PoS(CORFU2017)200 https://pos.sissa.it/ -flux background in R ∗ Speaker. We review two known in theis literature the exemples phase of space non-associative star star product representing products. quantization The of first non-geometric one closed string theory. The second is thethe octonionic quasi-Poisson star algebra product isomorphic which provides to the thein quantization Malcev of details algebra the of construction, imaginary properties octonions. andular, We physical discuss we applications consider of the these quantization star ofexemples products. we non-geometric In M-theory formulate partic- background. the minimal Basednon-associative set on deformation of quantization. these physically two motivated conditions for the definition of ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Vladislav G. Kupriyanov Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, München, Germany CMCC-Universidade Federal do ABC, Santo André,Tomsk SP, State Brazil University, Tomsk, Russia E-mail: [email protected] Non-associative star products and quantization of non-geometric backgrounds in string and M-theory Corfu Summer Institute 2017 ’School and2-28 Workshops September on 2017 Elementary Particle Physics andCorfu, Gravity’ Greece PoS(CORFU2017)200 ], B ~ 0, 4 6= (1.3) (1.2) (1.4) (1.5) (1.6) (1.1) -field, ) quasi- x B ]. In the ( 5 ) is called ρ = 1.2 B ~ . ) , . Vladislav G. Kupriyanov x ( 0 . There are also interest- 0 k , then div . R ) B = , one obtains the monopole x = 0 ( p } structure. i jk } j ρ ) = ε f 6= p x e , , (  i h → − ρ = ki p { , x { θ } l j g ∂ { . π jl , g and B ~ i + j θ π x i j ∂ , i.e., in general, { δ }} div + f ] reads: i j quasi-Poisson g → i 3 i j , θ = ∂ i jk , p ], that the presence of a non-constant f θ ε } ) l 2 8 j { e x , ∂ , , , i j ( p 1 7 kl h 1 , δ = i j i , { θ , which is bi-linear, antisymmetric and satisfies the x θ 6 } = k } { + + g π = } j , jk , j }} } f π ). To start with, let us remind that the magnetic field θ h g π , l { , i , , , ∂ i x f g 1.1 il π k { { { , { p θ f  -flux algebra [ { i jk R 1 3 ] and references therein. R , 1. Note that making, = 9 2 : 3 s 0 = ¯ ` h : } = h = = + i jk , } g j } Π j , x 123 f x , i ε , { i x x { { , and bracket of two functions i jk ε R = bracket Poisson i jk , for which the Jacobi identity does not hold as a R } The problem we are going to discuss in this contribution to the Proceedings of the Corfu However, there are several physical situations when the system is formulated in terms of The standard Hamiltonian description of the classical mechanical system is based on the notion Another exemple quasi-Poisson brackets comes from the closed string theory with non-geometric g , f meaning the violation of the Jacobi identity. Sometimes in the literature the algebra ( If one admits the presence of the magnetic charges of the density The Jacobiator of the three covariant momenta reads with which will be discussed in the details in the Section 4, as well as its magnetic analogous [ Poisson without imposing the Jacobi identity on the bi-vector Summer Institute is thethe consistent deformation quantization quantization. of More quasi-Poisson precisely, structures we in are the interested framework in of the quantization of the bracket violating the Jacobi identity ( An important advantage of thisof description two is integrals a of Poisson motion Theorem, isPoisson stating again bracket that an one the integral may Poisson of formulate motion bracket is the (not that necessarily algebra the new), of standard i.e., classical Hamiltonian in observables. formalism terms is The of compatible the other with essential the point canonical quantization scheme. as a monopole algebra. Throughout this{ paper we will also call the antisymmetric bi-linear bracket algebra corresponding to a constant magneticing charge distribution exemples of non-Jacobian structure related to the non-geometric backgrounds in M-theorycontext [ of open string theoryresults it was in found the in non-associativity [ ofnon-associative structures a see star [ product. For specific exemples of non-commutative and Jacobi identity: of the fluxes. In particular, the constant Non-associative star products 1. Introduction can be introduced in the theory via the algebra of coordinates and covariant momenta: PoS(CORFU2017)200 ) ) in 1.9 (1.9) (1.8) (1.7) (1.11) (1.10) 1.9 ). 1.5 , is defined from ) f , , can be killed by a ). Or, in other words, g ) ( f Vladislav G. Kupriyanov − . are related by the gauge 1 , 1.7 g C } 0 ( g − , + 1 ], which also states that each f , , in the direction of a given bi- C ) = { g h ) = 0 11 · , is associative and represents the ) is trivial. In the first order one g f g 0, the associativity equation ( , ) = ? ( , f g 1 1.9 ) = 6= ) ( , h g 2. The associativity equation ( , f − 1 , ? / f C ] the star product is defined as a formal i jk (
f C , } g + g 1 ( Π g ( 10 } − r , C g ) D ? f C , , r f ? f 1 { ). There are natural "gauge" transformations ) gh f ¯ h , { C − i f ( 1.8 D , h ( = 1 ) = 0 , and the question is which condition should be 1 denote some bi-differential operators. The “initial ∞ ? g = ? 2 ¯ h 1 ? ∑ r , C ] ) ) − ¯ f h g g g i + − ( , → , D 2 ? f ) and ( ) f g − is also associative and gives quantization of the same 1 [ f ? · h ( = 0 C : , r f 0 1.7 ? g C , which should satisfy the condition of associativity, → f g 0 = ( D g lim ¯ h = : ( ? ? 1 ) g f ) to restrict the higher order terms in ( f ) provides the quantization of the classical bracket ( h C ? , ) should vanish. It happens that no other non-trivial conditions , then f g + 1.9 } 1.7 , , g h 1.1 f ) , appear. The existence of the associative star product for any Poisson ( ) f g g , A } { , f g ), while the antisymmetric one, f ( , which quantize the same Poisson bracket ( 1 f 0 1 ), and is given by , implies the consistency condition requiring that the classical Jacobiator of ? C { . C 1.10
