isomorphic phase and space of describe M-theory. the unique Here we expect theall dimensionless 3-brackets quantity vanish inoff. the limit This algebratrue is of the symmetric 3-algebra under describing the M-waves in the absence of magnetic charge, As observed in [ constraint imposed on the momentum a parameter one can setthe it free to phase zero in space the 3-algebra 3-algebra. structure In of this M2-branes limit with the the brackets non-vanishing ( 3-brackets 3.3 Free M-theory phase space JHEP10(2017)144 . a of in ~y 1 A 4 x S (3.24) ] (2010) 6= 0, and 12 4 J. Phys. p , that is asymp- with fibre 1 JHEP 3 − , R K A Non-geometric Fluxes, non-vanishing. 6= 0 and arXiv:1202.6366 (2011) 385401 [ R R and is the least common multiple A 44 4 p D (2012) 121 04 J. Phys. , singularity at each monopole location . , where 1 ]. D JHEP − in the full eight-dimensional M-wave phase Z ) with both , = 0 N / 4 4 A x 4 2.8 R – 19 – SPIRE should now reinstate the local coordinate , this allows for both V , becomes Asymmetric Orbifolds, Non-Geometric Fluxes and IN R p z Nonassociative Gravity in String Theory? ][ νραβ = → ∞ ]. 4 it becomes an ALE space of type R ε x V ), which permits any use, distribution and reproduction in = ]. ). SPIRE → ∞ IN 2.17 11 ,νραβ ][ SPIRE , together with an . In the limit of dense monopole distribution, this reduction to 4 R IN K R K Z arXiv:1010.1263 ][ quotient singularity all coalesce, the two-cycles collapse to the same point and the space CC-BY 4.0 / [ 4 1 a and ~y ), which near infinity looks like a circle bundle over − R This article is distributed under the terms of the Creative Commons D -flux is N A 3.13 R T-duality and closed string non-commutative (doubled) geometry ]. , in the limit arXiv:1010.1361 11 [ (2011) 015401 SPIRE R arXiv:1106.0316 IN Asymmetric Strings and Nonassociative[ Geometry Non-Commutativity in Closed String[ Theory 44 084 Dually, in the non-geometric Kaluza-Klein monopole background, the decompactifica- R. Blumenhagen, A. Deser, D. L¨ust,E. Plauschinn and F. Rennecke, C. Condeescu, I. Florakis and D. L¨ust, R. Blumenhagen and E. Plauschinn, D. L¨ust, [3] [4] [1] [2] Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. 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