} 2 1.8 g ). ¯ h , being an invertible differential operator. If , and ) f , i.e., it should satisfy ( ¯ ? h ) { O ( 1.10 x ( O i j follows from the Formality Theorem by Kontsevich [ + θ i j 1 θ is a deformation parameter, and = ¯ h D In the zero order in deformation parameter the condition ( If now we relax the Jacoby identity, i.e., suppose that Remind that in the standard setting the problem of deformation quantization is the problem of and represent the deformation of the ordinaryvector point-wise product, field The symmetric part ofgauge the transformation ( bi-differential operator bi-vector transformation ( bracket. Physically gauge equivalentclassical star bracket products represent different quantizations of the same any three functions defined inon ( the classical bracket with quantization of the bracket obtains the equation on the second order two star products will be violated already in the second order in how to keep the violation of the associativity under control? used instead of the associativity ( expansion, guarantees that the star product ( the initial condition ( acting on star products and defined as: the construction of the star product Non-associative star products In the standard approach to the deformation quantization [ condition”, consisting in the requirement that inreproduce the quasi-classical the limit given the quasi-Poisson star bracket, commutator should where PoS(CORFU2017)200 , ]. ), ]. ] it 13 15 16 1.4 (2.1) (2.4) (2.2) (2.3) ) first 18 1.4 as f alternativity . ) p , x ) in the form Vladislav G. Kupriyanov 1.4 to a function ) = ( I ˆ f x = ( ] we proved that neither ( x 17 ] and the octonionic star product [ -flux algebra ( , , R 19 J ) , ]. The significance and utility of this star ∂ with x ) ( ) x 20 g x ( ( . f ] the quantization of non-geometric M-theory -flux [ IJ ˆ f . R Θ I 22 ! ˆ i j 3 x ¯ h 2 i δ ) = 0 = x + − f I )( k x g ? p I ? = i j x f i jk I δ ˆ ( x R 2 3 s ¯ ` h

]. ) = ). From the practical point of view it means that non-trivial (non- x ( 23 1.1 , IJ Θ 22 ]. Remind that the multiplication is called alternative if the corresponding , = 12 denotes the action of a differential operator on a function. In particular one has 21 } J , . x ] we will formulate the physically motivated minimal set of requirements for the -flux in closed string theory and quantization , 19 ] where it was derived using the Kontsevich formula for deformation quantization I ) is alternating, i.e., antisymmetric in all arguments. In this case the associator is R 17 x 19 { 1.9 ] it was demonstrated that for alternative deformation quantization the quasi-classical limit . ) Originally the star product providing the quantization of the quasi-Poisson structure ( For the convenience let us first represent the constant In this notes we consider two known in the literature examples of the non-associative star prod- The previous attempt to answer this question consisted in the requirement of the x 16 ( ]. However, later on it was shown that the monopole star products are not alternative [ i j A star product is defined by associating a (formal) differential operator appeared in [ product in understanding non-geometric stringexemplified theory in [ and non-associative quantum mechanics is 2. Constant of twisted Poisson structures.based In on these associative notes algebra to of construct differential the operators [ star product we follow an approach In [ nor the quasi-Poisson structureMalcev-Poisson isomorphic structure to and the consequently algebra dowas of shown not more the allow stronger the imaginary statement: alternative octonions theif star quasi-Poisson fail it structure products. to is satisfies the be In Poisson Malcev [ obeying identity only ( Discussing the properties of thesetion star of products and the some non-associative physical quantum applications, mechanics like [ the formula- non-associative deformation quantization. Finally we brieflystructing discuss a how one star may product proceed satisfying inθ con- these conditions for the arbitrary given quasi-Poisson structure where the symbol of the star commutator has to satisfy the Malcev identity. In [ ucts: the phase space star product for the constant associative) alternative star products do not exist. backgrounds [ proportional to the jacobiator.functions The is alternativity useful, of the e.g.,14 star for product the at definition least of on states the some in class non-associative of quantum mechanics [ Non-associative star products of the star product [ associator ( where the operators PoS(CORFU2017)200 = M ∂ (2.9) (2.5) (2.8) (2.6) (2.7) ) can LM defined 2.1 Θ ˆ f J ∂ IJ , . It should be Θ ) as a definition n L . i jk k ) R I 2.4 denote the Fourier x 3 s k ( , ` ) g k
3 , J ( Vladislav G. Kupriyanov ˜ f − n ∂ −→ J ··· ) and ( ) ) x of degree = . ∂ ( f −→ ? n IJ n 2.3 ] J ? Θ S k ∂ . Let ··· ) I x I n 1 ( . Then , ∂ . x ). J ←− j σ 3 ··· I ¯ I h 2 i x ˆ x x 1 ∂ R −→ , J I ( e 2.1 , i n , ∂ ik ) x , the differential operator J ? ∈
[ n , meaning also that the star product n x J f − I k J ˆ ) ( ∂ g e i n on the left and on the right correspond- ∂ ) I f ··· l Θ ) x ◦ ¯ I h ( k ? Θ i x ˆ f ∂ ( IJ ) ∂ ··· ˜ 2 = ( f ( Θ + ) = 1 ··· l I σ J 6= x k I k , 1 1 6 and ˆ ( I J x ¯ p ) h 2 ) k g 1 i g I e 6 Θ π k ? . ? x n d 2 Θ I · d 4 I f ) i jk ( ˆ n x 1 I ik ( = ( I ∂ R ←− k σ IJ − I ik 3 s Z k ¯ e h ` x − i Θ ··· ··· n e ¯ = h S 1 = 1 : i ) I I ∈ I 3. One may easily check that and k ∑ . Weyl star products satisfy k ) ˆ ] = x , σ ( ∂ ) j = f I ) ←− ˜ I f 2 l ˆ ! x I ( = ik ˆ , x , 1 , x n = − i between two arbitrary functions on phase space, we introduce n 1 6 J k ) do not close an algebra. Taking ( ˆ W ˆ e ) x ?
x k n [ g J = 6 J
= π ), in general, by requiring that, for any = ∂ J + ? x 2.4 d i 2 f ˆ f ( J f ) = ( IJ ∂ ˆ 2.5 − ? −→ x I I k J Θ for
ˆ Z IJ x n I x i I ( k p Θ ? x ∂ 2 1 = ( ∂ I I ) = x k ··· ∂ x ←− = 1 I = ) = )( 3 : x J g + ? x i stand for the action of the derivative ] 0 one can write ? I ∂ J , with I f x 0 it follows that ) x ( ( , x ∂ −→ ] = I ( = Weyl star product W L and x f k [ ∂ i ∂ x ∂ and ∂ JL lki I Θ R = ) can be obtained by symmetric ordering of the operators ˆ ∂ J l ←− denoting the standard Euclidean inner product of vectors. By the relation i k k · ∂ , i To define the star product To obtain an explicit form for the corresponding star product we first observe that since 2.2 I k x 3 s I 2 ¯ h k ell where One may also write ingly. Thus the Weyl star productbe representing written quantization in of terms the of quasi-Poisson a bracket bidifferential ( operator as the notion of [ which means that the operators ( by ( transform of where the sum runs over all permutations inshould the not symmetric be associative. group with Non-associative star products with of the star product, one may easily calculate for the star commutator and the star jacobiator which thereby provide a quantization of the classical brackets ( For example, stressed that because of the ( PoS(CORFU2017)200 A e (3.1) (3.2) (3.3) (3.4) (3.6) (3.5) (2.10) (2.11) (2.12) (2.13) . . Moreover f , , ?
D 0 1 l 4567 e . · , = k f 673 ABCD − , 0 3256 Vladislav G. Kupriyanov = η k , , 1 · 12 x 624 ? l · ) − f , 0 , k ¯ h 2 3247 , k , ( . 572 ]] = B , ) + C A i 0 ABDE e e e , k , η ) 1537 . ) 0 C 516 ε , e k + h ABC C , [ ( × e , ? η ˜ , g , and unital, B BD 2 k g ) g ∗ e δ ( ( A 471 k 1425 · f ABC e · ( = ? , f , ˜ f η ? A AE f A p k δ ∗ in integral form through the Fourier transforms e ]] + [ 6 + 0 R g 435 ) B Z g B + Z − k 1 ¯ h 3 s e e 1346 , π 6 ? ` 2 5 1 = , = 2 f , d = A 0 BE AB − ∗ ( − g e k δ δ ) h 123 [ B g 6 ? , p e ? − ) = k f · AD C ? = ) A 1267 6 e π δ e f g ) = X d 2 0 ( = ? l ( B is the identity element and the imaginary unit octonions = Z = e f : 1 ABC + ]] + [ ( ] A Z l C B e DEC e e Z -flux background , , η + ( ABCD R B ) = A can be written in the form x e x e [ [ · , ABC satisfy the contraction identity )( O 7, while ) A g η 0 e ∈ the usual cross product of three-dimensional vectors. ? k 1 for 0 l ABC f X ) is Hermitean, ), for the Jacobiator we get k = [ + ,..., ( η : j 1 k ] k = + 2.9 3.5 1 for C of octonions is the best known example of a nonassociative but alternative = e i jl , = ( ε : A O B ABC , = + e x = η , · R is neither commutative nor associative. The commutator algebra of the octonions is i A ) is a completely antisymmetric tensor of rank four with nonvanishing values ) e 0 ∈ 0 [ O k k ABCD is a completely antisymmetric tensor of rank three with non-vanishing values A , ε η k k , ( ABCD alone, × 0 ] for the proof. ABC k η g B k η ( For later use, let us rewrite the star product The algebra 22 and ˜ ˜ f given by where satisfy the multiplication law Here The algebra and 3-cyclic, meaning that the integrated associator vanishes, see [ The structure constants algebra. Every octonion Non-associative star products It is easy to see that ( where with where 3. Octonions and M-theory it is 2-cyclic or closed, Taking into account ( PoS(CORFU2017)200 ) 3.9 (4.3) (3.9) (4.1) (3.8) (4.2) (3.7) , see Section providing the , O . ) can be rewrit-
O coincide with the ξ , 3.4 i i ¯ ∈ h p p plays the role of the , λ Z , i Vladislav G. Kupriyanov − λ x Y 1234 and , , , 2 , 4 i 4 , X i λ x becomes a central element R , e σ 7 3 s 4 2 e ` 2 , R x = ¯ R h i λ R − 3 s f ∈ 3 s ` λ ` 2 ¯ h } 3 A 3 = i 2 . ). For simplicity first we quantize λ ξ λ p ] = D λ , i √ f 3.9 4 ξ q } ] = octonionic star product i x i , , x = e 7 { , , e σ ] is that the quasi-Poisson brackets ( 4 4 [ 7 x 4 x ABCD e , whose properties are essentially based on R [ { η with 3 s ). The classical Jacobiator reads: , ` i . 12 e λ 3.4 ) and − 2 λ k q 6 , f for any three octonions and implies that the Jacobiator is proportional to the − = C
and and ¯ h ξ ) 1 i jk η 2 O = ε k } i x C p = k ABC YZ − ξ ( e jk 7 η , k . i ξ k ~ , e 2 B e X p k i jk 0 we observe that the element i 3, the algebra of the imaginary octonions ( ξ Λ i j f ε p 4 , , λ ε − = i jk δ 2 2 -flux background A R ε ( → = i jk , Z η : i jk , 3 s + ξ deformed vector sum R 2 − 2 ` 1 4 ) ¯ h } -flux algebra to M-theory. In this sense { 4 λ
2 λ B R x R λ ε p = i j ξ 2 3 s , √ i , XY ¯ ` h − δ 4 ] = ] = ] = ( A j j j x ). The main conjecture of [ f f ξ = , e ) is the = = = for , , , i 6 { x i i i x 3 f 1.4 e e λ λ λ [ [ [ + 4.1 } } } i ] = j j j = e x Z p p
, , , , = A i i i : Y x x x p i , { f { { X [ = x ~ -flux algebra ( R In this section we discuss the problem how to construct the star product which in the quasi- To obtain the desired star product we then make the change of the coordinates, Introducing Now taking the contraction limit classical limit would reproduce the quasi-Poisson structure ( and can be taken to beconstant identity operator, while the algebra generatedprovide by the uplift of theM-theory radius. string 4. Quantization of M-theory the algebra of the quasi-Poisson brackets isomorphic to the algebra of imaginary octonions ( the notion of the seven-dimensional cross product4.1 related for to details. the algebra of octonions we obtain the quasi-Poisson algebra Non-associative star products and the alternative propertyassociator, of i.e., the algebra ten in components as The key mathematical tool usedquantizing for the the algebra construction ( of the Defining the coordinates and momenta in terms of the imaginary octonions as PoS(CORFU2017)200 ε ~ (4.4) (4.6) (4.5) (4.9) , . 1 C 0 0; (4.7) 0 − . From the classical , C enjoys the following 7 0 in the obtained star Λ
≥ ε ξ , define the map called 0 Vladislav G. Kupriyanov 0 3 2 ) = C ~ q . Using the properties of 0    → -flux algebra.
0 = R 00 B 0 : ε R 0 R q λ q A 3 s q 4 × ε ` ε · η σ 0 3 0 q × ~ q ; (4.8) λ 0 BB − q q − q 0 Λ p ε ( 2 − 0 q | ε 0 3 and 3 × 0 2 , 0 AA 1 | q ~ q 2 . q , 0 q 2 Λ q R | 1 2 1 q 3 s − | 0 0 q = − ` − | ε : 1 = 00 = 1 λ ~ i − | q q q q 1 q ε 2 p ABC from the unit ball in ε | , − 0 for λ ~ 3 + p 3 q ~ 1 q ) 0 3 7 q R 0 , + + ¯ q h − | i 2 ) = 0 0 0 q q 0 | q ξ 0 1 λ ε − q 2 0 q ). | with − ~ 0 = q − : q ( q q    i ε − | 3.9 ε σ 1 ¯ h ~ = − | 1 ), which according to the logic of the previous Section should ~ = ( 2 2 q : C ) 1 q | x 0 ) q 4.5 = ]. For each 0 p 00 q −
q , ε q ( 24 ABC q , , satisfying two main properties: it should be associative and its ) for the star product of the closed string 0 AB ~ λ 3 ε q 2 q Λ R , = , and q 0 2.12 = is given by ( − | q = q λ 1 A ε 1 in Λ } = Λ B belongs to the unit ball in ~ q x 0 , q ε | ≤ q A q ~ x ε | { 3 identity matrix, and by the rule: 0 ~ 1 being the sign of 7 matrix × q = ) one obtains the quasi-Poisson algebra ± × 0 ε = 4.1 0 ~ the 3 q , Vector It is associative, Commutator reproduces the cross product, 3 q 1 ε The solution was given in [ To warm up let us first discuss the following problem. How to define a binary operation The consistency check is the calculation of the contraction limit with brackets ( product giving the quantization of ( on the unit ball with properties. (V1) (V3) commutator should reproduce the cross product in three dimensions. which can be written in components as ( 4.1 Deformed vector sums vector star sum three-dimensional cross product one may easily check that the operation (V2) (V4)q reproduce the expression ( Non-associative star products where the 7 PoS(CORFU2017)200 H of ) 7 , due . For e (4.12) (4.13) (4.11) (4.10) (4.14) ε . , O i f ×
0 , l i e · ( k is non-zero if 0 0 k − ~ k ~ 0 η η k × · . × l )
, k , ~ ) 00 as l Vladislav G. Kupriyanov 00 k
~ k 0 4 0 ~ k O η k X gives ~ 0 ) i k × − ~ e 00 0 X k l ~ k ( ~ 4 k · ~ k , X + ( k ~ ) is equivalent to the statement that ( k + -like star product for the functions ~ 0 − 0 ) , to the algebra of octonions η − 00 . With respect to this decomposition, k . 2 0 k ~ 3 ( k , × η ε C ~ 0 X 0 l R is the Euclidean vector norm. ) (C2) . ) ) su k × × 0 00 ε 00 k  k ∈ l B ~ ~ 0 k according to the components k ~ · k is related to the algebra of quaternions k ~ X ) × ~ · i ) the Jacobiator is given by k − k X k l η ~ ~ 0 k ε ~ 0 on the corresponding three-dimensional sub- states that its norm calculates the area of the , X l k k ~ × ~ ( ABC × ε p 3.5 ε 0 X J = ( η  k + ( 3 2 ~ × = ) = ( k 2 1 8 (C3) 00 | 0 , = k : , k k ) = + ( ~ ~ i | , 0 = l A 0 00  η , 0 ) 0 00 k k k ~ l 0 ~ k × ~ ( 4 η k = ( η ~ X ) k × l , it is defined by, η 0 η 0 × k ; ~ k k − ~ 7 ~ × X ) × , where X ) 0 k η 2 R ) , k k 00 ) 0 4 ~ k ~ 0 ~ ( × 0 k k , ~ X η . However, unlike the three-dimensional cross product k k ~ 0  7 η , where ~ · ( + , × ) 4 1 l 0 from R , while property × 4 k ( 0 k ~ k ~ k ( ) 3 k k ~ = , ~ in A ε ( 0 ) − k 0 · k ; = ( , 00 × = k 2 : l 0 ~ k | ~ and , 0 0 k k k ~ ) ~ k is equivalent to the statement that the cross product ) the seven-dimensional cross product can be written in terms of the three- ~ k k , = ( ~ ~ | 00 − k = ( η ~ − 0 and ( k 2 0 ~ k k | η 4.1 × , l ~ ~ k ~ J 0 ~ k 0 ~ | ε k ) = X (C1) ~ k ]. , ~ 00 × and k = are linearly independent vectors, property − ~ k ( ~ l 26 ) 0 2 η | ) and ( η , A k = ~ ~ J 0 k 0 , × k = ~ 25 and observing that k k ) one may easily see that the Jacobiator ~ 0 ~ 0 η k 4.11 A k η ~ = ( ~ ( e × η k · × A ~ 4.14 k ) it does not obey the Jacobi identity: Using ( k ~ k k | , see [ × ~ ~ 3 The defining properties of cross products are The standard three-dimensional cross product k = ~ 3.6 R : k ~ (C2) (C3) (C1) In particular, reduction to the three-dimensional subspace spanned by yielding the expected three-dimensional cross product. space now vanishes, providing the well known identity for the cross product in three dimensions. by using ( As usual property and only if triangle spanned by to ( dimensional cross product as Consider the splitting of the seven-dimensional vector it is orthogonal to both Non-associative star products The above properties are essential foron the definition of the in the same way as the seven-dimensional cross product the previous Section, From ( vectors For better understanding of theuseful underlying to geometry of note the that seven-dimensional itX cross can product, be it expressed is in terms of the algebra of imaginary octonions by writing which can be represented through the associator on the octonion algebra PoS(CORFU2017)200 ) one 4.10 8 (4.20) (4.18) (4.17) (4.19) (4.15) (4.16) (4.21) R and the , ∈ p 0 ~ P η , we find 1, define . P ~ one may check ) 2 0 ~ 00 p | ≤ ,~ 1 p = (C3) p , p ~ ~ ,~ | – 0 p ε , ~
, such that for vectors , p 0 2 7 Vladislav G. Kupriyanov 0 | ,~ . = p ~ R p 0 p ~ ~ (C1) 0 ≥ ( p ~ ~ η ∈ η 2 η η × − | , ~ J 7
× 0 ~ 1 2 3 p B
p 1, we can easily translate this ~ ~ ) for the cross product ( 0 p p , ~ ~ X . P p The commutator reproduces the · − = , ) ¯ . X . Using 0 . 7 D + | 0 ) = 0 p 2 ± ~ 4.14 P , 00 p . Just like in the three-dimensional p B ~ ~ P 00 p p p ¯ ~ ~ ~ 7 | 0 p X − 2 X = p , · ~ η | B 0 η ), what is not the case of the seven- C 0 0 2 0 − 0 | 1 η p q ~ p ~ 0 p × p p P ~ ε ) ~ 4.9 p ~ B X p + 0 , i.e., ~ − | p ~ ~ 0 − 0 p 8 p 2 ~ 1 p P 2 ~ q − − | ( | R X − X 2 1 η q 0 p ⊂ ~ = = ( 2 1 from the unit ball p ABCD ~ q + 7 ) = ( 0 p 0 p 2 η ~ 9 ~ − | S ~ p ~ 0 + | = p , η 1 2 ,~ p 1 − ~ =
| 0 p 0 ~ ~
P , 00 ) = A p 0 0 0 p ~ X ) − | p p 2 ~ ~ q p p 0 | ~ ~ 00 ( 1 η 1 P − · − | p η ( − p ~ X ~ ) can be represented as ,~ q p 0 1 ~ 0 η ~ ) p ~ 0 p ). For this, we note that for arbitrary vectors ) − | − q ~ ,~ p η 0 ~ ): We fix a hemisphere 0 0 = Im 1 4.18 p p , ~ ~ 0 p η 2 ~ ( 0 ) to the seven-dimensional case corresponding to the algebra of p | = p also belongs to the unit ball 0 4.15 , ,~ η 0 p ~ ~ 0 p 0 p ~ q 4.15 A p p ~ ~ ε 4.6 p ( ~ p ~ η , i.e., η 7 η ~ = from the unit sphere = p ~ ~ = ( B 0 0 ~ ) is now associative. : X P p P ~ p p ) ∈ ~ ~ X 00 η p P ~ − | 4.14 p X ~ 1 ,~ − 0 we reproduce the seven-dimensional vector star sum through p ~ p is 7 1 is the sign of ,~ 7 B p ~ ± ) and ( B ( η = ∈ A ~ 2 4.18 p ~ p ) is nonassociative but alternative. The reduction of the seven-dimensional vector star ) on this three-dimensional subspace reproduces the three-dimensional vector star sum ,~ 1 p ~ ] ε 4.15 4.15 in the ball 27 As well as in the case of the seven-dimensional cross product the underlying geometry of Using the properties of the seven-dimensional cross product we may generalize the three- 0 , p ε ,~ p identity to the vector star sum ( By restricting to vectors The components of the associator ( the vector star sum can beisation better of understood the providing binary an alternative operation eight-dimensional ( character- sum ( ~ inverse of seven-dimensional cross product, Three-dimensional vector star product was associative ( that the corresponding associatorthrough is related to the Jacobiator ( has [ dimensional vector star sum ( Non-associative star products imaginary octonions. For any pair of vectors case, it admits an identity element given by the zero vector in dimensional on. However, taking into account the definition of the sign factor It is non-vanishing but totallysum antisymmetric, which ( means that the seven-dimensional vector star which by ( where meaning that the vector ~ PoS(CORFU2017)200 is a (4.23) (4.24) (4.22) O . , this star prod- | is the deformed ) k ~ 0 | ), ensures that the )
/ k ~ 0 ) ) , | k k k ~ 00 ~ | ~ , ( k ¯ h k ~ 4.20 ~ ( , , η Vladislav G. Kupriyanov ( 0 : k η ~ ξ ~ sin ~ B · ( . It can be regarded as a k ~ ~ ) ; . B η f 0 = k ~ p , ~ 1 ) ~ η B k

2 ~ ( 0 ? , ¯ h ). η = and p ( k ~ 1 . ~ ~ 2 B one finds immediately that: i f we introduce the map

η O e η ) R = η 4.16 ) ~ P V + ~ f 0 B ∈ 0 ~ X p ~ k k 0 ~ ~ ( A | − P = ˜ k g 0 η
X | ) 1 p ~ . ( 00 0 × k ~ k η η p ( ~ p ~ k ~ ˜ , ~ ? f Im ¯ ~ h η | deformed vector sum ) f

0 with p 2 7 p ~ ~ 0 ~ | k | | ) ~ + k , ~ p p 1 1 − ~ ~ π 7 2 k | | 0 ~ −

− 2 ( 10 d ¯ ¯ k h h ) ( ~ η P k sin ~ sin + ~ 7 X B 1 and ) ) 0 k k | ~ ~ 7 = P π k η = ~ | d of the deformed vector addition 2 | X k ) = | ~ ~ ( ¯ ( h k 0 ) = B ~ P , and unital, | ( 0 k ~ ∗ , we define the X ), which is the sign of the real part of ( k ¯ h f | = ~ Z Re , : , V

sin (B2) k ; k ~ η ) ~ ( ∈ ? ¯ h = 00 ) over the entire vector space ) 4.15 = η ) = 0 ; ( k ∗ 2 k ~ k and 0 ~ p ~ , ~ η ~

g B ξ , 0 ~ ) − P k k ~ k ~ ~ 4.15 , B )( ~ , X = 0 , g ¯ 0 k 1 h k 0 ~ from ( ~ ∗ ~ ( 1 we have (B1) P ( η ) 0 − ) and the properties of the operation η X = ? η p g ~ ( : = ,~

f A p ~ η ~ | η ) B ) we then immediately infer the identity ( ( ε 0 ? ~ P 4.23 B | k ~ f = , − ( k k 4.21 ~ ~ ( ). By definition it is the Weyl star product. η stands for the Fourier transform of the function ) = ~ ) = ˜ 0 B f k 0 4.23 ~ ~ , , ) and ( k k ~ ~ ( ( η η 4.20 ~ ~ is antisymmetric in all arguments. Perturbative expansion: B B The associator Due to the properties To extend the structure ( Let us define the star product as From the definition ( uct is Hermitean, (B3) (B1) (B2) (B4) where again Non-associative star products where the sign factor result of the vector star sumsum remains is in useful the for same hemisphere. derivingnormed various This algebra, properties. interpretation for of For the example, vector star since the algebra of octonions Then for each pair of vectors vector sum ( The inverse map is given by From ( 4.2 Octonionic star product PoS(CORFU2017)200 ) ε ? 4.23 (4.26) (4.27) (4.29) (4.30) (4.28) (4.25) . . ; in par- ) )  2 O 1 ( ξ ): Defining ~ -like one generate the ( − ) su i f on this three- 2 ) e 4.1 | ( . 0 ) reproduces the g k  , su ~ | f
¯ h 1 ( 4.24 ξ ~ − · Vladislav G. Kupriyanov , ) ) cot 0 | k ) 2 ~ , 0 / ¯ h 0 ~ k 1 ξ ~ ( ( ~ from ( | . η ), nor the ¯ O h 4 ) , ~ ). η B i 0 ¯  h }
+ ], , e ( g 0 2 0 D 4.2 0 4.24 ) 1 | , , 0 k , 26 0 f , 0 k 0 , A ξ k ~ = , ~ ( | { coth ). The solution to this problem was k D ˜ ) . f )( 0 2 25 ξ 0) reduces the nonassociative algebra l D g = / 0 , ~ 1 ξ ~ l ε , whose imaginary units ) + = g 2.10 = ( | k ? ~ , we arrive at 4 C 0 H ε

x f g 0 7 ~ f ABCD k ¯ 7 ∂ h e k ~ | D ~ ( from [ B η d f defined in ( ) ¯ with h 0 ) 2 1 = B ( ) underlying the octonion algebra 0 k η ¯ h − ξ i ~ ~ l Z ∂ , = f A ? , ). We have k l k cot 3.4 4 ~
12 = ( ( | ) ) is alternative on monomials and plane waves. ∂
11 ε 0 η
ABC g 0 A = k ) it follows that, for functions g , ~ | η 4.24 B ∂ η B η 0 ¯ ) A h • D ? , 4.24 ∂ ] 0 ξ ξ ~ l η 4.28 C D ( 2 AD ? ∇ ξ x f ξ ~ , δ , · 7 f ) = 7 B d 0 + 0 B ) ξ ~ ξ D k η , ~ ( , ∂ E ? π 7 0 0 Z ] A 1 2 7 (equivalently and integrating over d , C ¯ h k − g l . The corresponding reduction of the deformed vector sum ( ξ ( , ] for the quantization of the dual of the Lie algebra i − ξ , [ ( k ξ 6 ~ f . From ( e , D 26 [ Z 4 η 5 ADE , ABC (B2) 0 , i jk A = explicitly using ( η ~ − ε B η 4 ξ → 25 ¯ (B4) h f g lim ¯ ¯ is the flat space Laplacian in seven dimensions. In particular for the 2 h h = it provides a quantization of the quasi-Poisson bracket ( i = − η η , we have A f + • ? f + A ] = ∂ = f j η A A η A 0 ? e ξ ~ ξ (B3) ? ∂ , = A i  k g ~ e ξ

0 for [ = − = D to the associative algebra of quaternions ], by an appropriate choice of a gauge transformation 2 ξ ~ ) = g 0 f implies that the star product ( ∇ A k 26 η O ~ , e η A ? k = ? k ~ f ( A ∂ ξ (B4) ~ η ξ = 4 B Lie algebra η ∂ ? Setting Note that neither the octonionic star product Let us calculate ) ] 2 g ( , f Jacobiator one finds where which thereby provides a quantization of the classical 3-brackets ( one may construct a closed star product satisfying Taking into account property su reproduces the three-dimensional vector sum dimensional subspace, the corresponding star product proposed in [ ticular, by property [ of octonions with vanishing associator associative star product of [ are closed with respect to the integral, i.e. do not satisfy ( Non-associative star products quantization of the dual of the pre-Lie algebra ( Property and after some algebra one finds PoS(CORFU2017)200 ] ). 16 . 6 2.13 (4.34) (4.33) (4.35) (4.31) (4.32) ) with 
| ) 4.29 0 . | k ), ) ~
0 ) Λ k 0 ~ , ). One may also k ), and the gauge k 2.12 Λ ~ · , l Λ k ~ ( 4.24 p is central in the star 4.23 − η Λ 0 · ( 4 ~ l Vladislav G. Kupriyanov B η ) x | · 0 ) are simply unity in the ~ l ¯ h B k | ). For this, we define a star ( from ( + 4 l ) x 4.33 0 3.9 sin 2 , k ~ ¯ h
. , x + ( ~ | 1 k 4 0 6 ~ . − ( x k .  ~ ) + η Λ ) λ
0 x ) ~ = Λ 0 4 } ~ 2 ), since k 1 | x | B ξ ~ k / g − 0

¯ h 1 ξ ε ~ , )( k ) Λ ~ f + 4.5 · g × 4 ) as the original one ( ) 4 { ξ Λ ~ 0 ? ¯ h k k k sin 0. The coordinate ~ || ( f )( = ) dimensionally reduces to ( 4.1 k Λ
· , ~ Λ | k = + ( , this expression coincides exactly with ( λ ~ k g Λ ~ 4 • | x Λ 0 4 ] ( η sinh x obtained in the previous subsection we will now ¯ Λ k · 4.33 R p ¯ h h ) = ( g η | • 1 i , 12 on the Schwartz functions satisfies the 3-cyclicity 3 s ) -flux background x ~ η ~ ¯ h B − f ` 0 Λ i = 0; as before we may therefore set it to any non-zero • R
f [ η )( k e 2 4 • g ¯ ) h / 1 k 1. The extra factors in ( 0 2 + → 1 ξ ~ sin λ k k ~ • -direction; they reduce the Fourier space integrations in ( 0 λ = 4 −  ) as ) = ( ˜ 4 0 one needs to calculate, f g ¯ h x → x 4 constant ~ ) ( s × x = ( = lim 0 ,` k )( → ) the corresponding gauge transformation was found in [ ~ 2  ¯ h 4.32 ( g x ¯ h → ~ ˜ f / λ lim 1 λ λ = 3 s ` − • 7 4.24 0 ) f D k Λ ~ π 7 ( · 2 d ) ( 0 k 7 ~ . Using the deformed vector sum ) ) k ~ Λ 7 π , 0, and so the star product ( ξ d ~ 2 k ~ ( ) we can write ( Λ Λ → ( ( f Z η λ 4.31 ~ = B : ). ], that the closed star product 0 ) = ) x → ~ 17 lim λ ξ ~ )( 2.11 ( g Λ f λ • ) to the six-dimensional hyperplanes ) provides the quantization of the same algebra ( To study the contraction limit Using the closed octonionic star product f ( 4.33 4.31 Up to the occurance of the circle fibre coordinate In the dimensional reduction ofto M-theory those to IIA which string are theory constant we along restrict the the algebra ofproduct functions algebra of functions in the limit contraction limit ( constant value, which we take to be and has a form: transformation ( property ( 4.3 Phase space star product for M-theory construct the quantization of the quasi-Poisson structure defined in ( Non-associative star products For the octonionc star product ( As it was already( mentioned in the introduction,show, see the [ gauge equivalent star product ( product of functions on the seven-dimensional M-theory phase space by the prescription This star product provides a quantization of the brackets ( where PoS(CORFU2017)200 . ¯ h (5.1) (5.2) (4.36) ) f -structure, l ) ∂ 2 g ( . k ∂ su
h 4 j ¯ h ∂ i . ) ∂ g
O l 4 Vladislav G. Kupriyanov + ∂ ¯ h + i h l ∂ i ∂ f ) O f g k . k up to any desired order in ∂ m + ) ∂ ) ) ) ∂ f f h f  − x g j ] for the quantization of the dual ) ( l k λ m ) g f ∂ l ∂ ∂ • f i j ∂ i l j 26 , closed with respect to the integral k ∂ l n ∂ g θ f ∂ , f ∂ ∂ ∂ ∂ ( ( k k k h k g λ ), in section 4.2 it was shown that re- ∂ j i λ 25 ∂ ∂ ∂ j  ) one recovers the standard associative • i ∂ • ∂ g g i ∂ j j ∂ k j 1 ∂ g ( − ∂ ∂ i 2.9 θ 4.24 i i x f = , of [ g ∂ kl ~ m + ∂ ∂ 7 ) k f ∂ θ h ∂ 0 d j − l j − − , ∂ g ∂ ilm ∂ g = 0 g g l k j Z l i j , f h g ∂ m Π 1 ∂ l i ∂ ∂ k ξ θ 13 ∂ m k ∂ g = f ∂ λ ∂ 2 n 1 2 ∂ i ( ∂ g j )( m 3 • ¯ h h ∂ n k f ∂ g ∂ ∂ i j k f ∂ j − ∂ λ f i j f l ε j ∂ θ − j ∂ • i ∂ f l j ? θ i ∂ m ∂ ) g f j ∂ ¯ h f ∂ f θ l i ( ∂ i g k ( i ∂ ( ∂ n ∂ m i j ∂ i j ∂ h λ il ∂ j ∂ + ∂ k unital, ∂ • θ − ( θ ∂ g ∂  f , i mk f m h m · ikm ik g ∗ k ∂ l j ∂ ( j θ ∂ 0 in the star product ( m f f ∂ l θ n Π θ ) is compatible with the topological limit, since the corresponding i ∂ ∂ kn x ~ ∂ ∂ jm λ f ∂ 7 m jm g θ 1 2 → l i nl • ∂ d θ kl ml 5.1 θ ∂ ∂ ) = im ∗ n θ l R k θ + θ x g ∂ ∂ θ ikm 1 3 on the three-dimensional subspace endowed with the Z ∂ i jk i j is a Poisson one. nk jl )( f jkl nk ln  Π θ = g Π j g θ 3 θ θ θ θ , . These observations are in agreement with the topological limit in open 2 ∗ Π ∂ 2 ¯ 1 6 ? 2 h f + ) i ¯ 1 1 h 1 6 3 6 ) i ¯ m h f 2 ∂ g i j ], basically meaning that the star product should become associative if the ∂ 2 ( ( + + − + ( − + 8 λ θ mi su • , m jlm θ f 7 ∂ ) = ( h , Π , 6  lkm ik g 3 , θ Π ¯ h f 2 24  i ] it was proposed an iterative procedure that allows one to compute unital, Heritean, ( ) star product for any given quasi-Poisson structure − − A + 20 2.8 Here we discuss the reasonable physical conditions that one may impose on the non-associative In [ To this order the star product ( star products. The bothmiticity star reflects products the discussed reality of in physical thisof observables, notes while algebra. the are unitality Taking Hermitean is the and essential limit for unital. the definition The Her- This property can be regarded asity) the among absence cubic on-shell interactions of of nonassociativity (but fields. not noncommutativ- 5. Minimal set of requirements and 3-cyclic, Up to the third order it reads, associator, string theory [ Moyal product. In case ofstricting the the octonionic functions star product ( of the Lie algebra corresponding bi-vector Weyl ( Non-associative star products It is Hermitean, one obtains the associative star product PoS(CORFU2017)200 ¯ g h ]. , f 28 ) the (5.4) (5.5) (5.3) , 5.1 19 ) and the . 2.10 ] it was shown . Nevertheless
8 4 θ and the resulting ¯ h 2 ]. , corresponding to ¯ h i j in all orders. Since, O 16 θ , which were not taken θ Vladislav G. Kupriyanov + θ  kn θ l ) does not satisfy the closure ∂ m ∂ 4.24 , the integration by parts over jm θ i . ∂θ . ∂ ∗ Π 1 n f ∂ ? il ? or ) to restrict the higher order terms of the f is Poisson. To make an iterative procedure seems to be less trivial. In [ ∗ θ ]. For non-associative star product ( g Π 2 θ = θ 1.9 33 f = − θ∂ ∗ jk ) = 14 θ g f i ? ∂ ? l , like f ∂ 1 θ ( ]. The calculation of such a corrections is a highly non- mi θ 29 n ) for arbitrary ) already in the third order in the deformation parameter ∂ , corrections to the original bi-vector it definitely holds true, but possibly not for all. was proven in [ ¯ nl h θ 5.2 θ 2.11 θ m ∂  2 6 ¯ h ) just like the octonionic star product ( + ) for the star product representing quantization of a given quasi-Poisson jk 5.1 , and consequently vanishes if θ 1.7 i jk = ]. The first non-vanishing contribution appears in the order ]. We believe that more careful analysis of the 3-cyclicity condition is needed. Π 8 ). The latter means that the associativity holds true up to the surface terms. These 31 : ). However, one may follow the same logic as in the section 4.2 and search for the , ) jk quant x θ ( 30 2.11 i j 2.10 θ Unitality: Hermiticity: Topological limit. The condition of 3-cyclicity ( To conclude let us formulate the minimal set of the physically motivated requirements which The star product ( Two other important properties which the both star products from the Sections 2 and 4 enjoy may lead to the non-vanishing terms with three or more derivatives on ) ) ] for more details. ] compatible with the topological limit in higher orders one needs to introduce “quantum”, ) h ? ? ? 32 20 or into account in [ leading order of the corresponding gauge transformation was constructed in [ the expression for the associatorcontains the ( terms with two derivatives on (b (c For some quasi-Poisson structures formal expression ( structure (a can be used instead of the associativity condition ( the Kontsevich diagrams with wheels [ bi-vector reads, 3-cyclicity ( In string theory theassociativity property should of not 3-cyclicity manifest can[ itself be in interpreted on-shell as scattering the amplitudes, requirement but that only off-shell, the see non- gauge equivalent closed star product.star product At for least arbitrary in Poisson the associative case the existence of the closed that the closed non-associative star productthe compatible second with order the in topological the limitthis derivative expansion is does of 3-cyclic not up the prove to non-commutativity yet parameter the existence of the 3-cyclic star products for any trivial task [ properties are essential for the formulation of the non-associative quantum mechanics [ i.e., proportional to the powers of Non-associative star products is proportional to condition ( are the conditions of closure with respect to the appropriate integration measure ( [ PoS(CORFU2017)200 , 0403 (2011) 2015 44 (2010) 084, (2003) 157 (2016) 027 66 1012 1611 Vladislav G. Kupriyanov Workshop on Noncom- , em J. Phys. A , 61 (1978). , em JHEP 111 (2011) 385401 [arXiv:1106.0316 44 (2015) 093 [arXiv:1411.3710 1503 (2002) 33 [hep-th/0101219]. 225 15 , 144 (2017) [arXiv:1705.09639 [hep-th]]. 1710 Nonassociative gravity in string theory? at the Corfu Summer Institute 2017, for the nice and productive algebras,” arXiv:1803.00732 [hep-th]. ∞ (2015) 220402 [arXiv:1510.07559 [quant-ph]]. T-duality and closed string noncommutative (doubled) geometry (2001) 014 [hep-th/0106159]. 115 in curved backgrounds,” Commun. Math. Phys. 0109 [hep-th]]. Lett. Asymmetric Strings and Nonassociative Geometry,” J. Phys. A backgrounds, the algebra of octonions, and missing momentum modes,” JHEP M-theory R-flux backgrounds,” JHEP Theories from L Quantization. 1. Deformations of Symplectic Structures,” Annals Phys. 015401, [ arXiv:1010.1263]. [arXiv:1010.1361]. [hep-th]]. [arXiv:1607.06474 [hep-th]]. 003 (2004), [hep-th/0312043]. [arXiv:q-alg/9709040]. (2015) 129 [arXiv:1603.00218 [hep-th]]. Uncertainty relations and semiclassical evolution,” JHEP We thank Dieter Lüst, Patrizia Vitale, Peter Schupp, Ralph Blumenhagen and Richard Szabo [1] R. Blumenhagen and E. Plauschinn, [6] L. Cornalba and R. Schiappa, “Nonassociative star product deformations for D-brane world[7] volumes M. Herbst, A. Kling and M. Kreuzer, “Star products from open strings[8] in curvedM. backgrounds,” Herbst, JHEP A. Kling and M. Kreuzer, “Cyclicity of nonassociative products on D-branes,” JHEP [2] D. Lüst, [3] R. Blumenhagen, A. Deser, D. Lüst, E. Plauschinn and F. Rennecke, “Non-geometric Fluxes, [4] M. Günaydin, D. Lüst and E. Malek, “Nonassociativity in non-geometric string and M-theory [5] D. Lust, E. Malek and R. J. Szabo, “Non-geometric Kaluza-Klein monopoles and magnetic duals of [9] R. Blumenhagen, I. Brunner, V. Kupriyanov and D. Lüst, “Bootstrapping Non-commutative Gauge [11] M. Kontsevich, “Deformation quantization of Poisson manifolds,” Lett. Math. Phys. [14] M. Bojowald, S. Brahma and U. Büyükçam, “Testing nonassociative quantum mechanics,” Phys. Rev. [10] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, “Deformation Theory and for helpful discussions. Also we would like to thank the organizers of the [12] V. G. Kupriyanov, “Alternative multiplications and non-associativity in physics,” PoS CORFU [13] M. Bojowald, S. Brahma, U. Büyükçam and T. Strobl, “States in nonassociative quantum mechanics: Non-associative star products Acknowledgments mutative Field Theory and Gravity atmosphere. The author acknowledges the CAPES-Humboldt FellowshipGrant No. No. 0079/16-2 and 305372/2016-5. CNPq References PoS(CORFU2017)200 , , 58 338 -flux R (2015) 024 , 103 (2015) (2016) 240 -flux string R 1508 910 1509 Vladislav G. Kupriyanov (2013) 083508 54 (2014) 122301 [arXiv:1312.1621 [hep-th]]. via closed star product,” JHEP d 55 R , 141 (2014) [arXiv:1312.0719 [hep-th]]. 16 (2015) 012004 [arXiv:1504.03915 [hep-th]]. quantum gravity and noncommutative quantum field 1404 D -structures and quantization of non-geometric M-theory 634 2 (2012) 012 [arXiv:1207.0926 [hep-th]]. (2006) 221301 [arXiv:hep-th/0512113]. , 099 (2017), [arXiv:1701.02574 [hep-th]]. (2014) 171 [arXiv:1309.3172 [hep-th]]. 1209 96 1702 1401 (2018) no.1, 012041 [arXiv:1709.10080 [hep-th]]. , 028 (2017), [arXiv:1610.08359 [math-ph]]. 965 1704 (2000) 365 [arXiv:math.QA/9804022] nonassociative quantum mechanics,” J. Math. Phys. noncommutative Fourier transform for Lie groups,” J. Math. Phys. [arXiv:1506.02329 [hep-th]]. string vacua,” JHEP compactifications,” J. Phys. Conf. Ser. theory,” Phys. Rev. Lett. [arXiv:1301.7750 [math-ph]]. no. 2, 523 (2015). 227 627 (2008), [arXiv:0806.4615 [hep-th]]. Geometry in Double Field Theory,” JHEP arXiv:1802.05808 [math-ph]. flux backgrounds,” JHEP backgrounds,” JHEP [arXiv:1502.06544 [hep-th]]. Conf. Ser. [arXiv:1606.01409 [hep-th]]. JHEP [22] D. Mylonas, P. Schupp and R. J. Szabo, “Non-geometric fluxes, quasi-Hopf twist deformations and [25] C. Guedes, D. Oriti and M. Raasakka, “Quantization maps, algebra representation and [21] I. Bakas and D. Lüst, “3-cocycles, nonassociative star products and the magnetic paradigm of [26] V. G. Kupriyanov and P. Vitale, “Noncommutative [30] M. Penkava, P. Vanhaecke, “Deformation quantization of polynomial Poisson algebras.” J. Algebra [31] V. G. Kupriyanov and D. V. Vassilevich, “Star products made (somewhat) easier,” Eur. Phys.[32] J. C R. Blumenhagen, M. Fuchs, F. Hassler, D. Lüst and R. Sun, “Non-associative Deformations of [20] V. G. Kupriyanov and D. V. Vassilevich, “Nonassociative Weyl star products,” JHEP [23] P. Aschieri and R. J. Szabo, “Triproducts, nonassociative star products and geometry of [18] D. Vassilevich and F. M. C. Oliveira, “Nearly associative deformation quantization,” [19] D. Mylonas, P. Schupp and R.J. Szabo, “Membrane sigma-models and quantization of non-geometric [24] L. Freidel and E. R. Livine, “Effective 3 [29] G. Dito, “The Necessity of Wheels in Universal Quantization Formulas,” Commun. Math. Phys. [33] G. Felder and B. Shoikhet, “Deformation quantization with traces,” math/0002057 [math-qa]. [27] D. A. Salamon and T. Walpuski, “Notes[28] on theR. octonions,” J. arXiv:1005.2820 Szabo, [math.RA]. “Magnetic monopoles and nonassociative deformations of quantum theory,” J. Phys. [17] V. G. Kupriyanov and R. J. Szabo, “G Non-associative star products [15] M. Bojowald, S. Brahma, U. Buyukcam and T. Strobl, “Monopole star products[16] are non-alternative,” V. G. Kupriyanov, “Weak associativity and deformation quantization,” Nucl. Phys. B